diff --git a/parse/train/H1WgVz-AZ/H1WgVz-AZ.md b/parse/train/H1WgVz-AZ/H1WgVz-AZ.md new file mode 100644 index 0000000000000000000000000000000000000000..31f19a45307b325c389a9ea901939dc77a18fe41 --- /dev/null +++ b/parse/train/H1WgVz-AZ/H1WgVz-AZ.md @@ -0,0 +1,434 @@ +# LEARNING APPROXIMATE INFERENCE NETWORKS FOR STRUCTURED PREDICTION + +Lifu Tu Kevin Gimpel +Toyota Technological Institute at Chicago, Chicago, IL, 60637, USA +{lifu,kgimpel}@ttic.edu + +# ABSTRACT + +Structured prediction energy networks (SPENs; Belanger & McCallum 2016) use neural network architectures to define energy functions that can capture arbitrary dependencies among parts of structured outputs. Prior work used gradient descent for inference, relaxing the structured output to a set of continuous variables and then optimizing the energy with respect to them. We replace this use of gradient descent with a neural network trained to approximate structured argmax inference. This “inference network” outputs continuous values that we treat as the output structure. We develop large-margin training criteria for joint training of the structured energy function and inference network. On multi-label classification we report speed-ups of $1 0 { - } 6 0 \mathrm { x }$ compared to (Belanger et al., 2017) while also improving accuracy. For sequence labeling with simple structured energies, our approach performs comparably to exact inference while being much faster at test time. We then demonstrate improved accuracy by augmenting the energy with a “label language model” that scores entire output label sequences, showing it can improve handling of long-distance dependencies in part-of-speech tagging. Finally, we show how inference networks can replace dynamic programming for test-time inference in conditional random fields, suggestive for their general use for fast inference in structured settings. + +# 1 INTRODUCTION + +Energy-based modeling (LeCun et al., 2006) associates a scalar measure of compatibility to each configuration of input and output variables. Given an input $_ { \textbf { \em x } }$ , the predicted output $\hat { \textbf { \textit { y } } }$ is chosen by minimizing an energy function $E ( \pmb { x } , \hat { \pmb { y } } )$ . For structured prediction, the parameterization of the energy function can leverage domain knowledge about the structured output space. However, learning and prediction become complex. + +Structured prediction energy networks (SPENs; Belanger & McCallum 2016) use an energy function to score structured outputs, and perform inference by using gradient descent to iteratively optimize the energy with respect to the outputs. Belanger et al. (2017) develop an “end-to-end” method that unrolls an approximate energy minimization algorithm into a fixed-size computation graph that is trainable by gradient descent. After learning the energy function, however, they still must use gradient descent for test-time inference. + +We replace the gradient descent approach with a neural network trained to do inference, which we call an inference network. It can have any architecture such that it takes an input $_ { \textbf { \em x } }$ and returns an output interpretable as a $\textbf { { y } }$ . As in prior work, we relax $\textbf { { y } }$ from discrete to continuous. For multi-label classification, we use a feed-forward network that outputs a vector. We assign a single label to each dimension of the vector, interpreting its value as the probability of predicting that label. For sequence labeling, we output a distribution over predicted labels at each position in the sequence. We adapt the energy functions such that they can operate with both discrete ground truth outputs and outputs generated by our inference networks. + +We define large-margin training objectives to jointly train energy functions and inference networks. Our training objectives resemble the alternating optimization framework of generative adversarial networks (GANs; Goodfellow et al. 2014): the inference network is analogous to the generator and the energy function is analogous to the discriminator. Our approach avoids argmax computations, making training and test-time inference faster than standard SPENs. We experiment with multi-label classification using the same setup as Belanger & McCallum (2016), demonstrating speed-ups of $1 0 \mathrm { x }$ in training time and $6 0 \mathrm { x }$ in test-time inference while also improving accuracy. + +We then design a SPEN and inference network for sequence labeling by using recurrent neural networks (RNNs). We perform comparably to a conditional random field (CRF; Lafferty et al. 2001) when using the same energy function, with faster test-time inference. We also experiment with a richer energy that includes a “label language model” that scores entire output label sequences using an RNN, showing it can improve handling of long-distance dependencies in part-of-speech tagging. Finally, we show how inference networks can replace dynamic programming for test-time inference with CRFs, suggestive for the general use of inference networks to speed up inference in traditional structured prediction settings. + +# 2 STRUCTURED PREDICTION ENERGY NETWORKS + +We denote the space of inputs by $\mathcal { X }$ . For a given input $\mathbf { \boldsymbol { x } } \in \mathcal { X }$ , we denote the space of legal structured outputs by $\mathcal { V } ( \pmb { x } )$ . We denote the entire space of structured outputs by $\mathcal { V } = \cup _ { \pmb { x } \in \mathcal { X } } \mathcal { V } ( \pmb { x } )$ . A SPEN defines an energy function $E _ { \Theta } : \mathcal { X } \times \mathcal { Y } \mathbb { R }$ parameterized by $\Theta$ that uses a functional architecture to compute a scalar energy for an input/output pair. + +We describe the SPEN for multi-label classification (MLC) from Belanger & McCallum (2016). Here, $_ { \textbf { \em x } }$ is a fixed-length feature vector. We assume there are $L$ labels, each of which can be on or off for each input, so $\mathcal { V } ( \pmb { x } ) = \{ 0 , 1 \} ^ { L }$ for all $_ { \textbf { \em x } }$ . The energy function is the sum of two terms: $E _ { \Theta } ( { \pmb x } , { \pmb y } ) = E ^ { l o c } ( { \pmb x } , { \pmb y } ) + \dot { E } ^ { l a b } ( { \pmb y } )$ . $E ^ { l o c } ( { \pmb x } , { \pmb y } )$ is the sum of linear models: + +$$ +E ^ { l o c } ( { \pmb x } , { \pmb y } ) = \sum _ { i = 1 } ^ { L } y _ { i } b _ { i } ^ { \top } F ( { \pmb x } ) +$$ + +where $b _ { i }$ is a parameter vector for label $i$ and $F ( { \pmb x } )$ is a multi-layer perceptron computing a feature representation for the input $_ { \textbf { \em x } }$ . $E ^ { l a b } ( { \pmb y } )$ scores $\textbf { { y } }$ independent of $_ { \textbf { \em x } }$ : + +$$ +E ^ { l a b } ( { \pmb y } ) = c _ { 2 } ^ { \top } g ( C _ { 1 } { \pmb y } ) +$$ + +where $c _ { 2 }$ is a parameter vector, $g$ is an elementwise non-linearity function, and $C _ { 1 }$ is a parameter matrix. After learning the energy function, prediction minimizes energy: + +$$ +\pmb { \hat { y } } = \underset { \pmb { y } \in \mathscr { y } ( \pmb { x } ) } { \mathrm { a r g m i n } } E _ { \Theta } ( \pmb { x } , \pmb { y } ) +$$ + +However, solving Eq. (3) requires combinatorial algorithms because $\mathcal { V }$ is a discrete structured space. This becomes intractable when $E _ { \Theta }$ does not decompose into a sum over small “parts” of $\textbf { { y } }$ . Belanger & McCallum (2016) relax this problem by allowing the discrete vector $\textbf { { y } }$ to be continuous. We use $\mathcal { { V } } _ { R }$ to denote the relaxed output space. For MLC, $\begin{array} { r } { \breve { y } _ { R } ( { \pmb x } ) = [ 0 , 1 ] ^ { L } } \end{array}$ . They solve the relaxed problem by using gradient descent to iteratively optimize the energy with respect to $\textbf { { y } }$ . Since they train with a structured large-margin objective, repeated inference is required during learning. They note that using gradient descent for this inference step is time-consuming and makes learning less stable. So Belanger et al. (2017) propose an “end-to-end” learning procedure inspired by Domke (2012). This approach performs backpropagation through each step of gradient descent. We compare to both methods in our experiments below. + +# 3 INFERENCE NETWORKS FOR SPENS + +Belanger & McCallum (2016) relaxed $\textbf { { y } }$ from a discrete to a continuous vector and used gradient descent for inference. We also relax $\textbf { { y } }$ but we use a different strategy to approximate inference. We define an inference network $\mathbf { A } _ { \Psi } ( \pmb { x } )$ parameterized by $\Psi$ and train it with the goal that + +$$ +\mathbf { A } _ { \Psi } ( \pmb { x } ) \approx \underset { \pmb { y } \in \mathcal { V } _ { R } ( \pmb { x } ) } { \mathrm { a r g m i n } } E _ { \Theta } ( \pmb { x } , \pmb { y } ) +$$ + +Given an energy function $E _ { \Theta }$ and a dataset $X$ of inputs, we solve the following optimization problem: + +$$ +\hat { \Psi } \underset { \Psi } { \mathrm { a r g m i n } } \sum _ { \pmb { x } \in X } E _ { \Theta } ( \pmb { x } , \mathbf { A } _ { \Psi } ( \pmb { x } ) ) +$$ + +The architecture of $\mathbf { A } _ { \Psi }$ will depend on the task. For MLC, the same set of labels is applicable to every input, ${ \bf { S 0 } } \ y$ has the same length for all inputs. So, we can use a feed-forward network for $\mathbf { A } _ { \Psi }$ with a vector output, treating each dimension as the prediction for a single label. For sequence labeling, each $_ { \textbf { \em x } }$ (and therefore each $\textbf { { y } }$ ) can have a different length, so we must use a network architecture for $\mathbf { A } _ { \Psi }$ that permits different lengths of predictions. We use an RNN that returns a vector at each position of $_ { \textbf { \em x } }$ . We interpret this vector as a probability distribution over output labels at that position. + +We note that the output of $\mathbf { A } _ { \Psi }$ must be compatible with the energy function, which is typically defined in terms of the original discrete output space $\mathcal { V }$ . This may require generalizing the energy function to be able to operate both on elements of $\mathcal { V }$ and $\mathcal { V } _ { R }$ . For MLC, no change is required. For sequence labeling, the change is straightforward and is described below in Section 7.2.1. + +# 4 JOINT TRAINING OF SPENS AND INFERENCE NETWORKS + +Belanger & McCallum (2016) propose a structured hinge loss for training SPENs: + +$$ +\operatorname* { m i n } _ { \Theta } \sum _ { \langle x _ { i } , y _ { i } \rangle \in \mathcal { D } } \left[ \operatorname* { m a x } _ { y \in \mathcal { V } _ { R } ( \pmb { x } ) } \left( \triangle ( \pmb { y } , \pmb { y } _ { i } ) - E _ { \Theta } ( \pmb { x } _ { i } , \pmb { y } ) + E _ { \Theta } ( \pmb { x } _ { i } , \pmb { y } _ { i } ) \right) \right] _ { + } +$$ + +where $\mathcal { D }$ is the set of training pairs, $[ f ] _ { + } = \operatorname* { m a x } ( 0 , f )$ , and $\triangle ( \pmb { y } , \pmb { y } ^ { \prime } )$ is a structured cost function that returns a nonnegative value indicating the difference between $\textbf { { y } }$ and $\boldsymbol { y } ^ { \prime }$ . This loss is often referred to as “margin-rescaled” structured hinge loss (Taskar et al., 2004; Tsochantaridis et al., 2005). + +However, this loss is expensive to minimize for structured models because of the “cost-augmented” inference step $( \operatorname* { m a x } _ { \pmb { y } \in \mathcal { y } _ { R } ( \pmb { x } ) } )$ . In prior work with SPENs, this step used gradient descent. We replace this with a cost-augmented inference network ${ \bf A } _ { \Phi } ( { \pmb x } )$ . As suggested by the notation, the cost-augmented inference network $\mathbf { A } _ { \Phi }$ and the inference network $\mathbf { A } _ { \Psi }$ will typically have the same functional form, but use different parameters $\Phi$ and $\Psi$ . We write our new optimization problem as: + +$$ +\operatorname* { m i n } _ { \Theta } \operatorname* { m a x } _ { \Phi } \sum _ { \langle \pmb { x } _ { i } , \pmb { y } _ { i } \rangle \in \mathcal { D } } \left[ \triangle ( \mathbf { A } _ { \Phi } ( \pmb { x } _ { i } ) , \pmb { y } _ { i } ) - E _ { \Theta } ( \pmb { x } _ { i } , \mathbf { A } _ { \Phi } ( \pmb { x } _ { i } ) ) + E _ { \Theta } ( \pmb { x } _ { i } , \pmb { y } _ { i } ) \right] _ { + } +$$ + +We treat this optimization problem as a minimax game and find a saddle point for the game. Following Goodfellow et al. (2014), we implement this using an iterative numerical approach. We alternatively optimize $\Phi$ and $\Theta$ , holding the other fixed. Optimizing $\Phi$ to completion in the inner loop of training is computationally prohibitive and may lead to overfitting. So we alternate between one mini-batch for optimizing $\Phi$ and one for optimizing $\Theta$ . We also add $L _ { 2 }$ regularization terms for $\Theta$ and $\Phi$ . + +The objective for the cost-augmented inference network is: + +$$ +\hat { \Phi } \underset { \Phi } { \mathrm { a r g m a x } } [ \bigtriangleup ( \mathbf { A } _ { \Phi } ( \boldsymbol { x } _ { i } ) , \pmb { y } _ { i } ) - E _ { \Theta } ( \boldsymbol { x } _ { i } , \mathbf { A } _ { \Phi } ( \boldsymbol { x } ) _ { i } ) + E _ { \Theta } ( \boldsymbol { x } _ { i } , \pmb { y } _ { i } ) ] _ { + } +$$ + +That is, we update $\Phi$ so that $\mathbf { A } _ { \Phi }$ yields an output that has low energy and high cost, in order to mimic cost-augmented inference. The energy parameters $\Theta$ are kept fixed. There is an analogy here to the generator in GANs: $\mathbf { A } _ { \Phi }$ is trained to produce a high-cost structured output that is also appealing to the current energy function. To help stabilize training of $\Phi$ , we add several terms to this objective, discussed below in Section 5. + +The objective for the energy function is: + +$$ +\hat { \Theta } \underset { \Theta } { \mathrm { a r g m i n } } [ \triangle ( \mathbf { A } _ { \Phi } ( \pmb { x } _ { i } ) , \pmb { y } _ { i } ) - E _ { \Theta } ( \pmb { x } _ { i } , \mathbf { A } _ { \Phi } ( \pmb { x } _ { i } ) ) + E _ { \Theta } ( \pmb { x } _ { i } , \pmb { y } _ { i } ) ] _ { + } + \lambda \| \Theta \| _ { 2 } ^ { 2 } +$$ + +That is, we update $\Theta$ so as to widen the gap between the cost-augmented and ground truth outputs. There is an analogy here to the discriminator in GANs. The energy function is updated so as to enable it to distinguish “fake” outputs produced by $\mathbf { A } _ { \Phi }$ from real outputs $\mathbf { \nabla } _ { \mathbf { \psi } _ { j } } \mathbf { \sigma } _ { j } \mathbf { \sigma } _ { j } $ . + +Training iterates between updating $\Phi$ and $\Theta$ using the objectives above. + +# 4.1 TEST-TIME INFERENCE + +After training, we want to use an inference network $\mathbf { A } _ { \Psi }$ defined in Eq. (4). However, training only gives us a cost-augmented inference network $\mathbf { A } _ { \Phi }$ . Since $\mathbf { A } _ { \Psi }$ and $\mathbf { A } _ { \Phi }$ have the same functional form, we can use $\Phi$ to initialize $\Psi$ , then do additional training on $\mathbf { A } _ { \Psi }$ as in Eq. (5) where $X$ is the training or validation set. This step helps the resulting inference network to produce outputs with lower energy, as it is no longer affected by the cost function. Since this procedure does not use the output labels of the $_ { \textbf { \em x } }$ ’s in $X$ , it could also be applied to the test data in a transductive setting. + +# 4.2 VARIATIONS AND SPECIAL CASES + +This approach also permits us to use large-margin structured prediction with slack rescaling (Tsochantaridis et al., 2005). Slack rescaling can yield higher accuracies than margin rescaling, but requires “cost-scaled” inference during training which is intractable for many classes of output structures. However, we can use our notion of inference networks to circumvent this tractability issue and approximately optimize the slack-rescaled hinge loss, yielding the following optimization problem: + +$$ +\operatorname* { m i n } _ { \Theta } \operatorname* { m a x } _ { \Phi } \sum _ { \langle \mathbf { x } _ { i } , \mathbf { y } _ { i } \rangle \in \mathcal { D } } \bigtriangleup ( \mathbf { A } _ { \Phi } ( \mathbf { x } _ { i } ) , \mathbf { y } _ { i } ) [ 1 - E _ { \Theta } ( \mathbf { x } _ { i } , \mathbf { A } _ { \Phi } ( \mathbf { x } _ { i } ) ) + E _ { \Theta } ( \mathbf { x } _ { i } , \mathbf { y } _ { i } ) ] _ { + } +$$ + +Using the same argument as above, we can also break this into alternating optimization of $\Phi$ and $\Theta$ . + +We can optimize a structured perceptron (Collins, 2002) version by using the margin-rescaled hinge loss (Eq. (7)) and fixing $\begin{array} { r } { \triangle ( \mathbf { A } _ { \Phi } ( \pmb { x } _ { i } ) , \pmb { y } _ { i } ) = 0 } \end{array}$ . When using this loss, the cost-augmented inference network is actually a test-time inference network, because the cost is always zero, so using this loss may lessen the need to retune the inference network after training. + +When we fix $\begin{array} { r } { \triangle ( \mathbf { A } _ { \Phi } ( \pmb { x } _ { i } ) , \pmb { y } _ { i } ) = 1 } \end{array}$ , then margin-rescaled hinge is equivalent to slack-rescaled hinge. While using $\triangle = 1$ is not useful in standard max-margin training with exact argmax inference (because the cost has no impact on optimization when fixed to a positive constant), it is potentially useful in our setting. Consider our SPEN objectives with $\triangle = 1$ : + +$$ +[ 1 - E _ { \Theta } ( { \pmb x } _ { i } , { \pmb A } _ { \Phi } ( { \pmb x } _ { i } ) ) + E _ { \Theta } ( { \pmb x } _ { i } , { \pmb y } _ { i } ) ] _ { + } +$$ + +There will always be a nonzero difference between the two energies because ${ \bf A } _ { \Phi } ( { \pmb x } _ { i } )$ will never exactly equal the discrete vector $\mathbf { \nabla } _ { \mathbf { \mathcal { Y } } _ { i } }$ . Since there is no explicit minimization over all discrete vectors $\textbf { { y } }$ , this case is more similar to a “contrastive” hinge loss which seeks to make the energy of the true output lower than the energy of a particular “negative sample” by a margin of at least 1. + +In our experiments, we will compare four hinge losses for training SPENs: margin-rescaled (Eq. (7)), slack-rescaled (Eq. (10)), perceptron (margin-rescaled with $\triangle = 0$ ), and contrastive $\triangle = 1$ ). + +# 5 IMPROVING TRAINING FOR INFERENCE NETWORKS + +We found that the alternating nature of the optimization led to difficulties during training. Similar observations have been noted about other alternative optimization settings, especially those underlying generative adversarial networks (Salimans et al., 2016). Below we describe several techniques we found to help stabilize training, which are optional terms added to the objective in Eq. (8). + +$L _ { 2 }$ Regularization: We use $L _ { 2 }$ regularization, adding the penalty term $\| \Phi \| _ { 2 } ^ { 2 }$ with coefficient $\lambda _ { 1 }$ + +Entropy Regularization: We add an entropy-based regularizer $\mathrm { l o s s } _ { \mathrm { H } } ( \mathbf { A } _ { \Phi } ( \pmb { x } ) )$ defined for the problem under consideration. For MLC, the output of $\mathbf { A } _ { \Phi } ( \pmb { x } )$ is a vector of scalars in [0, 1], one for each label, where the scalar is interpreted as a label probability. The entropy regularizer $\mathrm { l o s s } _ { \mathrm { H } }$ is the sum of the entropies over these label binary distributions. For sequence labeling, where the length of $_ { \textbf { \em x } }$ is $N$ and where there are $L$ unique labels, the output of ${ \bf A } _ { \Phi } ( { \pmb x } )$ is a length- $N$ sequence of length- $L$ vectors, each of which represents the distribution over the $L$ labels at that position in $_ { \textbf { \em x } }$ . Then, $\mathrm { l o s s } _ { \mathrm { H } }$ is the sum of entropies of these label distributions across positions in the sequence. + +When tuning the coefficient $\lambda _ { 2 }$ for this regularizer, we consider both positive and negative values, permitting us to favor either low- or high-entropy distributions as the task prefers.1 + +Local Cross Entropy Loss: We add a local (non-structured) cross entropy $\mathrm { l o s s } _ { \mathrm { C E } } ( \mathbf { A } _ { \Phi } ( \pmb { x } _ { i } ) , \pmb { y } _ { i } )$ defined for the problem under consideration. We only experiment with this loss for sequence labeling. + +It is the sum of the label cross entropy losses over all positions in the sequence. This loss provides more explicit feedback to the inference network, helping the optimization procedure to find a solution that minimizes the energy function while also correctly classifying individual labels. It can also be viewed as a multi-task loss for the inference network. + +Regularization Toward Pretrained Inference Network: We add the penalty $\lVert \Phi - \Phi _ { 0 } \rVert _ { 2 } ^ { 2 }$ where $\Phi _ { 0 }$ is a pretrained network, e.g., a local classifier trained to independently predict each part of $\textbf { { y } }$ . + +Each additional term has its own tunable hyperparameter. Finally we obtain: + +$$ +\begin{array} { r } { \hat { \Phi } \underset { \Phi } { \operatorname { a r g m a x } } \ [ \triangle ( \mathbf { A } _ { \Phi } ( \boldsymbol { x } _ { i } ) , \boldsymbol { y } _ { i } ) - E _ { \Theta } ( \boldsymbol { x } _ { i } , \mathbf { A } _ { \Phi } ( \boldsymbol { x } _ { i } ) ) + E _ { \Theta } ( \boldsymbol { x } _ { i } , \boldsymbol { y } _ { i } ) ] _ { + } - \lambda _ { 1 } \| \Phi \| _ { 2 } ^ { 2 } } \\ { + \lambda _ { 2 } \mathrm { l o s s } _ { \mathrm { H } } ( \mathbf { A } _ { \Phi } ( \boldsymbol { x } _ { i } ) ) - \lambda _ { 3 } \mathrm { l o s s } _ { \mathrm { C E } } ( \mathbf { A } _ { \Phi } ( \boldsymbol { x } _ { i } ) , \boldsymbol { y } _ { i } ) - \lambda _ { 4 } \| \Phi - \Phi _ { 0 } \| _ { 2 } ^ { 2 } } \end{array} +$$ + +# 6 RELATED WORK + +Our methods are reminiscent of other alternating optimization problems like that underlying generative adversarial networks (GANs; Goodfellow et al. 2014). GANs are based on a minimax game and have a value function that one agent (a discriminator $D$ ) seeks to maximize and another (a generator $G$ ) seeks to minimize. By their analysis, a log loss discriminator converges to a degenerate uniform solution. When using hinge loss, we can get a non-degenerate discriminator while matching the data distribution (Dai et al., 2017; Zhao et al., 2016). Our formulation is closer to this hinge loss version of the GAN. + +Our approach is also related to knowledge distillation (Ba & Caruana, 2014; Hinton et al., 2015), which refers to strategies in which one model (a “student”) is trained to mimic another (a “teacher”). Typically, the teacher is a larger, more accurate model but which is too computationally expensive to use at test time. Urban et al. (2016) train shallow networks using image classification data labeled by an ensemble of deep teacher nets. Geras et al. (2016) train a convolutional network to mimic an LSTM for speech recognition. Others have explored knowledge distillation for sequence-to-sequence learning (Kim & Rush, 2016) and parsing (Kuncoro et al., 2016). + +Since we train a single inference network for an entire dataset, our approach is also related to “amortized inference” (Srikumar et al., 2012; Gershman & Goodman, 2014; Paige & Wood, 2016; Chang et al., 2015). Such methods precompute or save solutions to subproblems for faster overall computation. Our inference networks likely devote more modeling capacity to the most frequent substructures in the data. A kind of inference network is used in variational autoencoders (Kingma & Welling, 2013) to approximate posterior inference in generative models. + +Our methods are also related to work in structured prediction that seeks to approximate structured models with factorized ones, e.g., mean-field approximations in graphical models (Koller & Friedman, 2009; Krähenbühl & Koltun, 2011). Like our use of inference networks, there have been efforts in designing differentiable approximations of combinatorial search procedures (Martins & Kreutzer, 2017; Goyal et al., 2018) and structured losses for training with them (Wiseman & Rush, 2016). Since we relax discrete output variables to be continuous, there is also a connection to recent work that focuses on structured prediction with continuous valued output variables (Wang et al., 2016). They also propose a formulation that yields an alternating optimization problem, but it is based on proximal methods. + +There are other settings in which gradient descent is used for inference, e.g., image generation applications like DeepDream (Mordvintsev et al., 2015) and neural style transfer (Gatys et al., 2015), as well as machine translation (Hoang et al., 2017). In these and related settings, gradient descent has started to be replaced by inference networks, especially for image transformation tasks (Johnson et al., 2016; Li & Wand, 2016). Our results below provide more evidence for making this transition. An alternative to what we pursue here would be to obtain an easier convex optimization problem for inference via input convex neural networks (Amos et al., 2017). + +Table 1: Test F1 when comparing methods on multi-label classification datasets. + +
BibtexBookmarksDeliciousavg.
MLP38.933.837.836.8
SPEN (BM16)42.234.437.538.0
SPEN (E2E)38.133.934.435.5
SPEN (InfNet)42.237.637.539.1
+ +# 7 EXPERIMENTS + +In Sec. 7.1 we compare our approach to previous work on training SPENs for MLC. We compare accuracy and speed, finding our approach to outperform prior work. We then perform experiments with sequence labeling tasks in Sec. 7.2. + +# 7.1 MULTI-LABEL CLASSIFICATION + +We use the MLC datasets used by Belanger & McCallum (2016): Bibtex, Delicious, and Bookmarks. Dataset statistics are shown in Table 7 in the Appendix. For Bibtex and Delicious, we follow Belanger and McCallum and tune the hyperparameters using a different sampling of train and test data, then use the standard train/test split for final experimentation using the tuned hyperparameters. For Bookmarks, we use the same train/dev/test split as (Belanger & McCallum, 2016). For evaluation, we report the example averaged (macro averaged) F1 measure. + +We use the SPEN for MLC described in Section 2 and also used by Belanger & McCallum (2016). For the feature representation network $F ( { \pmb x } )$ , we use feed-forward networks with two hidden layers, using their same layer widths: 150 for Bibtex/Bookmarks and 250 for Delicious. We pretrain the feature networks $F ( { \dot { \mathbf { x } } } )$ by minimizing independent-label cross entropy for 10 epochs using Adam (Kingma & Ba, 2014) with learning rate 0.001. While training SPENs, we only update the parameters of the energy function $( \Theta )$ and the inference network $( \Phi )$ , keeping the feature network parameters $F ( { \pmb x } )$ fixed. We use Adam with learning rate 0.001 to train $\Theta$ and $\Phi$ . + +The inference networks are feed-forward networks with two hidden layers, using the same architectures as the feature networks $F ( { \pmb x } )$ . This permits us to initialize inference network parameters $\Phi$ using pretrained feature network parameters. For the output, we use an affine transformation layer with a sigmoid nonlinearity function, so the output values are in the range $( 0 , 1 )$ . We interpret each value as the probability of predicting the corresponding label. We obtain discrete predictions by thresholding at a threshold $\tau$ tuned to maximize F1 on the development data. We add three terms to the inference network objective from Section 5: $L _ { 2 }$ regularization, entropy regularization, and regularization toward the pretrained feature network. Margin rescaling and slack rescaling use squared $L _ { 2 }$ distance for $\triangle$ . Additional details are provided in Sec. 9.1 in the appendix. + +Comparison to Prior Work. Table 1 shows results comparing to prior work. The MLP and “SPEN (BM16)” baseline results are taken from (Belanger & McCallum, 2016). We obtained the “SPEN (E2E)” (Belanger et al., 2017) results by running the code available from the authors on these datasets. This method constructs a recurrent neural network that performs gradient-based minimization of the energy with respect to $\textbf { { y } }$ . They noted in their software release that, while this method is more stable, it is prone to overfitting and actually performs worse than the original SPEN. We indeed find this to be the case, as SPEN (E2E) underperforms SPEN (BM16) on all three datasets. + +Our method (“SPEN (InfNet)”) achieves the best average performance across the three datasets. It performs especially well on Bookmarks, which is the largest of the three. Our results use the contrastive hinge loss and retune the inference network on the development data after the energy is trained; these decisions were made based on the tuning described in Sec. 9.1, but all four hinge losses led to similarly strong results. + +Speed Comparison. Table 2 compares training and test-time inference speed among the different methods. We only report speeds of methods that we ran.2 The SPEN (E2E) times were obtained using code obtained from Belanger and McCallum. We suspect that SPEN (BM16) training would be comparable to or slower than SPEN (E2E). Our method can process examples during training about 10 times as fast as the end-to-end SPEN, and 60-130 times as fast during test-time inference. In fact, at test time, our method is roughly the same speed as the MLP baseline, since our inference networks use the same architecture as the feature networks which form the MLP baseline. Compared to the MLP, the training of our method takes significantly more time overall because of joint training of the energy function and inference network, but fortunately the test-time inference is comparable. + +Table 2: Training and test-time inference speed comparison (examples/sec). + +
Training Speed (examples/sec)Testing Speed (examples/sec)
BibtexBookmarksDeliciousBibtexBookmarksDelicious
MLP2167019591261589070692307113750
SPEN (E2E)55155938314201401832
SPEN (InfNet)5533546746679419488888112148
+ +# 7.2 SEQUENCE LABELING + +We also evaluate our methods on sequence labeling. We report experiments with Twitter part-ofspeech (POS) tagging here. Named entity recognition experiments are reported in the Appendix. + +# 7.2.1 ENERGY FUNCTIONS FOR SEQUENCE LABELING + +The input space $\mathcal { X }$ is now the set of all sequences of symbols drawn from a vocabulary. For an input sequence $_ { \textbf { \em x } }$ of length $N$ , where there are $L$ possible output labels for each position in $_ { \textbf { \em x } }$ , the output space $\mathcal { V } ( \pmb { x } )$ is $[ L ] ^ { \widetilde { N } }$ , where the notation $[ q ]$ represents the set containing the first $q$ positive integers. We define $\pmb { y } = \langle y _ { 1 } , y _ { 2 } , . . , y _ { N } \rangle$ where each $y _ { i }$ ranges over possible output labels, i.e., $y _ { i } \in [ L ]$ . + +When defining our energy for sequence labeling, we take inspiration from bidirectional LSTMs (BLSTMs; Hochreiter $\&$ Schmidhuber 1997) and conditional random fields (CRFs; Lafferty et al. 2001). A “linear chain” CRF uses two types of features: one capturing the connection between an output label and $_ { \textbf { \em x } }$ and the other capturing the dependence between neighboring output labels. We use a BLSTM to compute feature representations for $_ { \textbf { \em x } }$ . We use $f ( \pmb { x } , t ) \in \mathbb { R } ^ { d }$ to denote the “input feature vector” for position $t$ , defining it to be the $d$ -dimensional BLSTM hidden vector at $t$ . + +We then define the following energy function: + +$$ +E _ { \Theta } ( \pmb { x } , \pmb { y } ) = - \left( \sum _ { t } U _ { y _ { t } } ^ { \top } f ( \pmb { x } , t ) + \sum _ { t } W _ { y _ { t - 1 } , y _ { t } } \right) +$$ + +where $U _ { i } \in \mathbb { R } ^ { d }$ is a parameter vector for label $i$ and the parameter matrix $W \in \mathbb { R } ^ { L \times L }$ contains label pair parameters. The full set of parameters $\Theta$ includes the $U _ { i }$ vectors, $W$ , and the parameters of the BLSTM. The above energy only permits discrete $\textbf { { y } }$ . For the general case that permits relaxing $\textbf { { y } }$ to be continuous, we treat each $y _ { t }$ as a vector. It will be one-hot for the ground truth $\textbf { { y } }$ and will be a vector of label probabilities for relaxed $\textbf { { y } }$ ’s. Then the general energy function is: + +$$ +E _ { \Theta } ( \pmb { x } , \pmb { y } ) = - \left( \sum _ { t } \sum _ { i = 1 } ^ { L } y _ { t , i } \left( U _ { i } ^ { \top } f ( \pmb { x } , t ) \right) + \sum _ { t } y _ { t - 1 } ^ { \top } W y _ { t } \right) +$$ + +where $y _ { t , i }$ is the $i$ th entry of the vector $y _ { t }$ . In the discrete case, this entry is 1 for a single $i$ and 0 for all others, so this energy reduces to Eq. (12) in that case. In the continuous case, this scalar indicates the probability of the tth position being labeled with label $i$ . For the label pair terms in this general energy function, we use a bilinear product between the vectors $y _ { t - 1 }$ and $y _ { t }$ using parameter matrix $W$ , which also reduces to Eq. (12) when they are one-hot vectors. + +Tag Language Model. In order to capture long-distance dependencies in an entire sequence of labels, we train a “tag language model” on a large corpus of automatically-tagged tweets, then include a term in the energy function representing the log-probability of the given tag sequence under this tag language model. Details are provided below in Section 7.2.4. + +Table 3: Comparison of SPEN hinge losses and showing the impact of retuning (Twitter POS validation accuracies). Inference networks are trained with the cross entropy term. + +
SPEN hinge lossvalidation accuracy (%)
-retuning 89.1+retuning
margin rescaling slack rescaling89.489.3
perceptron (MR,△= 0)89.289.6
88.889.4
contrastive (△= 1)89.0
+ +Table 4: Twitter POS accuracies of BLSTM, CRF, and SPEN (InfNet), using our tuned SPEN configuration (slack-rescaled hinge, inference network trained with cross entropy term). Though slowest to train, the SPEN matches the test-time speed of the BLSTM while achieving the highest accuracies. + +
validation accuracy (%)test accuracy (%)training speed (examples/sec)testing speed (examples/sec)
BLSTM88.688.83851250
CRF89.189.2250500
SPEN (InfNet)89.689.81251250
+ +# 7.2.2 EXPERIMENTAL SETUP + +For Twitter part-of-speech (POS) tagging, we use the annotated data from Gimpel et al. (2011) and Owoputi et al. (2013) which contains $L = 2 5$ POS tags. For training, we combine the 1000- tweet OCT27TRAIN set and the 327-tweet OCT27DEV set. For validation, we use the 500-tweet OCT27TEST set and for testing we use the 547-tweet DAILY547 test set. We use 100-dimensional skip-gram embeddings trained on 56 million English tweets with word2vec (Mikolov et al., 2013).3 + +We use a BLSTM to compute the “input feature vector” $f ( { \pmb x } , t )$ for each position $t$ , using hidden vectors of dimensionality $d = 1 0 0$ . We also use BLSTMs for the inference networks. The output layer of the inference network is a softmax function, so at every position, the inference network produces a distribution over labels at that position. We train inference networks using stochastic gradient descent (SGD) with momentum and train the energy parameters using Adam. For $\triangle$ , we use $L _ { 1 }$ distance. We tune hyperparameters on the validation set; full details of tuning are provided in the appendix. We found that the cross entropy stabilization term worked well for this setting; details and an empirical comparison are provided in Section 9.2.1. + +We compare to standard BLSTM and CRF baselines. We train the BLSTM baseline to minimize per-token log loss; this is often called a “BLSTM tagger”. We train a CRF baseline using the energy in Eq. (12) with the standard conditional log-likelihood objective using the standard dynamic programming algorithms (forward-backward) to compute gradients during training. Further details are provided in the appendix. + +# 7.2.3 RESULTS + +Loss Function Comparison. Table 3 shows results when comparing SPEN training objectives. We see a larger difference among losses here than for MLC tasks. When using the perceptron loss, there is no margin, which leads to overfitting: 89.4 on validation, 88.6 on test (not shown in the table). The contrastive loss, which strives to achieve a margin of 1, does better on test (89.0). We also see here that margin rescaling and slack rescaling both outperform the contrastive hinge, unlike the MLC tasks. We suspect that in the case in which each input/output has a different length, using a cost that captures length is more important. + +Comparison to Standard Baselines. Table 4 compares our final tuned SPEN configuration to two standard baselines: a BLSTM tagger and a CRF. The SPEN achieves higher validation and test accuracies with faster test-time inference. While our method is slower than the baselines during training, it is faster than the CRF at test time, operating at essentially the same speed as the BLSTM baseline while being more accurate. + +Table 5: Twitter POS validation/test accuracies when adding tag language model (TLM) energy term to a SPEN trained with margin-rescaled hinge. + +
val. accuracy (%)test accuracy (%)
-TLM89.889.6
+TLM89.990.2
+ +Here, the SPEN and CRF are using the same functional form for their energy functions, namely the energy given in Eq. (13). We note that the SPEN outperforms the CRF, despite using the same form for the energy. There are two factors that can explain this. First, the losses are different. The CRF uses conditional log-likelihood while the SPEN results here use slack-rescaled hinge, which outperforms the other hinge loss variants (Table 3). Second, the stabilization terms used when training the inference network may be providing a regularizing effect for the model. Our motivation for these experiments was to show the impact of these differences while keeping the form of the energy function fixed. We now turn to richer energies. + +7.2.4 TOWARDS GLOBAL ENERGIES: TAG LANGUAGE MODELS FOR TWITTER POS TAGGING + +The above results only use the pairwise energy; no results used the tag language model (TLM). To compute the TLM energy term, we first automatically tag unlabeled tweets, then train an LSTM language model on the automatic tag sequences. When doing so, we define the input tag embeddings to be $L$ -dimensional one-hot vectors specifying the tags in the training sequences. This is nonstandard compared to standard language modeling. In standard language modeling, we train on observed sequences and compute likelihoods of other fully-observed sequences. However, in our case, we train on tag sequences but we want to use the same model on sequences of tag distributions produced by an inference network. We train the TLM on sequences of one-hot vectors and then use it to compute likelihoods of sequences of tag distributions. Further details about training are provided in Section 9.2.2 in the appendix. + +We define an additional energy term $E ^ { \mathrm { T L M } } ( y )$ based on the pretrained TLM. If the argument $\textbf { { y } }$ consisted of one-hot vectors, we could simply compute its likelihood. However, to support relaxed $\textbf { { y } }$ ’s, we need to define a more general function: + +$$ +E ^ { \mathrm { T L M } } ( \pmb { y } ) = - \sum _ { t = 1 } ^ { | \pmb { y } | + 1 } \log ( \pmb { y } _ { t } ^ { \top } \mathrm { T L M } ( \langle \pmb { y } _ { 0 } , . . . , \pmb { y } _ { t - 1 } \rangle ) ) +$$ + +where $y _ { 0 }$ is the start-of-sequence symbol, $y _ { \vert \pmb { y } \vert + 1 }$ is the end-of-sequence symbol, and $\mathrm { T L M } \big ( \langle y _ { 0 } , . . . , y _ { t - 1 } \rangle \big )$ returns the softmax distribution over tags at position $t$ (under the pretrained tag language model) given the preceding tag vectors. When each $y _ { t }$ is a one-hot vector, this energy reduces to the negative log-likelihood of the tag sequence specified by $\textbf { { y } }$ . + +We define the new joint energy as the sum of the energy function in Eq. (13) and the TLM energy function in Eq. (14). During learning, we keep the TLM parameters fixed to their pretrained values, but we tune the weight of the TLM energy (over the set $\{ 0 . 1 , 0 . 2 , 0 . 5 \} )$ in the joint energy. We train SPENs with the new joint energy using the margin-rescaled hinge, training the inference network with the cross entropy term. + +Table 5 shows results.4 Adding the TLM energy leads to a gain of 0.6 on the test set. Other settings showed more variance; when using slack-rescaled hinge, we found a small drop on test, while when simply training inference networks for a fixed, pretrained joint energy with tuned mixture coefficient, we found a gain of 0.3 on test when adding the TLM energy. We investigated the improvements and found some to involve corrections that seemingly stem from handling non-local dependencies better. Table 10 in the appendix shows examples in which the model with the TLM appears to be better at using the broader context when making tagging decisions. These results suggest that our method of training inference networks can be used to add rich features to structured prediction, though we leave a thorough exploration of global energies to future work. + +Table 6: Comparison of test-time inference algorithms for a trained CRF (Twitter POS tagging). We show the test accuracy for the inference network setting that does best on validation. All inference networks use the same architecture and therefore have essentially the same speed. + +
test-time inference algorithmval. accuracy (%)test accuracy (%)speed (examples/sec)
Viterbialgorithm89.189.2500
Inference network + cross entropy89.789.51250
Inference network+ entropy89.6
Inference network + squared L2 distance88.9
+ +# 7.2.5 BEYOND SPENS: INFERENCE NETWORKS FOR STRUCTURED PREDICTION + +We note that inference networks can be used for any prediction problem. We now explore the use of an inference network to approximate test-time inference for a trained CRF. The results are shown in Table 6. All results use the same trained CRF energy function (Eq. (12)), trained to minimize log loss using the forward-backward algorithm for exact inference during training. The first row shows accuracy and speed when using Viterbi for test-time inference, which is the same setting as the “CRF” row in Table 4. Subsequent rows show results when training inference networks to mimic Viterbi with various stabilization terms. When training these inference networks, we train them on the training set and tune based on early stopping on the validation set. The energy stays fixed while inference networks are trained. + +When using either entropy or cross entropy, our inference networks outperform Viterbi while doubling its speed. When using the squared $L _ { 2 }$ distance term (which regularizes the inference network toward the pretrained BLSTM), the accuracy reduces to be closer to that of the BLSTM, which reaches $8 8 . 6 \%$ on validation (see Table 4). When using no stabilization terms for the inference network, learning fails, reaching $1 3 . 7 \%$ on the development set, showing the importance of using some stabilization term while training the inference network. + +These results show promise for training inference networks to speed up combinatorial algorithms for structured prediction and other domains. + +# 8 CONCLUSIONS AND FUTURE WORK + +We presented ways to jointly train structured energy functions and inference networks using largemargin objectives. The energy function captures arbitrary dependencies among the labels, while the inference networks learns to capture the properties of the energy in an efficient manner, yielding fast test-time inference. Future work includes exploring the space of network architectures for inference networks to balance accuracy and efficiency, experimenting with additional global terms in structured energy functions, and exploring richer structured output spaces such as trees and sentences. + +# ACKNOWLEDGMENTS + +We thank the anonymous reviewers, David Belanger, Weiran Wang and Zheng Cai. We also thank NVIDIA Corporation for donating GPUs used in this research. + +# REFERENCES + +Brandon Amos, Lei Xu, and J. Zico Kolter. Input convex neural networks. In Proc. of ICML, 2017. + +Jimmy Ba and Rich Caruana. Do deep nets really need to be deep? In Advances in NIPS, 2014. + +David Belanger and Andrew McCallum. Structured prediction energy networks. 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In NIPS Deep Learning Workshop, 2015. + +Cong Duy Vu Hoang, Gholamreza Haffari, and Trevor Cohn. Towards decoding as continuous optimisation in neural machine translation. In Proc. of EMNLP, 2017. + +Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural Computation, 1997. + +Justin Johnson, Alexandre Alahi, and Li Fei-Fei. Perceptual losses for real-time style transfer and super-resolution. In Proc. of ECCV, 2016. + +Yoon Kim and Alexander M. Rush. Sequence-level knowledge distillation. In Proc. of EMNLP, 2016. + +Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. CoRR, abs/1412.6980, 2014. + +Diederik Kingma and Max Welling. Auto-encoding variational Bayes. CoRR, abs/1312.6114, 2013. + +Daphne Koller and Nir Friedman. Probabilistic Graphical Models: Principles and Techniques. 2009. + +Philipp Krähenbühl and Vladlen Koltun. Efficient inference in fully connected CRFs with Gaussian edge potentials. 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In Proc. of RepL4NLP, 2017. + +Gregor Urban, Krzysztof J. Geras, Samira Ebrahimi Kahou, Ozlem Aslan, Shengjie Wang, Rich Caruana, Abdel-rahman Mohamed, Matthai Philipose, and Matthew Richardson. Do deep convolutional nets really need to be deep? arXiv preprint arXiv:1603.05691, 2016. + +Shenlong Wang, Sanja Fidler, and Raquel Urtasun. Proximal deep structured models. In Advances in NIPS, 2016. + +Sam Wiseman and Alexander M. Rush. Sequence-to-sequence learning as beam-search optimization. In Proc. of EMNLP, 2016. + +Junbo Jake Zhao, Michaël Mathieu, and Yann LeCun. Energy-based generative adversarial network. CoRR, 2016. + +Table 7: Statistics of the multi-label classification datasets. + +
#labels# features#train#dev#test
Bibtex15918364836-2515
Bookmarks2082151480001200027856
Delicious98250112896-3185
+ +Table 8: Development F1 for Bookmarks when comparing hinge losses for SPEN (InfNet) and whether to retune the inference network. + +
hinge loss-retuning+retuning
margin rescaling38.5138.68
slack rescaling38.5738.62
perceptron (MR,△= 0)38.5538.70
contrastive (△= 1)38.8038.88
+ +# 9 APPENDIX + +# 9.1 MULTI-LABEL CLASSIFICATION + +Table 7 shows dataset statistics for the multi-label classification datasets. + +Hyperparameter Tuning. We tune $\lambda$ (the $L _ { 2 }$ regularization strength for $\Theta$ ) over the set $\{ 0 . 0 1 , 0 . 0 0 1 , 0 . 0 0 0 1 \}$ . The classification threshold $\tau$ is chosen from $[ 0 , 0 . 0 1 , 0 . 0 2 , 0 . 0 3 , 0 . 0 4 , 0 . 0 5 , 0 . 1 , 0 . 1 5 , 0 . 2 , 0 . 2 5 ,$ 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, 0.75] as also done by Belanger & McCallum (2016). We tune the coefficients for the three stabilization terms for the inference network objective from Section 5 over the follow ranges: $L _ { 2 }$ regularization $( \lambda _ { 1 } ~ \in ~ \{ 0 . 0 1 , 0 . 0 0 1 , 0 . 0 0 0 1 \} )$ ), entropy regularization $\mathbf { \lambda } ) _ { 2 } ~ = ~ 1 )$ , and regularization toward the pretrained feature network $( \dot { \lambda } _ { 4 } \in \{ 0 , 1 , 1 0 \} )$ ). + +Comparison of Loss Functions and Impact of Inference Network Retuning. Table 8 shows results comparing the four loss functions from Section 4.2 on the development set for Bookmarks, the largest of the three datasets. We find performance to be highly similar across the losses, with the contrastive loss appearing slightly better than the others. + +After training, we “retune” the inference network as specified by Eq. (5) on the development set for 20 epochs using a smaller learning rate of 0.00001. Table 8 shows slightly higher F1 for all losses with retuning. We were surprised to see that the final cost-augmented inference network performs well as a test-time inference network. This suggests that by the end of training, the cost-augmented network may be approaching the argmin and that there may not be much need for retuning. + +When using $\triangle = 0$ or 1, retuning leads to the same small gain as when using the margin-rescaled or slack-rescaled losses. Here the gain is presumably from adjusting the inference network for other inputs rather than from converting it from a cost-augmented to a test-time inference network. + +# 9.2 TWITTER POS TAGGING + +# 9.2.1 HYPERPARAMETER TUNING + +When training inference networks and SPENs for Twitter POS tagging, we use the following hyperparameter tuning. We tune the inference network learning rate $( \{ 0 . 1 , 0 . 0 5 , 0 . 0 2 , 0 . 0 1 , 0 . 0 0 5 , 0 . 0 0 1 \} )$ ), $L _ { 2 }$ regularization $( \lambda _ { 1 } \in \{ 0 , 1 { \mathrm { e } } - 3 , 1 { \mathrm { e } } - 4 , 1 { \mathrm { e } } - 5 , 1 { \mathrm { e } } - { \bar { 6 } } , 1 { \mathrm { e } } - { \bar { 7 } } \} )$ ), the entropy regularization term $( \lambda _ { 2 } \in \{ 0 . 1 , 0 . 5 , 1 , 2 , 5 , 1 0 \} )$ , the cross entropy regularization term $( \lambda _ { 3 } \in \{ 0 . 1 , 0 . 5 , 1 , 2 , 5 , 1 0 \} )$ , and the squared L2 distance $( \bar { \lambda } _ { 4 } \in \{ 0 , 0 . 1 , 0 . 2 , \bar { 0 . 5 } , 1 , 2 , 1 0 \} )$ ). We train the energy functions with Adam with a learning rate of 0.001 and $L _ { 2 }$ regularization $( \lambda _ { 1 } \in \{ 0 , 1 \mathrm { { e } - 3 , 1 \mathrm { { e } - 4 , 1 \mathrm { { e } - 5 } , 1 \mathrm { { e } - 6 } , 1 \mathrm { { e } - 7 } \} ) } }$ . + +Table 9 compares the use of the cross entropy and entropy stabilization terms when training inference networks for a SPEN with margin-rescaled hinge. Cross entropy works better than entropy in this setting, though retuning permits the latter to bridge the gap more than halfway. + +Table 9: Comparison of inference network stabilization terms and showing impact of retuning when training SPENs with margin-rescaled hinge (Twitter POS validation accuracies). + +
validation accuracy (%)
inference network stabilization terms-retuning+retuning
cross entropy89.189.3
entropy84.286.8
+ +Table 10: Examples of improvements in Twitter POS tagging when using tag language model (TLM). In all of these examples, the predicted tag when using the TLM matches the gold standard. + +
predicted tags
#tweet (target word in bold)-TLM+TLM
1... that's a t-17, technically . does that count as top-25 ?determinerpronoun
2... lol you know im down like 4 flats on a cadillac ... lol...adjectivepreposition
3... them who he is : he wants her to like him for his pers..prepositionverb
4I wonder when Nic Cage is going to film " Another Somethingnounverb
5Something Las Vegas " . Cut my hair, gag and bore menounverb
6 7... they had their fun,we hd ours !;) lmaooo " Logic will get you from A to B . Imagination will take youproper noun verbverb
everywhere ." - Albert Einstein .noun
8lmao I'm not a sheep who listens to it cos everyone else does ..verbpreposition
9Noo its not cuss you have swag andd you wont look dumb !..nouncoord. conj.
+ +When training CRFs, we use SGD with momentum. We tune the learning rate (over $\{ 0 . 1 , 0 . 0 5 , 0 . 0 2 , 0 . 0 1 , 0 . 0 0 5 , 0 . 0 0 1 \}$ ) and $L _ { 2 }$ regularization coefficient (over $\{ 0 , 1 \mathrm { e } - 3 , 1 \mathrm { e } - 4 , 1 \mathrm { e } -$ $5 , 1 \mathrm { e } - 6 , 1 \mathrm { e } - 7 \}$ ). For all methods, we use early stopping based on validation accuracy. + +# 9.2.2 TAG LANGUAGE MODEL DETAILS AND ANALYSIS + +To obtain training data for training the tag language model, we run the Twitter POS tagger from Owoputi et al. (2013) on a dataset of 303K randomly-sampled English tweets. We train the tag language model on 300K tweets and use the remaining 3K for tuning hyperparameters and early stopping. We train an LSTM language model on the tag sequences using stochastic gradient descent with momentum and early stopping on the validation set. We used a dropout rate of 0.5 for the LSTM hidden layer. We tune the learning rate $( \{ 0 . 1 , 0 . 2 , 0 . 5 , 1 . 0 \} )$ , the number of LSTM layers $( \{ 1 , 2 \} )$ , and the hidden layer size $( \{ 5 0 , 1 0 0 , 2 0 0 \} )$ ). + +Table 10 shows examples in which our SPEN that includes the TLM appears to be using broader context when making tagging decisions. These are examples from the test set labeled by two models: the SPEN without the TLM (which achieves $8 9 . 6 \%$ accuracy, as shown in Table 5) and the SPEN with the TLM (which reaches $9 0 . 2 \%$ accuracy). In example 1, the token “that” is predicted to be a determiner based on local context, but is correctly labeled a pronoun when using the TLM. This example is difficult because of the noun/verb tag ambiguity of the next word (“count”) and its impact on the tag for “that”. Examples 2 and 3 show two corrections for the token “like”, which is a highly ambiguous word in Twitter POS tagging. The broader context makes it much clearer which tag is intended. + +The next two examples (4 and 5) are cases of noun/verb ambiguity that are resolvable with larger context. The last four examples show improvements for nonstandard word forms. The shortened form of “had” (example 6) is difficult to tag due to its collision with “HD” (high-definition), but the model with the TLM is able to tag it correctly. In example 7, the ambiguous token “b” is frequently used as a short form of “be” on Twitter, and since it comes after “to” in this context, the verb interpretation is encouraged. However, the broader context makes it clear that it is not a verb and the TLM-enriched model tags it correctly. The words in the last two examples are nonstandard word forms that were not observed in the training data, which is likely the reason for their erroneous predictions. When using the TLM, we can better handle these rare forms based on the broader context. + +Table 11: Named entity recognition F1 of BLSTM, CRF, and SPEN (InfNet) with slack-rescaled hinge where inference networks used cross entropy stabilization term. Though slowest to train, the SPEN matches the test-time speed of the BLSTM while improving F1 by 2 points, though it lags behind the CRF. + +
validation F1test F1training speed (examples/sec)testing speed (examples/sec)
BLSTM88.3083.023851042
CRF91.3187.15222454
SPEN (InfNet)89.9885.061181025
+ +# 9.2.3 LEARNED PAIRWISE POTENTIAL MATRIX + +![](images/28cdae28d8e710e4818ba3c338be0b1442bf330857c6db3da42ee019ad439d1e.jpg) +Figure 1: Learned pairwise potential matrix for Twitter POS tagging. + +Figure 1 shows the learned pairwise potential matrix $W$ in Twitter POS tagging. We can see strong correlations between labels in neighborhoods. For example, an adjective (A) is more likely to be followed by a noun (N) than a verb (V) (see row labeled “A” in the figure). + +# 9.3 NAMED ENTITY RECOGNITION + +For named entity recognition (NER), we perform experiments on the English data from the CoNLL 2003 shared task (Tjong Kim Sang & De Meulder, 2003). This task contains sentences annotated with named entities and their types, containing 14987 training sentences, 3466 in the development set, and 3684 in the test set. There are four named entity types: PERSON, LOCATION, ORGANIZATION, and MISC. We use the BIOES tagging scheme instead of the original BIO2, following prior work (Ratinov & Roth, 2009; Ma & Hovy, 2016). There are $L = 1 7$ classes. We use 100-dimensional pretrained GloVe (Pennington et al., 2014) embeddings trained on 6 billion words from Wikipedia and web text, which work better than other pretrained embeddings (Ma & Hovy, 2016). + +Results are shown in Table 11. We see a large 4-point gap between the BLSTM and CRF, suggesting the importance of structured information for this problem. Though the SPEN still lags behind the CRF in F1, it matches the test-time speed of the BLSTM while improving F1 by 2 points. \ No newline at end of file diff --git a/parse/train/H1WgVz-AZ/H1WgVz-AZ_content_list.json b/parse/train/H1WgVz-AZ/H1WgVz-AZ_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..17ffc8ebe7adaa3007f12e3a9c7e9eab63866e04 --- /dev/null +++ b/parse/train/H1WgVz-AZ/H1WgVz-AZ_content_list.json @@ -0,0 +1,2263 @@ +[ + { + "type": "text", + "text": "LEARNING APPROXIMATE INFERENCE NETWORKS FOR STRUCTURED PREDICTION ", + "text_level": 1, + "bbox": [ + 176, + 99, + 779, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Lifu Tu Kevin Gimpel \nToyota Technological Institute at Chicago, Chicago, IL, 60637, USA \n{lifu,kgimpel}@ttic.edu ", + "bbox": [ + 183, + 169, + 633, + 212 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 250, + 544, + 263 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Structured prediction energy networks (SPENs; Belanger & McCallum 2016) use neural network architectures to define energy functions that can capture arbitrary dependencies among parts of structured outputs. Prior work used gradient descent for inference, relaxing the structured output to a set of continuous variables and then optimizing the energy with respect to them. We replace this use of gradient descent with a neural network trained to approximate structured argmax inference. This “inference network” outputs continuous values that we treat as the output structure. We develop large-margin training criteria for joint training of the structured energy function and inference network. On multi-label classification we report speed-ups of $1 0 { - } 6 0 \\mathrm { x }$ compared to (Belanger et al., 2017) while also improving accuracy. For sequence labeling with simple structured energies, our approach performs comparably to exact inference while being much faster at test time. We then demonstrate improved accuracy by augmenting the energy with a “label language model” that scores entire output label sequences, showing it can improve handling of long-distance dependencies in part-of-speech tagging. Finally, we show how inference networks can replace dynamic programming for test-time inference in conditional random fields, suggestive for their general use for fast inference in structured settings. ", + "bbox": [ + 232, + 279, + 766, + 527 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 551, + 336, + 568 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Energy-based modeling (LeCun et al., 2006) associates a scalar measure of compatibility to each configuration of input and output variables. Given an input $_ { \\textbf { \\em x } }$ , the predicted output $\\hat { \\textbf { \\textit { y } } }$ is chosen by minimizing an energy function $E ( \\pmb { x } , \\hat { \\pmb { y } } )$ . For structured prediction, the parameterization of the energy function can leverage domain knowledge about the structured output space. However, learning and prediction become complex. ", + "bbox": [ + 174, + 582, + 823, + 651 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Structured prediction energy networks (SPENs; Belanger & McCallum 2016) use an energy function to score structured outputs, and perform inference by using gradient descent to iteratively optimize the energy with respect to the outputs. Belanger et al. (2017) develop an “end-to-end” method that unrolls an approximate energy minimization algorithm into a fixed-size computation graph that is trainable by gradient descent. After learning the energy function, however, they still must use gradient descent for test-time inference. ", + "bbox": [ + 174, + 659, + 825, + 742 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We replace the gradient descent approach with a neural network trained to do inference, which we call an inference network. It can have any architecture such that it takes an input $_ { \\textbf { \\em x } }$ and returns an output interpretable as a $\\textbf { { y } }$ . As in prior work, we relax $\\textbf { { y } }$ from discrete to continuous. For multi-label classification, we use a feed-forward network that outputs a vector. We assign a single label to each dimension of the vector, interpreting its value as the probability of predicting that label. For sequence labeling, we output a distribution over predicted labels at each position in the sequence. We adapt the energy functions such that they can operate with both discrete ground truth outputs and outputs generated by our inference networks. ", + "bbox": [ + 174, + 750, + 825, + 861 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We define large-margin training objectives to jointly train energy functions and inference networks. Our training objectives resemble the alternating optimization framework of generative adversarial networks (GANs; Goodfellow et al. 2014): the inference network is analogous to the generator and the energy function is analogous to the discriminator. Our approach avoids argmax computations, making training and test-time inference faster than standard SPENs. We experiment with multi-label classification using the same setup as Belanger & McCallum (2016), demonstrating speed-ups of $1 0 \\mathrm { x }$ in training time and $6 0 \\mathrm { x }$ in test-time inference while also improving accuracy. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 146 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We then design a SPEN and inference network for sequence labeling by using recurrent neural networks (RNNs). We perform comparably to a conditional random field (CRF; Lafferty et al. 2001) when using the same energy function, with faster test-time inference. We also experiment with a richer energy that includes a “label language model” that scores entire output label sequences using an RNN, showing it can improve handling of long-distance dependencies in part-of-speech tagging. Finally, we show how inference networks can replace dynamic programming for test-time inference with CRFs, suggestive for the general use of inference networks to speed up inference in traditional structured prediction settings. ", + "bbox": [ + 173, + 152, + 825, + 265 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 STRUCTURED PREDICTION ENERGY NETWORKS ", + "text_level": 1, + "bbox": [ + 174, + 284, + 607, + 300 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We denote the space of inputs by $\\mathcal { X }$ . For a given input $\\mathbf { \\boldsymbol { x } } \\in \\mathcal { X }$ , we denote the space of legal structured outputs by $\\mathcal { V } ( \\pmb { x } )$ . We denote the entire space of structured outputs by $\\mathcal { V } = \\cup _ { \\pmb { x } \\in \\mathcal { X } } \\mathcal { V } ( \\pmb { x } )$ . A SPEN defines an energy function $E _ { \\Theta } : \\mathcal { X } \\times \\mathcal { Y } \\mathbb { R }$ parameterized by $\\Theta$ that uses a functional architecture to compute a scalar energy for an input/output pair. ", + "bbox": [ + 174, + 314, + 825, + 371 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We describe the SPEN for multi-label classification (MLC) from Belanger & McCallum (2016). Here, $_ { \\textbf { \\em x } }$ is a fixed-length feature vector. We assume there are $L$ labels, each of which can be on or off for each input, so $\\mathcal { V } ( \\pmb { x } ) = \\{ 0 , 1 \\} ^ { L }$ for all $_ { \\textbf { \\em x } }$ . The energy function is the sum of two terms: $E _ { \\Theta } ( { \\pmb x } , { \\pmb y } ) = E ^ { l o c } ( { \\pmb x } , { \\pmb y } ) + \\dot { E } ^ { l a b } ( { \\pmb y } )$ . $E ^ { l o c } ( { \\pmb x } , { \\pmb y } )$ is the sum of linear models: ", + "bbox": [ + 174, + 377, + 825, + 434 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/baf6f71e967caca1d316d7ddd7201209229ab95076d159c5f9173901c32ca808.jpg", + "text": "$$\nE ^ { l o c } ( { \\pmb x } , { \\pmb y } ) = \\sum _ { i = 1 } ^ { L } y _ { i } b _ { i } ^ { \\top } F ( { \\pmb x } )\n$$", + "text_format": "latex", + "bbox": [ + 403, + 438, + 594, + 481 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $b _ { i }$ is a parameter vector for label $i$ and $F ( { \\pmb x } )$ is a multi-layer perceptron computing a feature representation for the input $_ { \\textbf { \\em x } }$ . $E ^ { l a b } ( { \\pmb y } )$ scores $\\textbf { { y } }$ independent of $_ { \\textbf { \\em x } }$ : ", + "bbox": [ + 173, + 484, + 825, + 513 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/bf0061c1693a8bf3aed1da60ab8f002d6a79e952808a4d8623a007d8b16a07d2.jpg", + "text": "$$\nE ^ { l a b } ( { \\pmb y } ) = c _ { 2 } ^ { \\top } g ( C _ { 1 } { \\pmb y } )\n$$", + "text_format": "latex", + "bbox": [ + 424, + 517, + 573, + 536 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $c _ { 2 }$ is a parameter vector, $g$ is an elementwise non-linearity function, and $C _ { 1 }$ is a parameter matrix. After learning the energy function, prediction minimizes energy: ", + "bbox": [ + 171, + 539, + 821, + 568 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/55227d60b498df46d076e076cb18459a35d7bed1eceed53a64bfd7480da624be.jpg", + "text": "$$\n\\pmb { \\hat { y } } = \\underset { \\pmb { y } \\in \\mathscr { y } ( \\pmb { x } ) } { \\mathrm { a r g m i n } } E _ { \\Theta } ( \\pmb { x } , \\pmb { y } )\n$$", + "text_format": "latex", + "bbox": [ + 421, + 570, + 575, + 599 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "However, solving Eq. (3) requires combinatorial algorithms because $\\mathcal { V }$ is a discrete structured space. This becomes intractable when $E _ { \\Theta }$ does not decompose into a sum over small “parts” of $\\textbf { { y } }$ . Belanger & McCallum (2016) relax this problem by allowing the discrete vector $\\textbf { { y } }$ to be continuous. We use $\\mathcal { { V } } _ { R }$ to denote the relaxed output space. For MLC, $\\begin{array} { r } { \\breve { y } _ { R } ( { \\pmb x } ) = [ 0 , 1 ] ^ { L } } \\end{array}$ . They solve the relaxed problem by using gradient descent to iteratively optimize the energy with respect to $\\textbf { { y } }$ . Since they train with a structured large-margin objective, repeated inference is required during learning. They note that using gradient descent for this inference step is time-consuming and makes learning less stable. So Belanger et al. (2017) propose an “end-to-end” learning procedure inspired by Domke (2012). This approach performs backpropagation through each step of gradient descent. We compare to both methods in our experiments below. ", + "bbox": [ + 173, + 602, + 825, + 742 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3 INFERENCE NETWORKS FOR SPENS ", + "text_level": 1, + "bbox": [ + 174, + 761, + 509, + 777 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Belanger & McCallum (2016) relaxed $\\textbf { { y } }$ from a discrete to a continuous vector and used gradient descent for inference. We also relax $\\textbf { { y } }$ but we use a different strategy to approximate inference. We define an inference network $\\mathbf { A } _ { \\Psi } ( \\pmb { x } )$ parameterized by $\\Psi$ and train it with the goal that ", + "bbox": [ + 174, + 791, + 826, + 834 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/427e08864aacf82c0e47a677521569f8238eff6352c48ab6feb01ca7bd8dc8b1.jpg", + "text": "$$\n\\mathbf { A } _ { \\Psi } ( \\pmb { x } ) \\approx \\underset { \\pmb { y } \\in \\mathcal { V } _ { R } ( \\pmb { x } ) } { \\mathrm { a r g m i n } } E _ { \\Theta } ( \\pmb { x } , \\pmb { y } )\n$$", + "text_format": "latex", + "bbox": [ + 401, + 838, + 596, + 866 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Given an energy function $E _ { \\Theta }$ and a dataset $X$ of inputs, we solve the following optimization problem: ", + "bbox": [ + 179, + 869, + 825, + 885 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/3bf9815a6140b121415a70d85ca5f0ab9bb0e286e7ee824337a7cea13a6eaa31.jpg", + "text": "$$\n\\hat { \\Psi } \\underset { \\Psi } { \\mathrm { a r g m i n } } \\sum _ { \\pmb { x } \\in X } E _ { \\Theta } ( \\pmb { x } , \\mathbf { A } _ { \\Psi } ( \\pmb { x } ) )\n$$", + "text_format": "latex", + "bbox": [ + 383, + 887, + 612, + 921 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The architecture of $\\mathbf { A } _ { \\Psi }$ will depend on the task. For MLC, the same set of labels is applicable to every input, ${ \\bf { S 0 } } \\ y$ has the same length for all inputs. So, we can use a feed-forward network for $\\mathbf { A } _ { \\Psi }$ with a vector output, treating each dimension as the prediction for a single label. For sequence labeling, each $_ { \\textbf { \\em x } }$ (and therefore each $\\textbf { { y } }$ ) can have a different length, so we must use a network architecture for $\\mathbf { A } _ { \\Psi }$ that permits different lengths of predictions. We use an RNN that returns a vector at each position of $_ { \\textbf { \\em x } }$ . We interpret this vector as a probability distribution over output labels at that position. ", + "bbox": [ + 173, + 102, + 825, + 188 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We note that the output of $\\mathbf { A } _ { \\Psi }$ must be compatible with the energy function, which is typically defined in terms of the original discrete output space $\\mathcal { V }$ . This may require generalizing the energy function to be able to operate both on elements of $\\mathcal { V }$ and $\\mathcal { V } _ { R }$ . For MLC, no change is required. For sequence labeling, the change is straightforward and is described below in Section 7.2.1. ", + "bbox": [ + 174, + 194, + 825, + 251 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "4 JOINT TRAINING OF SPENS AND INFERENCE NETWORKS ", + "text_level": 1, + "bbox": [ + 176, + 270, + 686, + 286 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Belanger & McCallum (2016) propose a structured hinge loss for training SPENs: ", + "bbox": [ + 173, + 300, + 712, + 315 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/8b09b16075467b88bf5b8bf7af453f3340a24e7d9e02967171df4ed33283c98a.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\Theta } \\sum _ { \\langle x _ { i } , y _ { i } \\rangle \\in \\mathcal { D } } \\left[ \\operatorname* { m a x } _ { y \\in \\mathcal { V } _ { R } ( \\pmb { x } ) } \\left( \\triangle ( \\pmb { y } , \\pmb { y } _ { i } ) - E _ { \\Theta } ( \\pmb { x } _ { i } , \\pmb { y } ) + E _ { \\Theta } ( \\pmb { x } _ { i } , \\pmb { y } _ { i } ) \\right) \\right] _ { + }\n$$", + "text_format": "latex", + "bbox": [ + 277, + 319, + 717, + 361 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\mathcal { D }$ is the set of training pairs, $[ f ] _ { + } = \\operatorname* { m a x } ( 0 , f )$ , and $\\triangle ( \\pmb { y } , \\pmb { y } ^ { \\prime } )$ is a structured cost function that returns a nonnegative value indicating the difference between $\\textbf { { y } }$ and $\\boldsymbol { y } ^ { \\prime }$ . This loss is often referred to as “margin-rescaled” structured hinge loss (Taskar et al., 2004; Tsochantaridis et al., 2005). ", + "bbox": [ + 173, + 364, + 825, + 409 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "However, this loss is expensive to minimize for structured models because of the “cost-augmented” inference step $( \\operatorname* { m a x } _ { \\pmb { y } \\in \\mathcal { y } _ { R } ( \\pmb { x } ) } )$ . In prior work with SPENs, this step used gradient descent. We replace this with a cost-augmented inference network ${ \\bf A } _ { \\Phi } ( { \\pmb x } )$ . As suggested by the notation, the cost-augmented inference network $\\mathbf { A } _ { \\Phi }$ and the inference network $\\mathbf { A } _ { \\Psi }$ will typically have the same functional form, but use different parameters $\\Phi$ and $\\Psi$ . We write our new optimization problem as: ", + "bbox": [ + 174, + 412, + 826, + 486 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/3cd9ffa2b7b4e03e6ad212e519e8fd303dc78c32edbda1b39426a6f957ea3fca.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\Theta } \\operatorname* { m a x } _ { \\Phi } \\sum _ { \\langle \\pmb { x } _ { i } , \\pmb { y } _ { i } \\rangle \\in \\mathcal { D } } \\left[ \\triangle ( \\mathbf { A } _ { \\Phi } ( \\pmb { x } _ { i } ) , \\pmb { y } _ { i } ) - E _ { \\Theta } ( \\pmb { x } _ { i } , \\mathbf { A } _ { \\Phi } ( \\pmb { x } _ { i } ) ) + E _ { \\Theta } ( \\pmb { x } _ { i } , \\pmb { y } _ { i } ) \\right] _ { + }\n$$", + "text_format": "latex", + "bbox": [ + 258, + 489, + 738, + 526 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We treat this optimization problem as a minimax game and find a saddle point for the game. Following Goodfellow et al. (2014), we implement this using an iterative numerical approach. We alternatively optimize $\\Phi$ and $\\Theta$ , holding the other fixed. Optimizing $\\Phi$ to completion in the inner loop of training is computationally prohibitive and may lead to overfitting. So we alternate between one mini-batch for optimizing $\\Phi$ and one for optimizing $\\Theta$ . We also add $L _ { 2 }$ regularization terms for $\\Theta$ and $\\Phi$ . ", + "bbox": [ + 173, + 536, + 825, + 608 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The objective for the cost-augmented inference network is: ", + "bbox": [ + 174, + 613, + 560, + 628 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/81b3451fee569d726e87a3eefef544d0a83f9edbf6416c0fbd76d4ca9b61468c.jpg", + "text": "$$\n\\hat { \\Phi } \\underset { \\Phi } { \\mathrm { a r g m a x } } [ \\bigtriangleup ( \\mathbf { A } _ { \\Phi } ( \\boldsymbol { x } _ { i } ) , \\pmb { y } _ { i } ) - E _ { \\Theta } ( \\boldsymbol { x } _ { i } , \\mathbf { A } _ { \\Phi } ( \\boldsymbol { x } ) _ { i } ) + E _ { \\Theta } ( \\boldsymbol { x } _ { i } , \\pmb { y } _ { i } ) ] _ { + }\n$$", + "text_format": "latex", + "bbox": [ + 276, + 632, + 720, + 659 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "That is, we update $\\Phi$ so that $\\mathbf { A } _ { \\Phi }$ yields an output that has low energy and high cost, in order to mimic cost-augmented inference. The energy parameters $\\Theta$ are kept fixed. There is an analogy here to the generator in GANs: $\\mathbf { A } _ { \\Phi }$ is trained to produce a high-cost structured output that is also appealing to the current energy function. To help stabilize training of $\\Phi$ , we add several terms to this objective, discussed below in Section 5. ", + "bbox": [ + 173, + 662, + 826, + 733 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The objective for the energy function is: ", + "bbox": [ + 176, + 739, + 436, + 755 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/6e4c463e6ac65a7e85e44a613b15c85229d4923b6dd4c9b25b7d1273aa2be9da.jpg", + "text": "$$\n\\hat { \\Theta } \\underset { \\Theta } { \\mathrm { a r g m i n } } [ \\triangle ( \\mathbf { A } _ { \\Phi } ( \\pmb { x } _ { i } ) , \\pmb { y } _ { i } ) - E _ { \\Theta } ( \\pmb { x } _ { i } , \\mathbf { A } _ { \\Phi } ( \\pmb { x } _ { i } ) ) + E _ { \\Theta } ( \\pmb { x } _ { i } , \\pmb { y } _ { i } ) ] _ { + } + \\lambda \\| \\Theta \\| _ { 2 } ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 245, + 758, + 751, + 786 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "That is, we update $\\Theta$ so as to widen the gap between the cost-augmented and ground truth outputs. There is an analogy here to the discriminator in GANs. The energy function is updated so as to enable it to distinguish “fake” outputs produced by $\\mathbf { A } _ { \\Phi }$ from real outputs $\\mathbf { \\nabla } _ { \\mathbf { \\psi } _ { j } } \\mathbf { \\sigma } _ { j } \\mathbf { \\sigma } _ { j } $ . ", + "bbox": [ + 173, + 790, + 826, + 833 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Training iterates between updating $\\Phi$ and $\\Theta$ using the objectives above. ", + "bbox": [ + 174, + 838, + 640, + 853 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "4.1 TEST-TIME INFERENCE ", + "text_level": 1, + "bbox": [ + 174, + 869, + 379, + 883 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "After training, we want to use an inference network $\\mathbf { A } _ { \\Psi }$ defined in Eq. (4). However, training only gives us a cost-augmented inference network $\\mathbf { A } _ { \\Phi }$ . Since $\\mathbf { A } _ { \\Psi }$ and $\\mathbf { A } _ { \\Phi }$ have the same functional form, we can use $\\Phi$ to initialize $\\Psi$ , then do additional training on $\\mathbf { A } _ { \\Psi }$ as in Eq. (5) where $X$ is the training or validation set. This step helps the resulting inference network to produce outputs with lower energy, as it is no longer affected by the cost function. Since this procedure does not use the output labels of the $_ { \\textbf { \\em x } }$ ’s in $X$ , it could also be applied to the test data in a transductive setting. ", + "bbox": [ + 173, + 895, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 160 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.2 VARIATIONS AND SPECIAL CASES ", + "text_level": 1, + "bbox": [ + 176, + 176, + 450, + 190 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This approach also permits us to use large-margin structured prediction with slack rescaling (Tsochantaridis et al., 2005). Slack rescaling can yield higher accuracies than margin rescaling, but requires “cost-scaled” inference during training which is intractable for many classes of output structures. However, we can use our notion of inference networks to circumvent this tractability issue and approximately optimize the slack-rescaled hinge loss, yielding the following optimization problem: ", + "bbox": [ + 173, + 202, + 826, + 272 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/c38a7125f1caf5dee8eb0dcc6338c0a0ca5a1baa5f9921e518f50e97c7405cd5.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\Theta } \\operatorname* { m a x } _ { \\Phi } \\sum _ { \\langle \\mathbf { x } _ { i } , \\mathbf { y } _ { i } \\rangle \\in \\mathcal { D } } \\bigtriangleup ( \\mathbf { A } _ { \\Phi } ( \\mathbf { x } _ { i } ) , \\mathbf { y } _ { i } ) [ 1 - E _ { \\Theta } ( \\mathbf { x } _ { i } , \\mathbf { A } _ { \\Phi } ( \\mathbf { x } _ { i } ) ) + E _ { \\Theta } ( \\mathbf { x } _ { i } , \\mathbf { y } _ { i } ) ] _ { + }\n$$", + "text_format": "latex", + "bbox": [ + 253, + 275, + 743, + 311 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Using the same argument as above, we can also break this into alternating optimization of $\\Phi$ and $\\Theta$ . ", + "bbox": [ + 178, + 315, + 825, + 330 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We can optimize a structured perceptron (Collins, 2002) version by using the margin-rescaled hinge loss (Eq. (7)) and fixing $\\begin{array} { r } { \\triangle ( \\mathbf { A } _ { \\Phi } ( \\pmb { x } _ { i } ) , \\pmb { y } _ { i } ) = 0 } \\end{array}$ . When using this loss, the cost-augmented inference network is actually a test-time inference network, because the cost is always zero, so using this loss may lessen the need to retune the inference network after training. ", + "bbox": [ + 174, + 335, + 825, + 392 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "When we fix $\\begin{array} { r } { \\triangle ( \\mathbf { A } _ { \\Phi } ( \\pmb { x } _ { i } ) , \\pmb { y } _ { i } ) = 1 } \\end{array}$ , then margin-rescaled hinge is equivalent to slack-rescaled hinge. While using $\\triangle = 1$ is not useful in standard max-margin training with exact argmax inference (because the cost has no impact on optimization when fixed to a positive constant), it is potentially useful in our setting. Consider our SPEN objectives with $\\triangle = 1$ : ", + "bbox": [ + 173, + 398, + 825, + 455 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/e57faeecede0e0329d0b02faa574cae0a413eb640a5c47c7dca92999b9f2e2e6.jpg", + "text": "$$\n[ 1 - E _ { \\Theta } ( { \\pmb x } _ { i } , { \\pmb A } _ { \\Phi } ( { \\pmb x } _ { i } ) ) + E _ { \\Theta } ( { \\pmb x } _ { i } , { \\pmb y } _ { i } ) ] _ { + }\n$$", + "text_format": "latex", + "bbox": [ + 369, + 459, + 629, + 478 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "There will always be a nonzero difference between the two energies because ${ \\bf A } _ { \\Phi } ( { \\pmb x } _ { i } )$ will never exactly equal the discrete vector $\\mathbf { \\nabla } _ { \\mathbf { \\mathcal { Y } } _ { i } }$ . Since there is no explicit minimization over all discrete vectors $\\textbf { { y } }$ , this case is more similar to a “contrastive” hinge loss which seeks to make the energy of the true output lower than the energy of a particular “negative sample” by a margin of at least 1. ", + "bbox": [ + 174, + 481, + 825, + 537 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In our experiments, we will compare four hinge losses for training SPENs: margin-rescaled (Eq. (7)), slack-rescaled (Eq. (10)), perceptron (margin-rescaled with $\\triangle = 0$ ), and contrastive $\\triangle = 1$ ). ", + "bbox": [ + 174, + 544, + 826, + 573 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "5 IMPROVING TRAINING FOR INFERENCE NETWORKS ", + "text_level": 1, + "bbox": [ + 174, + 593, + 635, + 609 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We found that the alternating nature of the optimization led to difficulties during training. Similar observations have been noted about other alternative optimization settings, especially those underlying generative adversarial networks (Salimans et al., 2016). Below we describe several techniques we found to help stabilize training, which are optional terms added to the objective in Eq. (8). ", + "bbox": [ + 173, + 623, + 825, + 680 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "$L _ { 2 }$ Regularization: We use $L _ { 2 }$ regularization, adding the penalty term $\\| \\Phi \\| _ { 2 } ^ { 2 }$ with coefficient $\\lambda _ { 1 }$ ", + "bbox": [ + 173, + 685, + 805, + 702 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Entropy Regularization: We add an entropy-based regularizer $\\mathrm { l o s s } _ { \\mathrm { H } } ( \\mathbf { A } _ { \\Phi } ( \\pmb { x } ) )$ defined for the problem under consideration. For MLC, the output of $\\mathbf { A } _ { \\Phi } ( \\pmb { x } )$ is a vector of scalars in [0, 1], one for each label, where the scalar is interpreted as a label probability. The entropy regularizer $\\mathrm { l o s s } _ { \\mathrm { H } }$ is the sum of the entropies over these label binary distributions. For sequence labeling, where the length of $_ { \\textbf { \\em x } }$ is $N$ and where there are $L$ unique labels, the output of ${ \\bf A } _ { \\Phi } ( { \\pmb x } )$ is a length- $N$ sequence of length- $L$ vectors, each of which represents the distribution over the $L$ labels at that position in $_ { \\textbf { \\em x } }$ . Then, $\\mathrm { l o s s } _ { \\mathrm { H } }$ is the sum of entropies of these label distributions across positions in the sequence. ", + "bbox": [ + 173, + 707, + 825, + 805 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "When tuning the coefficient $\\lambda _ { 2 }$ for this regularizer, we consider both positive and negative values, permitting us to favor either low- or high-entropy distributions as the task prefers.1 ", + "bbox": [ + 174, + 811, + 821, + 840 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Local Cross Entropy Loss: We add a local (non-structured) cross entropy $\\mathrm { l o s s } _ { \\mathrm { C E } } ( \\mathbf { A } _ { \\Phi } ( \\pmb { x } _ { i } ) , \\pmb { y } _ { i } )$ defined for the problem under consideration. We only experiment with this loss for sequence labeling. ", + "bbox": [ + 174, + 848, + 823, + 876 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "It is the sum of the label cross entropy losses over all positions in the sequence. This loss provides more explicit feedback to the inference network, helping the optimization procedure to find a solution that minimizes the energy function while also correctly classifying individual labels. It can also be viewed as a multi-task loss for the inference network. ", + "bbox": [ + 174, + 103, + 823, + 160 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Regularization Toward Pretrained Inference Network: We add the penalty $\\lVert \\Phi - \\Phi _ { 0 } \\rVert _ { 2 } ^ { 2 }$ where $\\Phi _ { 0 }$ is a pretrained network, e.g., a local classifier trained to independently predict each part of $\\textbf { { y } }$ . ", + "bbox": [ + 171, + 166, + 823, + 195 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Each additional term has its own tunable hyperparameter. Finally we obtain: ", + "bbox": [ + 174, + 202, + 674, + 217 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/fc64f56ebe399567e7f1d7d37808748ed8b7c86a56715673a34127347f2f0f18.jpg", + "text": "$$\n\\begin{array} { r } { \\hat { \\Phi } \\underset { \\Phi } { \\operatorname { a r g m a x } } \\ [ \\triangle ( \\mathbf { A } _ { \\Phi } ( \\boldsymbol { x } _ { i } ) , \\boldsymbol { y } _ { i } ) - E _ { \\Theta } ( \\boldsymbol { x } _ { i } , \\mathbf { A } _ { \\Phi } ( \\boldsymbol { x } _ { i } ) ) + E _ { \\Theta } ( \\boldsymbol { x } _ { i } , \\boldsymbol { y } _ { i } ) ] _ { + } - \\lambda _ { 1 } \\| \\Phi \\| _ { 2 } ^ { 2 } } \\\\ { + \\lambda _ { 2 } \\mathrm { l o s s } _ { \\mathrm { H } } ( \\mathbf { A } _ { \\Phi } ( \\boldsymbol { x } _ { i } ) ) - \\lambda _ { 3 } \\mathrm { l o s s } _ { \\mathrm { C E } } ( \\mathbf { A } _ { \\Phi } ( \\boldsymbol { x } _ { i } ) , \\boldsymbol { y } _ { i } ) - \\lambda _ { 4 } \\| \\Phi - \\Phi _ { 0 } \\| _ { 2 } ^ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 236, + 248, + 761, + 299 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "6 RELATED WORK ", + "text_level": 1, + "bbox": [ + 174, + 344, + 344, + 361 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Our methods are reminiscent of other alternating optimization problems like that underlying generative adversarial networks (GANs; Goodfellow et al. 2014). GANs are based on a minimax game and have a value function that one agent (a discriminator $D$ ) seeks to maximize and another (a generator $G$ ) seeks to minimize. By their analysis, a log loss discriminator converges to a degenerate uniform solution. When using hinge loss, we can get a non-degenerate discriminator while matching the data distribution (Dai et al., 2017; Zhao et al., 2016). Our formulation is closer to this hinge loss version of the GAN. ", + "bbox": [ + 173, + 395, + 825, + 492 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Our approach is also related to knowledge distillation (Ba & Caruana, 2014; Hinton et al., 2015), which refers to strategies in which one model (a “student”) is trained to mimic another (a “teacher”). Typically, the teacher is a larger, more accurate model but which is too computationally expensive to use at test time. Urban et al. (2016) train shallow networks using image classification data labeled by an ensemble of deep teacher nets. Geras et al. (2016) train a convolutional network to mimic an LSTM for speech recognition. Others have explored knowledge distillation for sequence-to-sequence learning (Kim & Rush, 2016) and parsing (Kuncoro et al., 2016). ", + "bbox": [ + 173, + 498, + 825, + 597 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Since we train a single inference network for an entire dataset, our approach is also related to “amortized inference” (Srikumar et al., 2012; Gershman & Goodman, 2014; Paige & Wood, 2016; Chang et al., 2015). Such methods precompute or save solutions to subproblems for faster overall computation. Our inference networks likely devote more modeling capacity to the most frequent substructures in the data. A kind of inference network is used in variational autoencoders (Kingma & Welling, 2013) to approximate posterior inference in generative models. ", + "bbox": [ + 173, + 603, + 825, + 688 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Our methods are also related to work in structured prediction that seeks to approximate structured models with factorized ones, e.g., mean-field approximations in graphical models (Koller & Friedman, 2009; Krähenbühl & Koltun, 2011). Like our use of inference networks, there have been efforts in designing differentiable approximations of combinatorial search procedures (Martins & Kreutzer, 2017; Goyal et al., 2018) and structured losses for training with them (Wiseman & Rush, 2016). Since we relax discrete output variables to be continuous, there is also a connection to recent work that focuses on structured prediction with continuous valued output variables (Wang et al., 2016). They also propose a formulation that yields an alternating optimization problem, but it is based on proximal methods. ", + "bbox": [ + 173, + 694, + 826, + 819 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "There are other settings in which gradient descent is used for inference, e.g., image generation applications like DeepDream (Mordvintsev et al., 2015) and neural style transfer (Gatys et al., 2015), as well as machine translation (Hoang et al., 2017). In these and related settings, gradient descent has started to be replaced by inference networks, especially for image transformation tasks (Johnson et al., 2016; Li & Wand, 2016). Our results below provide more evidence for making this transition. An alternative to what we pursue here would be to obtain an easier convex optimization problem for inference via input convex neural networks (Amos et al., 2017). ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/8b72a4945f4a7f3180daa392e3b744eef87f57e685021a59c525807ba1899d6a.jpg", + "table_caption": [ + "Table 1: Test F1 when comparing methods on multi-label classification datasets. " + ], + "table_footnote": [], + "table_body": "
BibtexBookmarksDeliciousavg.
MLP38.933.837.836.8
SPEN (BM16)42.234.437.538.0
SPEN (E2E)38.133.934.435.5
SPEN (InfNet)42.237.637.539.1
", + "bbox": [ + 310, + 131, + 687, + 199 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "7 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 229, + 326, + 244 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In Sec. 7.1 we compare our approach to previous work on training SPENs for MLC. We compare accuracy and speed, finding our approach to outperform prior work. We then perform experiments with sequence labeling tasks in Sec. 7.2. ", + "bbox": [ + 174, + 262, + 825, + 303 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "7.1 MULTI-LABEL CLASSIFICATION ", + "text_level": 1, + "bbox": [ + 176, + 321, + 436, + 337 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We use the MLC datasets used by Belanger & McCallum (2016): Bibtex, Delicious, and Bookmarks. Dataset statistics are shown in Table 7 in the Appendix. For Bibtex and Delicious, we follow Belanger and McCallum and tune the hyperparameters using a different sampling of train and test data, then use the standard train/test split for final experimentation using the tuned hyperparameters. For Bookmarks, we use the same train/dev/test split as (Belanger & McCallum, 2016). For evaluation, we report the example averaged (macro averaged) F1 measure. ", + "bbox": [ + 173, + 348, + 826, + 433 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We use the SPEN for MLC described in Section 2 and also used by Belanger & McCallum (2016). For the feature representation network $F ( { \\pmb x } )$ , we use feed-forward networks with two hidden layers, using their same layer widths: 150 for Bibtex/Bookmarks and 250 for Delicious. We pretrain the feature networks $F ( { \\dot { \\mathbf { x } } } )$ by minimizing independent-label cross entropy for 10 epochs using Adam (Kingma & Ba, 2014) with learning rate 0.001. While training SPENs, we only update the parameters of the energy function $( \\Theta )$ and the inference network $( \\Phi )$ , keeping the feature network parameters $F ( { \\pmb x } )$ fixed. We use Adam with learning rate 0.001 to train $\\Theta$ and $\\Phi$ . ", + "bbox": [ + 174, + 439, + 825, + 536 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The inference networks are feed-forward networks with two hidden layers, using the same architectures as the feature networks $F ( { \\pmb x } )$ . This permits us to initialize inference network parameters $\\Phi$ using pretrained feature network parameters. For the output, we use an affine transformation layer with a sigmoid nonlinearity function, so the output values are in the range $( 0 , 1 )$ . We interpret each value as the probability of predicting the corresponding label. We obtain discrete predictions by thresholding at a threshold $\\tau$ tuned to maximize F1 on the development data. We add three terms to the inference network objective from Section 5: $L _ { 2 }$ regularization, entropy regularization, and regularization toward the pretrained feature network. Margin rescaling and slack rescaling use squared $L _ { 2 }$ distance for $\\triangle$ . Additional details are provided in Sec. 9.1 in the appendix. ", + "bbox": [ + 173, + 544, + 825, + 667 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Comparison to Prior Work. Table 1 shows results comparing to prior work. The MLP and “SPEN (BM16)” baseline results are taken from (Belanger & McCallum, 2016). We obtained the “SPEN (E2E)” (Belanger et al., 2017) results by running the code available from the authors on these datasets. This method constructs a recurrent neural network that performs gradient-based minimization of the energy with respect to $\\textbf { { y } }$ . They noted in their software release that, while this method is more stable, it is prone to overfitting and actually performs worse than the original SPEN. We indeed find this to be the case, as SPEN (E2E) underperforms SPEN (BM16) on all three datasets. ", + "bbox": [ + 173, + 675, + 825, + 773 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Our method (“SPEN (InfNet)”) achieves the best average performance across the three datasets. It performs especially well on Bookmarks, which is the largest of the three. Our results use the contrastive hinge loss and retune the inference network on the development data after the energy is trained; these decisions were made based on the tuning described in Sec. 9.1, but all four hinge losses led to similarly strong results. ", + "bbox": [ + 174, + 780, + 825, + 849 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Speed Comparison. Table 2 compares training and test-time inference speed among the different methods. We only report speeds of methods that we ran.2 The SPEN (E2E) times were obtained using code obtained from Belanger and McCallum. We suspect that SPEN (BM16) training would be comparable to or slower than SPEN (E2E). Our method can process examples during training about 10 times as fast as the end-to-end SPEN, and 60-130 times as fast during test-time inference. In fact, at test time, our method is roughly the same speed as the MLP baseline, since our inference networks use the same architecture as the feature networks which form the MLP baseline. Compared to the MLP, the training of our method takes significantly more time overall because of joint training of the energy function and inference network, but fortunately the test-time inference is comparable. ", + "bbox": [ + 176, + 857, + 823, + 885 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/3a9a43483713f0298592fe74c282cc354bc5beea38f2261988b0e2867be2a061.jpg", + "table_caption": [ + "Table 2: Training and test-time inference speed comparison (examples/sec). " + ], + "table_footnote": [], + "table_body": "
Training Speed (examples/sec)Testing Speed (examples/sec)
BibtexBookmarksDeliciousBibtexBookmarksDelicious
MLP2167019591261589070692307113750
SPEN (E2E)55155938314201401832
SPEN (InfNet)5533546746679419488888112148
", + "bbox": [ + 223, + 131, + 776, + 199 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 228, + 825, + 327 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "7.2 SEQUENCE LABELING ", + "text_level": 1, + "bbox": [ + 176, + 343, + 369, + 357 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We also evaluate our methods on sequence labeling. We report experiments with Twitter part-ofspeech (POS) tagging here. Named entity recognition experiments are reported in the Appendix. ", + "bbox": [ + 176, + 368, + 825, + 397 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "7.2.1 ENERGY FUNCTIONS FOR SEQUENCE LABELING ", + "text_level": 1, + "bbox": [ + 173, + 411, + 565, + 426 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The input space $\\mathcal { X }$ is now the set of all sequences of symbols drawn from a vocabulary. For an input sequence $_ { \\textbf { \\em x } }$ of length $N$ , where there are $L$ possible output labels for each position in $_ { \\textbf { \\em x } }$ , the output space $\\mathcal { V } ( \\pmb { x } )$ is $[ L ] ^ { \\widetilde { N } }$ , where the notation $[ q ]$ represents the set containing the first $q$ positive integers. We define $\\pmb { y } = \\langle y _ { 1 } , y _ { 2 } , . . , y _ { N } \\rangle$ where each $y _ { i }$ ranges over possible output labels, i.e., $y _ { i } \\in [ L ]$ . ", + "bbox": [ + 174, + 436, + 826, + 494 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "When defining our energy for sequence labeling, we take inspiration from bidirectional LSTMs (BLSTMs; Hochreiter $\\&$ Schmidhuber 1997) and conditional random fields (CRFs; Lafferty et al. 2001). A “linear chain” CRF uses two types of features: one capturing the connection between an output label and $_ { \\textbf { \\em x } }$ and the other capturing the dependence between neighboring output labels. We use a BLSTM to compute feature representations for $_ { \\textbf { \\em x } }$ . We use $f ( \\pmb { x } , t ) \\in \\mathbb { R } ^ { d }$ to denote the “input feature vector” for position $t$ , defining it to be the $d$ -dimensional BLSTM hidden vector at $t$ . ", + "bbox": [ + 173, + 500, + 826, + 584 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We then define the following energy function: ", + "bbox": [ + 174, + 590, + 475, + 604 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/a2de9a95dbf1faa7fdd406f922cdf30d4c171fb6c35d5e95f61b51265db017fc.jpg", + "text": "$$\nE _ { \\Theta } ( \\pmb { x } , \\pmb { y } ) = - \\left( \\sum _ { t } U _ { y _ { t } } ^ { \\top } f ( \\pmb { x } , t ) + \\sum _ { t } W _ { y _ { t - 1 } , y _ { t } } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 333, + 609, + 665, + 654 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $U _ { i } \\in \\mathbb { R } ^ { d }$ is a parameter vector for label $i$ and the parameter matrix $W \\in \\mathbb { R } ^ { L \\times L }$ contains label pair parameters. The full set of parameters $\\Theta$ includes the $U _ { i }$ vectors, $W$ , and the parameters of the BLSTM. The above energy only permits discrete $\\textbf { { y } }$ . For the general case that permits relaxing $\\textbf { { y } }$ to be continuous, we treat each $y _ { t }$ as a vector. It will be one-hot for the ground truth $\\textbf { { y } }$ and will be a vector of label probabilities for relaxed $\\textbf { { y } }$ ’s. Then the general energy function is: ", + "bbox": [ + 174, + 659, + 825, + 729 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/cb4ca5a91be813d2a1984f6c9163203c55b0b5b5df1adf4d42d7c5c8a4c4da79.jpg", + "text": "$$\nE _ { \\Theta } ( \\pmb { x } , \\pmb { y } ) = - \\left( \\sum _ { t } \\sum _ { i = 1 } ^ { L } y _ { t , i } \\left( U _ { i } ^ { \\top } f ( \\pmb { x } , t ) \\right) + \\sum _ { t } y _ { t - 1 } ^ { \\top } W y _ { t } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 299, + 734, + 699, + 779 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $y _ { t , i }$ is the $i$ th entry of the vector $y _ { t }$ . In the discrete case, this entry is 1 for a single $i$ and 0 for all others, so this energy reduces to Eq. (12) in that case. In the continuous case, this scalar indicates the probability of the tth position being labeled with label $i$ . For the label pair terms in this general energy function, we use a bilinear product between the vectors $y _ { t - 1 }$ and $y _ { t }$ using parameter matrix $W$ , which also reduces to Eq. (12) when they are one-hot vectors. ", + "bbox": [ + 173, + 782, + 825, + 853 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Tag Language Model. In order to capture long-distance dependencies in an entire sequence of labels, we train a “tag language model” on a large corpus of automatically-tagged tweets, then include a term in the energy function representing the log-probability of the given tag sequence under this tag language model. Details are provided below in Section 7.2.4. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/7902a62cd03e34d65119c0ce5c2887af77c0114eb6d439f00ba232fee30b3ac2.jpg", + "table_caption": [ + "Table 3: Comparison of SPEN hinge losses and showing the impact of retuning (Twitter POS validation accuracies). Inference networks are trained with the cross entropy term. " + ], + "table_footnote": [], + "table_body": "
SPEN hinge lossvalidation accuracy (%)
-retuning 89.1+retuning
margin rescaling slack rescaling89.489.3
perceptron (MR,△= 0)89.289.6
88.889.4
contrastive (△= 1)89.0
", + "bbox": [ + 333, + 146, + 663, + 226 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/29f2914da3874eb87cf5a63ba8a63ca0c718be7cbf302e423cefbdb9f035a4c7.jpg", + "table_caption": [ + "Table 4: Twitter POS accuracies of BLSTM, CRF, and SPEN (InfNet), using our tuned SPEN configuration (slack-rescaled hinge, inference network trained with cross entropy term). Though slowest to train, the SPEN matches the test-time speed of the BLSTM while achieving the highest accuracies. " + ], + "table_footnote": [], + "table_body": "
validation accuracy (%)test accuracy (%)training speed (examples/sec)testing speed (examples/sec)
BLSTM88.688.83851250
CRF89.189.2250500
SPEN (InfNet)89.689.81251250
", + "bbox": [ + 238, + 314, + 758, + 381 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "7.2.2 EXPERIMENTAL SETUP ", + "text_level": 1, + "bbox": [ + 174, + 411, + 390, + 426 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "For Twitter part-of-speech (POS) tagging, we use the annotated data from Gimpel et al. (2011) and Owoputi et al. (2013) which contains $L = 2 5$ POS tags. For training, we combine the 1000- tweet OCT27TRAIN set and the 327-tweet OCT27DEV set. For validation, we use the 500-tweet OCT27TEST set and for testing we use the 547-tweet DAILY547 test set. We use 100-dimensional skip-gram embeddings trained on 56 million English tweets with word2vec (Mikolov et al., 2013).3 ", + "bbox": [ + 174, + 435, + 825, + 506 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We use a BLSTM to compute the “input feature vector” $f ( { \\pmb x } , t )$ for each position $t$ , using hidden vectors of dimensionality $d = 1 0 0$ . We also use BLSTMs for the inference networks. The output layer of the inference network is a softmax function, so at every position, the inference network produces a distribution over labels at that position. We train inference networks using stochastic gradient descent (SGD) with momentum and train the energy parameters using Adam. For $\\triangle$ , we use $L _ { 1 }$ distance. We tune hyperparameters on the validation set; full details of tuning are provided in the appendix. We found that the cross entropy stabilization term worked well for this setting; details and an empirical comparison are provided in Section 9.2.1. ", + "bbox": [ + 174, + 512, + 825, + 625 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We compare to standard BLSTM and CRF baselines. We train the BLSTM baseline to minimize per-token log loss; this is often called a “BLSTM tagger”. We train a CRF baseline using the energy in Eq. (12) with the standard conditional log-likelihood objective using the standard dynamic programming algorithms (forward-backward) to compute gradients during training. Further details are provided in the appendix. ", + "bbox": [ + 174, + 631, + 825, + 702 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "7.2.3 RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 715, + 292, + 731 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Loss Function Comparison. Table 3 shows results when comparing SPEN training objectives. We see a larger difference among losses here than for MLC tasks. When using the perceptron loss, there is no margin, which leads to overfitting: 89.4 on validation, 88.6 on test (not shown in the table). The contrastive loss, which strives to achieve a margin of 1, does better on test (89.0). We also see here that margin rescaling and slack rescaling both outperform the contrastive hinge, unlike the MLC tasks. We suspect that in the case in which each input/output has a different length, using a cost that captures length is more important. ", + "bbox": [ + 174, + 739, + 825, + 838 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Comparison to Standard Baselines. Table 4 compares our final tuned SPEN configuration to two standard baselines: a BLSTM tagger and a CRF. The SPEN achieves higher validation and test accuracies with faster test-time inference. While our method is slower than the baselines during training, it is faster than the CRF at test time, operating at essentially the same speed as the BLSTM baseline while being more accurate. ", + "bbox": [ + 174, + 844, + 825, + 887 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/8537d0717da40f4c1c2018e46b6eaad0d33c747c6a87d714b36ba1dacfa0a6d9.jpg", + "table_caption": [ + "Table 5: Twitter POS validation/test accuracies when adding tag language model (TLM) energy term to a SPEN trained with margin-rescaled hinge. " + ], + "table_footnote": [], + "table_body": "
val. accuracy (%)test accuracy (%)
-TLM89.889.6
+TLM89.990.2
", + "bbox": [ + 346, + 145, + 650, + 186 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 218, + 823, + 246 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Here, the SPEN and CRF are using the same functional form for their energy functions, namely the energy given in Eq. (13). We note that the SPEN outperforms the CRF, despite using the same form for the energy. There are two factors that can explain this. First, the losses are different. The CRF uses conditional log-likelihood while the SPEN results here use slack-rescaled hinge, which outperforms the other hinge loss variants (Table 3). Second, the stabilization terms used when training the inference network may be providing a regularizing effect for the model. Our motivation for these experiments was to show the impact of these differences while keeping the form of the energy function fixed. We now turn to richer energies. ", + "bbox": [ + 173, + 253, + 825, + 364 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7.2.4 TOWARDS GLOBAL ENERGIES: TAG LANGUAGE MODELS FOR TWITTER POS TAGGING ", + "bbox": [ + 178, + 380, + 823, + 395 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The above results only use the pairwise energy; no results used the tag language model (TLM). To compute the TLM energy term, we first automatically tag unlabeled tweets, then train an LSTM language model on the automatic tag sequences. When doing so, we define the input tag embeddings to be $L$ -dimensional one-hot vectors specifying the tags in the training sequences. This is nonstandard compared to standard language modeling. In standard language modeling, we train on observed sequences and compute likelihoods of other fully-observed sequences. However, in our case, we train on tag sequences but we want to use the same model on sequences of tag distributions produced by an inference network. We train the TLM on sequences of one-hot vectors and then use it to compute likelihoods of sequences of tag distributions. Further details about training are provided in Section 9.2.2 in the appendix. ", + "bbox": [ + 173, + 404, + 825, + 544 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We define an additional energy term $E ^ { \\mathrm { T L M } } ( y )$ based on the pretrained TLM. If the argument $\\textbf { { y } }$ consisted of one-hot vectors, we could simply compute its likelihood. However, to support relaxed $\\textbf { { y } }$ ’s, we need to define a more general function: ", + "bbox": [ + 174, + 549, + 825, + 592 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/5eb397eae23ba8116a03e4aa804b4d4ce2322a93afe9110101cb66e890e1379c.jpg", + "text": "$$\nE ^ { \\mathrm { T L M } } ( \\pmb { y } ) = - \\sum _ { t = 1 } ^ { | \\pmb { y } | + 1 } \\log ( \\pmb { y } _ { t } ^ { \\top } \\mathrm { T L M } ( \\langle \\pmb { y } _ { 0 } , . . . , \\pmb { y } _ { t - 1 } \\rangle ) )\n$$", + "text_format": "latex", + "bbox": [ + 331, + 598, + 666, + 643 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "where $y _ { 0 }$ is the start-of-sequence symbol, $y _ { \\vert \\pmb { y } \\vert + 1 }$ is the end-of-sequence symbol, and $\\mathrm { T L M } \\big ( \\langle y _ { 0 } , . . . , y _ { t - 1 } \\rangle \\big )$ returns the softmax distribution over tags at position $t$ (under the pretrained tag language model) given the preceding tag vectors. When each $y _ { t }$ is a one-hot vector, this energy reduces to the negative log-likelihood of the tag sequence specified by $\\textbf { { y } }$ . ", + "bbox": [ + 173, + 648, + 825, + 707 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We define the new joint energy as the sum of the energy function in Eq. (13) and the TLM energy function in Eq. (14). During learning, we keep the TLM parameters fixed to their pretrained values, but we tune the weight of the TLM energy (over the set $\\{ 0 . 1 , 0 . 2 , 0 . 5 \\} )$ in the joint energy. We train SPENs with the new joint energy using the margin-rescaled hinge, training the inference network with the cross entropy term. ", + "bbox": [ + 173, + 713, + 825, + 784 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Table 5 shows results.4 Adding the TLM energy leads to a gain of 0.6 on the test set. Other settings showed more variance; when using slack-rescaled hinge, we found a small drop on test, while when simply training inference networks for a fixed, pretrained joint energy with tuned mixture coefficient, we found a gain of 0.3 on test when adding the TLM energy. We investigated the improvements and found some to involve corrections that seemingly stem from handling non-local dependencies better. Table 10 in the appendix shows examples in which the model with the TLM appears to be better at using the broader context when making tagging decisions. These results suggest that our method of training inference networks can be used to add rich features to structured prediction, though we leave a thorough exploration of global energies to future work. ", + "bbox": [ + 173, + 790, + 825, + 887 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/561df4aea97afbad078266a90ebf6a32391e3c2f361ff2f78b753fbb2276ec13.jpg", + "table_caption": [ + "Table 6: Comparison of test-time inference algorithms for a trained CRF (Twitter POS tagging). We show the test accuracy for the inference network setting that does best on validation. All inference networks use the same architecture and therefore have essentially the same speed. " + ], + "table_footnote": [], + "table_body": "
test-time inference algorithmval. accuracy (%)test accuracy (%)speed (examples/sec)
Viterbialgorithm89.189.2500
Inference network + cross entropy89.789.51250
Inference network+ entropy89.6
Inference network + squared L2 distance88.9
", + "bbox": [ + 174, + 159, + 823, + 227 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 260, + 823, + 287 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "7.2.5 BEYOND SPENS: INFERENCE NETWORKS FOR STRUCTURED PREDICTION", + "text_level": 1, + "bbox": [ + 178, + 305, + 741, + 320 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We note that inference networks can be used for any prediction problem. We now explore the use of an inference network to approximate test-time inference for a trained CRF. The results are shown in Table 6. All results use the same trained CRF energy function (Eq. (12)), trained to minimize log loss using the forward-backward algorithm for exact inference during training. The first row shows accuracy and speed when using Viterbi for test-time inference, which is the same setting as the “CRF” row in Table 4. Subsequent rows show results when training inference networks to mimic Viterbi with various stabilization terms. When training these inference networks, we train them on the training set and tune based on early stopping on the validation set. The energy stays fixed while inference networks are trained. ", + "bbox": [ + 174, + 330, + 825, + 455 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "When using either entropy or cross entropy, our inference networks outperform Viterbi while doubling its speed. When using the squared $L _ { 2 }$ distance term (which regularizes the inference network toward the pretrained BLSTM), the accuracy reduces to be closer to that of the BLSTM, which reaches $8 8 . 6 \\%$ on validation (see Table 4). When using no stabilization terms for the inference network, learning fails, reaching $1 3 . 7 \\%$ on the development set, showing the importance of using some stabilization term while training the inference network. ", + "bbox": [ + 174, + 463, + 825, + 546 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "These results show promise for training inference networks to speed up combinatorial algorithms for structured prediction and other domains. ", + "bbox": [ + 174, + 554, + 823, + 582 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "8 CONCLUSIONS AND FUTURE WORK ", + "text_level": 1, + "bbox": [ + 174, + 604, + 504, + 621 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We presented ways to jointly train structured energy functions and inference networks using largemargin objectives. The energy function captures arbitrary dependencies among the labels, while the inference networks learns to capture the properties of the energy in an efficient manner, yielding fast test-time inference. Future work includes exploring the space of network architectures for inference networks to balance accuracy and efficiency, experimenting with additional global terms in structured energy functions, and exploring richer structured output spaces such as trees and sentences. ", + "bbox": [ + 174, + 637, + 825, + 720 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 739, + 326, + 752 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We thank the anonymous reviewers, David Belanger, Weiran Wang and Zheng Cai. We also thank NVIDIA Corporation for donating GPUs used in this research. ", + "bbox": [ + 173, + 763, + 823, + 792 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 815, + 287, + 830 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Brandon Amos, Lei Xu, and J. Zico Kolter. Input convex neural networks. In Proc. of ICML, 2017. 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#labels# features#train#dev#test
Bibtex15918364836-2515
Bookmarks2082151480001200027856
Delicious98250112896-3185
", + "bbox": [ + 299, + 128, + 699, + 184 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/fa2d86b3c7ba7b4ae31a000e1a9f9501718a743a237f4a4feeaeffb78aae75ad.jpg", + "table_caption": [ + "Table 8: Development F1 for Bookmarks when comparing hinge losses for SPEN (InfNet) and whether to retune the inference network. " + ], + "table_footnote": [], + "table_body": "
hinge loss-retuning+retuning
margin rescaling38.5138.68
slack rescaling38.5738.62
perceptron (MR,△= 0)38.5538.70
contrastive (△= 1)38.8038.88
", + "bbox": [ + 338, + 247, + 658, + 314 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "9 APPENDIX ", + "text_level": 1, + "bbox": [ + 173, + 347, + 294, + 363 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "9.1 MULTI-LABEL CLASSIFICATION ", + "text_level": 1, + "bbox": [ + 176, + 381, + 436, + 396 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Table 7 shows dataset statistics for the multi-label classification datasets. ", + "bbox": [ + 174, + 409, + 648, + 422 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Hyperparameter Tuning. We tune $\\lambda$ (the $L _ { 2 }$ regularization strength for $\\Theta$ ) over the set $\\{ 0 . 0 1 , 0 . 0 0 1 , 0 . 0 0 0 1 \\}$ . The classification threshold $\\tau$ is chosen from $[ 0 , 0 . 0 1 , 0 . 0 2 , 0 . 0 3 , 0 . 0 4 , 0 . 0 5 , 0 . 1 , 0 . 1 5 , 0 . 2 , 0 . 2 5 ,$ 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, 0.75] as also done by Belanger & McCallum (2016). We tune the coefficients for the three stabilization terms for the inference network objective from Section 5 over the follow ranges: $L _ { 2 }$ regularization $( \\lambda _ { 1 } ~ \\in ~ \\{ 0 . 0 1 , 0 . 0 0 1 , 0 . 0 0 0 1 \\} )$ ), entropy regularization $\\mathbf { \\lambda } ) _ { 2 } ~ = ~ 1 )$ , and regularization toward the pretrained feature network $( \\dot { \\lambda } _ { 4 } \\in \\{ 0 , 1 , 1 0 \\} )$ ). ", + "bbox": [ + 173, + 430, + 825, + 529 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Comparison of Loss Functions and Impact of Inference Network Retuning. Table 8 shows results comparing the four loss functions from Section 4.2 on the development set for Bookmarks, the largest of the three datasets. We find performance to be highly similar across the losses, with the contrastive loss appearing slightly better than the others. ", + "bbox": [ + 174, + 535, + 825, + 590 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "After training, we “retune” the inference network as specified by Eq. (5) on the development set for 20 epochs using a smaller learning rate of 0.00001. Table 8 shows slightly higher F1 for all losses with retuning. We were surprised to see that the final cost-augmented inference network performs well as a test-time inference network. This suggests that by the end of training, the cost-augmented network may be approaching the argmin and that there may not be much need for retuning. ", + "bbox": [ + 174, + 597, + 825, + 667 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "When using $\\triangle = 0$ or 1, retuning leads to the same small gain as when using the margin-rescaled or slack-rescaled losses. Here the gain is presumably from adjusting the inference network for other inputs rather than from converting it from a cost-augmented to a test-time inference network. ", + "bbox": [ + 174, + 674, + 825, + 717 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "9.2 TWITTER POS TAGGING ", + "text_level": 1, + "bbox": [ + 174, + 737, + 385, + 752 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "9.2.1 HYPERPARAMETER TUNING ", + "text_level": 1, + "bbox": [ + 176, + 765, + 423, + 780 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "When training inference networks and SPENs for Twitter POS tagging, we use the following hyperparameter tuning. We tune the inference network learning rate $( \\{ 0 . 1 , 0 . 0 5 , 0 . 0 2 , 0 . 0 1 , 0 . 0 0 5 , 0 . 0 0 1 \\} )$ ), $L _ { 2 }$ regularization $( \\lambda _ { 1 } \\in \\{ 0 , 1 { \\mathrm { e } } - 3 , 1 { \\mathrm { e } } - 4 , 1 { \\mathrm { e } } - 5 , 1 { \\mathrm { e } } - { \\bar { 6 } } , 1 { \\mathrm { e } } - { \\bar { 7 } } \\} )$ ), the entropy regularization term $( \\lambda _ { 2 } \\in \\{ 0 . 1 , 0 . 5 , 1 , 2 , 5 , 1 0 \\} )$ , the cross entropy regularization term $( \\lambda _ { 3 } \\in \\{ 0 . 1 , 0 . 5 , 1 , 2 , 5 , 1 0 \\} )$ , and the squared L2 distance $( \\bar { \\lambda } _ { 4 } \\in \\{ 0 , 0 . 1 , 0 . 2 , \\bar { 0 . 5 } , 1 , 2 , 1 0 \\} )$ ). We train the energy functions with Adam with a learning rate of 0.001 and $L _ { 2 }$ regularization $( \\lambda _ { 1 } \\in \\{ 0 , 1 \\mathrm { { e } - 3 , 1 \\mathrm { { e } - 4 , 1 \\mathrm { { e } - 5 } , 1 \\mathrm { { e } - 6 } , 1 \\mathrm { { e } - 7 } \\} ) } }$ . ", + "bbox": [ + 173, + 790, + 826, + 876 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Table 9 compares the use of the cross entropy and entropy stabilization terms when training inference networks for a SPEN with margin-rescaled hinge. Cross entropy works better than entropy in this setting, though retuning permits the latter to bridge the gap more than halfway. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/52e56f1f7963ce24dd015d0b30495b629bbf3b1fbb7952dcbcd1238e1cc48616.jpg", + "table_caption": [ + "Table 9: Comparison of inference network stabilization terms and showing impact of retuning when training SPENs with margin-rescaled hinge (Twitter POS validation accuracies). " + ], + "table_footnote": [], + "table_body": "
validation accuracy (%)
inference network stabilization terms-retuning+retuning
cross entropy89.189.3
entropy84.286.8
", + "bbox": [ + 299, + 146, + 697, + 199 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/6825bbc3e98b08c9213a525503588606eea7ad0c4d07b08a4ef13e3a4ee9ddcc.jpg", + "table_caption": [ + "Table 10: Examples of improvements in Twitter POS tagging when using tag language model (TLM). In all of these examples, the predicted tag when using the TLM matches the gold standard. " + ], + "table_footnote": [], + "table_body": "
predicted tags
#tweet (target word in bold)-TLM+TLM
1... that's a t-17, technically . does that count as top-25 ?determinerpronoun
2... lol you know im down like 4 flats on a cadillac ... lol...adjectivepreposition
3... them who he is : he wants her to like him for his pers..prepositionverb
4I wonder when Nic Cage is going to film " Another Somethingnounverb
5Something Las Vegas " . Cut my hair, gag and bore menounverb
6 7... they had their fun,we hd ours !;) lmaooo " Logic will get you from A to B . Imagination will take youproper noun verbverb
everywhere ." - Albert Einstein .noun
8lmao I'm not a sheep who listens to it cos everyone else does ..verbpreposition
9Noo its not cuss you have swag andd you wont look dumb !..nouncoord. conj.
", + "bbox": [ + 174, + 265, + 825, + 449 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "When training CRFs, we use SGD with momentum. We tune the learning rate (over $\\{ 0 . 1 , 0 . 0 5 , 0 . 0 2 , 0 . 0 1 , 0 . 0 0 5 , 0 . 0 0 1 \\}$ ) and $L _ { 2 }$ regularization coefficient (over $\\{ 0 , 1 \\mathrm { e } - 3 , 1 \\mathrm { e } - 4 , 1 \\mathrm { e } -$ $5 , 1 \\mathrm { e } - 6 , 1 \\mathrm { e } - 7 \\}$ ). For all methods, we use early stopping based on validation accuracy. ", + "bbox": [ + 174, + 479, + 825, + 522 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "9.2.2 TAG LANGUAGE MODEL DETAILS AND ANALYSIS ", + "text_level": 1, + "bbox": [ + 173, + 537, + 575, + 551 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "To obtain training data for training the tag language model, we run the Twitter POS tagger from Owoputi et al. (2013) on a dataset of 303K randomly-sampled English tweets. We train the tag language model on 300K tweets and use the remaining 3K for tuning hyperparameters and early stopping. We train an LSTM language model on the tag sequences using stochastic gradient descent with momentum and early stopping on the validation set. We used a dropout rate of 0.5 for the LSTM hidden layer. We tune the learning rate $( \\{ 0 . 1 , 0 . 2 , 0 . 5 , 1 . 0 \\} )$ , the number of LSTM layers $( \\{ 1 , 2 \\} )$ , and the hidden layer size $( \\{ 5 0 , 1 0 0 , 2 0 0 \\} )$ ). ", + "bbox": [ + 174, + 563, + 825, + 660 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Table 10 shows examples in which our SPEN that includes the TLM appears to be using broader context when making tagging decisions. These are examples from the test set labeled by two models: the SPEN without the TLM (which achieves $8 9 . 6 \\%$ accuracy, as shown in Table 5) and the SPEN with the TLM (which reaches $9 0 . 2 \\%$ accuracy). In example 1, the token “that” is predicted to be a determiner based on local context, but is correctly labeled a pronoun when using the TLM. This example is difficult because of the noun/verb tag ambiguity of the next word (“count”) and its impact on the tag for “that”. Examples 2 and 3 show two corrections for the token “like”, which is a highly ambiguous word in Twitter POS tagging. The broader context makes it much clearer which tag is intended. ", + "bbox": [ + 173, + 666, + 825, + 791 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The next two examples (4 and 5) are cases of noun/verb ambiguity that are resolvable with larger context. The last four examples show improvements for nonstandard word forms. The shortened form of “had” (example 6) is difficult to tag due to its collision with “HD” (high-definition), but the model with the TLM is able to tag it correctly. In example 7, the ambiguous token “b” is frequently used as a short form of “be” on Twitter, and since it comes after “to” in this context, the verb interpretation is encouraged. However, the broader context makes it clear that it is not a verb and the TLM-enriched model tags it correctly. The words in the last two examples are nonstandard word forms that were not observed in the training data, which is likely the reason for their erroneous predictions. When using the TLM, we can better handle these rare forms based on the broader context. ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/11716e121864e9465a0fc3b640af9517d3676ddf28ab900b7c54a9eeea7a17fe.jpg", + "table_caption": [ + "Table 11: Named entity recognition F1 of BLSTM, CRF, and SPEN (InfNet) with slack-rescaled hinge where inference networks used cross entropy stabilization term. Though slowest to train, the SPEN matches the test-time speed of the BLSTM while improving F1 by 2 points, though it lags behind the CRF. " + ], + "table_footnote": [], + "table_body": "
validation F1test F1training speed (examples/sec)testing speed (examples/sec)
BLSTM88.3083.023851042
CRF91.3187.15222454
SPEN (InfNet)89.9885.061181025
", + "bbox": [ + 271, + 170, + 725, + 238 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "9.2.3 LEARNED PAIRWISE POTENTIAL MATRIX ", + "text_level": 1, + "bbox": [ + 173, + 268, + 516, + 284 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/28cdae28d8e710e4818ba3c338be0b1442bf330857c6db3da42ee019ad439d1e.jpg", + "image_caption": [ + "Figure 1: Learned pairwise potential matrix for Twitter POS tagging. " + ], + "image_footnote": [], + "bbox": [ + 210, + 330, + 813, + 607 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Figure 1 shows the learned pairwise potential matrix $W$ in Twitter POS tagging. We can see strong correlations between labels in neighborhoods. For example, an adjective (A) is more likely to be followed by a noun (N) than a verb (V) (see row labeled “A” in the figure). ", + "bbox": [ + 173, + 645, + 826, + 688 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "9.3 NAMED ENTITY RECOGNITION ", + "text_level": 1, + "bbox": [ + 176, + 704, + 429, + 718 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "For named entity recognition (NER), we perform experiments on the English data from the CoNLL 2003 shared task (Tjong Kim Sang & De Meulder, 2003). This task contains sentences annotated with named entities and their types, containing 14987 training sentences, 3466 in the development set, and 3684 in the test set. There are four named entity types: PERSON, LOCATION, ORGANIZATION, and MISC. We use the BIOES tagging scheme instead of the original BIO2, following prior work (Ratinov & Roth, 2009; Ma & Hovy, 2016). There are $L = 1 7$ classes. We use 100-dimensional pretrained GloVe (Pennington et al., 2014) embeddings trained on 6 billion words from Wikipedia and web text, which work better than other pretrained embeddings (Ma & Hovy, 2016). ", + "bbox": [ + 173, + 729, + 825, + 842 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Results are shown in Table 11. We see a large 4-point gap between the BLSTM and CRF, suggesting the importance of structured information for this problem. Though the SPEN still lags behind the CRF in F1, it matches the test-time speed of the BLSTM while improving F1 by 2 points. 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As in prior work, we relax", + "type": "text" + }, + { + "bbox": [ + 321, + 618, + 329, + 627 + ], + "score": 0.76, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "from discrete to continuous. For multi-label", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "classification, we use a feed-forward network that outputs a vector. We assign a single label to each", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "dimension of the vector, interpreting its value as the probability of predicting that label. 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This", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 140, + 285, + 471, + 300 + ], + "spans": [ + { + "bbox": [ + 140, + 285, + 471, + 300 + ], + "score": 1.0, + "content": "“inference network” outputs continuous values that we treat as the output structure.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 140, + 296, + 470, + 312 + ], + "spans": [ + { + "bbox": [ + 140, + 296, + 470, + 312 + ], + "score": 1.0, + "content": "We develop large-margin training criteria for joint training of the structured energy", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 307, + 469, + 322 + ], + "spans": [ + { + "bbox": [ + 141, + 307, + 469, + 322 + ], + "score": 1.0, + "content": "function and inference network. On multi-label classification we report speed-ups", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 318, + 470, + 333 + ], + "spans": [ + { + "bbox": [ + 141, + 318, + 154, + 333 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 155, + 319, + 185, + 330 + ], + "score": 0.35, + "content": "1 0 { - } 6 0 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 318, + 470, + 333 + ], + "score": 1.0, + "content": "compared to (Belanger et al., 2017) while also improving accuracy.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 330, + 469, + 344 + ], + "spans": [ + { + "bbox": [ + 141, + 330, + 469, + 344 + ], + "score": 1.0, + "content": "For sequence labeling with simple structured energies, our approach performs", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 342, + 469, + 353 + ], + "spans": [ + { + "bbox": [ + 141, + 342, + 469, + 353 + ], + "score": 1.0, + "content": "comparably to exact inference while being much faster at test time. We then", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 352, + 469, + 366 + ], + "spans": [ + { + "bbox": [ + 141, + 352, + 469, + 366 + ], + "score": 1.0, + "content": "demonstrate improved accuracy by augmenting the energy with a “label language", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 363, + 469, + 376 + ], + "spans": [ + { + "bbox": [ + 141, + 363, + 469, + 376 + ], + "score": 1.0, + "content": "model” that scores entire output label sequences, showing it can improve handling", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 374, + 469, + 387 + ], + "spans": [ + { + "bbox": [ + 141, + 374, + 469, + 387 + ], + "score": 1.0, + "content": "of long-distance dependencies in part-of-speech tagging. 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Given an input", + "type": "text" + }, + { + "bbox": [ + 356, + 474, + 364, + 483 + ], + "score": 0.72, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 473, + 454, + 485 + ], + "score": 1.0, + "content": ", the predicted output", + "type": "text" + }, + { + "bbox": [ + 455, + 473, + 462, + 484 + ], + "score": 0.85, + "content": "\\hat { \\textbf { \\textit { y } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "is chosen", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 483, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 249, + 497 + ], + "score": 1.0, + "content": "by minimizing an energy function", + "type": "text" + }, + { + "bbox": [ + 250, + 483, + 283, + 495 + ], + "score": 0.93, + "content": "E ( \\pmb { x } , \\hat { \\pmb { y } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 483, + 506, + 497 + ], + "score": 1.0, + "content": ". For structured prediction, the parameterization of the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 493, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 506, + 509 + ], + "score": 1.0, + "content": "energy function can leverage domain knowledge about the structured output space. 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We perform comparably to a conditional random field (CRF; Lafferty et al. 2001)", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 144, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 506, + 156 + ], + "score": 1.0, + "content": "when using the same energy function, with faster test-time inference. 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For a given input", + "type": "text" + }, + { + "bbox": [ + 320, + 250, + 348, + 261 + ], + "score": 0.9, + "content": "\\mathbf { \\boldsymbol { x } } \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 249, + 505, + 263 + ], + "score": 1.0, + "content": ", we denote the space of legal structured", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 260, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 152, + 274 + ], + "score": 1.0, + "content": "outputs by", + "type": "text" + }, + { + "bbox": [ + 152, + 261, + 175, + 273 + ], + "score": 0.92, + "content": "\\mathcal { V } ( \\pmb { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 260, + 393, + 274 + ], + "score": 1.0, + "content": ". 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This", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "score": 1.0, + "content": "approach performs backpropagation through each step of gradient descent. We compare to both", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 577, + 248, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 248, + 589 + ], + "score": 1.0, + "content": "methods in our experiments below.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32.5 + }, + { + "type": "title", + "bbox": [ + 107, + 603, + 312, + 616 + ], + "lines": [ + { + "bbox": [ + 105, + 602, + 312, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 312, + 618 + ], + "score": 1.0, + "content": "3 INFERENCE NETWORKS FOR SPENS", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 506, + 661 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 264, + 641 + ], + "score": 1.0, + "content": "Belanger & McCallum (2016) relaxed", + "type": "text" + }, + { + "bbox": [ + 265, + 630, + 272, + 639 + ], + "score": 0.8, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 627, + 506, + 641 + ], + "score": 1.0, + "content": "from a discrete to a continuous vector and used gradient", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 639, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 252, + 652 + ], + "score": 1.0, + "content": "descent for inference. 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This may require generalizing the energy", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 308, + 189 + ], + "score": 1.0, + "content": "function to be able to operate both on elements of", + "type": "text" + }, + { + "bbox": [ + 308, + 177, + 317, + 187 + ], + "score": 0.85, + "content": "\\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 176, + 335, + 189 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 335, + 176, + 349, + 187 + ], + "score": 0.88, + "content": "\\mathcal { V } _ { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 176, + 506, + 189 + ], + "score": 1.0, + "content": ". For MLC, no change is required. For", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 188, + 462, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 462, + 199 + ], + "score": 1.0, + "content": "sequence labeling, the change is straightforward and is described below in Section 7.2.1.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 108, + 214, + 420, + 227 + ], + "lines": [ + { + "bbox": [ + 105, + 213, + 420, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 420, + 229 + ], + "score": 1.0, + "content": "4 JOINT TRAINING OF SPENS AND INFERENCE NETWORKS", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 238, + 436, + 250 + ], + "lines": [ + { + "bbox": [ + 106, + 238, + 437, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 437, + 252 + ], + "score": 1.0, + "content": "Belanger & McCallum (2016) propose a structured hinge loss for training SPENs:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 253, + 439, + 286 + ], + "lines": [ + { + "bbox": [ + 170, + 253, + 439, + 286 + ], + "spans": [ + { + "bbox": [ + 170, + 253, + 439, + 286 + ], + "score": 0.92, + "content": "\\operatorname* { m i n } _ { \\Theta } \\sum _ { \\langle x _ { i } , y _ { i } \\rangle \\in \\mathcal { D } } \\left[ \\operatorname* { m a x } _ { y \\in \\mathcal { V } _ { R } ( \\pmb { x } ) } \\left( \\triangle ( \\pmb { y } , \\pmb { y } _ { i } ) - E _ { \\Theta } ( \\pmb { x } _ { i } , \\pmb { y } ) + E _ { \\Theta } ( \\pmb { x } _ { i } , \\pmb { y } _ { i } ) \\right) \\right] _ { + }", + "type": "interline_equation", + "image_path": "8b09b16075467b88bf5b8bf7af453f3340a24e7d9e02967171df4ed33283c98a.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 170, + 253, + 439, + 264.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 170, + 264.0, + 439, + 275.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 170, + 275.0, + 439, + 286.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 289, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 105, + 289, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 133, + 303 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 290, + 143, + 300 + ], + "score": 0.82, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 289, + 254, + 303 + ], + "score": 1.0, + "content": "is the set of training pairs,", + "type": "text" + }, + { + "bbox": [ + 254, + 290, + 329, + 302 + ], + "score": 0.92, + "content": "[ f ] _ { + } = \\operatorname* { m a x } ( 0 , f )", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 289, + 350, + 303 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 350, + 290, + 388, + 302 + ], + "score": 0.93, + "content": "\\triangle ( \\pmb { y } , \\pmb { y } ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 289, + 505, + 303 + ], + "score": 1.0, + "content": "is a structured cost function", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 364, + 312 + ], + "score": 1.0, + "content": "that returns a nonnegative value indicating the difference between", + "type": "text" + }, + { + "bbox": [ + 364, + 303, + 372, + 312 + ], + "score": 0.8, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 300, + 389, + 312 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 389, + 301, + 399, + 312 + ], + "score": 0.87, + "content": "\\boldsymbol { y } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 300, + 505, + 312 + ], + "score": 1.0, + "content": ". This loss is often referred", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 312, + 483, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 483, + 324 + ], + "score": 1.0, + "content": "to as “margin-rescaled” structured hinge loss (Taskar et al., 2004; Tsochantaridis et al., 2005).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 327, + 506, + 385 + ], + "lines": [ + { + "bbox": [ + 106, + 328, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 506, + 342 + ], + "score": 1.0, + "content": "However, this loss is expensive to minimize for structured models because of the “cost-augmented”", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 338, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 168, + 354 + ], + "score": 1.0, + "content": "inference step", + "type": "text" + }, + { + "bbox": [ + 168, + 340, + 227, + 353 + ], + "score": 0.9, + "content": "( \\operatorname* { m a x } _ { \\pmb { y } \\in \\mathcal { y } _ { R } ( \\pmb { x } ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 338, + 506, + 354 + ], + "score": 1.0, + "content": ". In prior work with SPENs, this step used gradient descent. We", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 332, + 365 + ], + "score": 1.0, + "content": "replace this with a cost-augmented inference network", + "type": "text" + }, + { + "bbox": [ + 333, + 351, + 362, + 363 + ], + "score": 0.92, + "content": "{ \\bf A } _ { \\Phi } ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 351, + 505, + 365 + ], + "score": 1.0, + "content": ". As suggested by the notation, the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 361, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 248, + 376 + ], + "score": 1.0, + "content": "cost-augmented inference network", + "type": "text" + }, + { + "bbox": [ + 248, + 363, + 264, + 374 + ], + "score": 0.88, + "content": "\\mathbf { A } _ { \\Phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 361, + 372, + 376 + ], + "score": 1.0, + "content": "and the inference network", + "type": "text" + }, + { + "bbox": [ + 372, + 363, + 389, + 374 + ], + "score": 0.89, + "content": "\\mathbf { A } _ { \\Psi }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 361, + 505, + 376 + ], + "score": 1.0, + "content": "will typically have the same", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 372, + 503, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 287, + 387 + ], + "score": 1.0, + "content": "functional form, but use different parameters", + "type": "text" + }, + { + "bbox": [ + 288, + 374, + 296, + 383 + ], + "score": 0.85, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 372, + 314, + 387 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 314, + 374, + 322, + 383 + ], + "score": 0.82, + "content": "\\Psi", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 372, + 503, + 387 + ], + "score": 1.0, + "content": ". We write our new optimization problem as:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 388, + 452, + 417 + ], + "lines": [ + { + "bbox": [ + 158, + 388, + 452, + 417 + ], + "spans": [ + { + "bbox": [ + 158, + 388, + 452, + 417 + ], + "score": 0.9, + "content": "\\operatorname* { m i n } _ { \\Theta } \\operatorname* { m a x } _ { \\Phi } \\sum _ { \\langle \\pmb { x } _ { i } , \\pmb { y } _ { i } \\rangle \\in \\mathcal { D } } \\left[ \\triangle ( \\mathbf { A } _ { \\Phi } ( \\pmb { x } _ { i } ) , \\pmb { y } _ { i } ) - E _ { \\Theta } ( \\pmb { x } _ { i } , \\mathbf { A } _ { \\Phi } ( \\pmb { x } _ { i } ) ) + E _ { \\Theta } ( \\pmb { x } _ { i } , \\pmb { y } _ { i } ) \\right] _ { + }", + "type": "interline_equation", + "image_path": "3cd9ffa2b7b4e03e6ad212e519e8fd303dc78c32edbda1b39426a6f957ea3fca.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 158, + 388, + 452, + 397.6666666666667 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 158, + 397.6666666666667, + 452, + 407.33333333333337 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 158, + 407.33333333333337, + 452, + 417.00000000000006 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 425, + 505, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 423, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 440 + ], + "score": 1.0, + "content": "We treat this optimization problem as a minimax game and find a saddle point for the game. Following", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 435, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 450 + ], + "score": 1.0, + "content": "Goodfellow et al. (2014), we implement this using an iterative numerical approach. We alternatively", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 446, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 144, + 462 + ], + "score": 1.0, + "content": "optimize", + "type": "text" + }, + { + "bbox": [ + 144, + 448, + 152, + 457 + ], + "score": 0.84, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 446, + 170, + 462 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 171, + 448, + 179, + 457 + ], + "score": 0.8, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 446, + 326, + 462 + ], + "score": 1.0, + "content": ", holding the other fixed. Optimizing", + "type": "text" + }, + { + "bbox": [ + 326, + 448, + 335, + 457 + ], + "score": 0.84, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 446, + 506, + 462 + ], + "score": 1.0, + "content": "to completion in the inner loop of training", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "score": 1.0, + "content": "is computationally prohibitive and may lead to overfitting. So we alternate between one mini-batch", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 469, + 483, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 166, + 483 + ], + "score": 1.0, + "content": "for optimizing", + "type": "text" + }, + { + "bbox": [ + 167, + 470, + 175, + 479 + ], + "score": 0.83, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 469, + 269, + 483 + ], + "score": 1.0, + "content": "and one for optimizing", + "type": "text" + }, + { + "bbox": [ + 270, + 470, + 279, + 480 + ], + "score": 0.7, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 469, + 334, + 483 + ], + "score": 1.0, + "content": ". We also add", + "type": "text" + }, + { + "bbox": [ + 334, + 470, + 346, + 480 + ], + "score": 0.88, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 469, + 444, + 483 + ], + "score": 1.0, + "content": "regularization terms for", + "type": "text" + }, + { + "bbox": [ + 444, + 470, + 453, + 479 + ], + "score": 0.82, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 469, + 471, + 483 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 471, + 470, + 479, + 479 + ], + "score": 0.84, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 469, + 483, + 483 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 486, + 343, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 344, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 344, + 499 + ], + "score": 1.0, + "content": "The objective for the cost-augmented inference network is:", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "interline_equation", + "bbox": [ + 169, + 501, + 441, + 522 + ], + "lines": [ + { + "bbox": [ + 169, + 501, + 441, + 522 + ], + "spans": [ + { + "bbox": [ + 169, + 501, + 441, + 522 + ], + "score": 0.91, + "content": "\\hat { \\Phi } \\underset { \\Phi } { \\mathrm { a r g m a x } } [ \\bigtriangleup ( \\mathbf { A } _ { \\Phi } ( \\boldsymbol { x } _ { i } ) , \\pmb { y } _ { i } ) - E _ { \\Theta } ( \\boldsymbol { x } _ { i } , \\mathbf { A } _ { \\Phi } ( \\boldsymbol { x } ) _ { i } ) + E _ { \\Theta } ( \\boldsymbol { x } _ { i } , \\pmb { y } _ { i } ) ] _ { + }", + "type": "interline_equation", + "image_path": "81b3451fee569d726e87a3eefef544d0a83f9edbf6416c0fbd76d4ca9b61468c.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 169, + 501, + 441, + 522 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 525, + 506, + 581 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 506, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 180, + 539 + ], + "score": 1.0, + "content": "That is, we update", + "type": "text" + }, + { + "bbox": [ + 180, + 526, + 189, + 536 + ], + "score": 0.84, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 525, + 218, + 539 + ], + "score": 1.0, + "content": "so that", + "type": "text" + }, + { + "bbox": [ + 218, + 526, + 234, + 537 + ], + "score": 0.91, + "content": "\\mathbf { A } _ { \\Phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 525, + 506, + 539 + ], + "score": 1.0, + "content": "yields an output that has low energy and high cost, in order to mimic", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 309, + 549 + ], + "score": 1.0, + "content": "cost-augmented inference. The energy parameters", + "type": "text" + }, + { + "bbox": [ + 309, + 537, + 318, + 546 + ], + "score": 0.83, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 537, + 505, + 549 + ], + "score": 1.0, + "content": "are kept fixed. 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In prior work with SPENs, this step used gradient descent. We", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 332, + 365 + ], + "score": 1.0, + "content": "replace this with a cost-augmented inference network", + "type": "text" + }, + { + "bbox": [ + 333, + 351, + 362, + 363 + ], + "score": 0.92, + "content": "{ \\bf A } _ { \\Phi } ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 351, + 505, + 365 + ], + "score": 1.0, + "content": ". 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We write our new optimization problem as:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 328, + 506, + 387 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 388, + 452, + 417 + ], + "lines": [ + { + "bbox": [ + 158, + 388, + 452, + 417 + ], + "spans": [ + { + "bbox": [ + 158, + 388, + 452, + 417 + ], + "score": 0.9, + "content": "\\operatorname* { m i n } _ { \\Theta } \\operatorname* { m a x } _ { \\Phi } \\sum _ { \\langle \\pmb { x } _ { i } , \\pmb { y } _ { i } \\rangle \\in \\mathcal { D } } \\left[ \\triangle ( \\mathbf { A } _ { \\Phi } ( \\pmb { x } _ { i } ) , \\pmb { y } _ { i } ) - E _ { \\Theta } ( \\pmb { x } _ { i } , \\mathbf { A } _ { \\Phi } ( \\pmb { x } _ { i } ) ) + E _ { \\Theta } ( \\pmb { x } _ { i } , \\pmb { y } _ { i } ) \\right] _ { + }", + "type": "interline_equation", + "image_path": "3cd9ffa2b7b4e03e6ad212e519e8fd303dc78c32edbda1b39426a6f957ea3fca.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 158, + 388, + 452, + 397.6666666666667 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 158, + 397.6666666666667, + 452, + 407.33333333333337 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 158, + 407.33333333333337, + 452, + 417.00000000000006 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 425, + 505, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 423, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 440 + ], + "score": 1.0, + "content": "We treat this optimization problem as a minimax game and find a saddle point for the game. Following", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 435, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 450 + ], + "score": 1.0, + "content": "Goodfellow et al. (2014), we implement this using an iterative numerical approach. We alternatively", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 446, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 144, + 462 + ], + "score": 1.0, + "content": "optimize", + "type": "text" + }, + { + "bbox": [ + 144, + 448, + 152, + 457 + ], + "score": 0.84, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 446, + 170, + 462 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 171, + 448, + 179, + 457 + ], + "score": 0.8, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 446, + 326, + 462 + ], + "score": 1.0, + "content": ", holding the other fixed. Optimizing", + "type": "text" + }, + { + "bbox": [ + 326, + 448, + 335, + 457 + ], + "score": 0.84, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 446, + 506, + 462 + ], + "score": 1.0, + "content": "to completion in the inner loop of training", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "score": 1.0, + "content": "is computationally prohibitive and may lead to overfitting. So we alternate between one mini-batch", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 469, + 483, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 166, + 483 + ], + "score": 1.0, + "content": "for optimizing", + "type": "text" + }, + { + "bbox": [ + 167, + 470, + 175, + 479 + ], + "score": 0.83, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 469, + 269, + 483 + ], + "score": 1.0, + "content": "and one for optimizing", + "type": "text" + }, + { + "bbox": [ + 270, + 470, + 279, + 480 + ], + "score": 0.7, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 469, + 334, + 483 + ], + "score": 1.0, + "content": ". We also add", + "type": "text" + }, + { + "bbox": [ + 334, + 470, + 346, + 480 + ], + "score": 0.88, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 469, + 444, + 483 + ], + "score": 1.0, + "content": "regularization terms for", + "type": "text" + }, + { + "bbox": [ + 444, + 470, + 453, + 479 + ], + "score": 0.82, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 469, + 471, + 483 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 471, + 470, + 479, + 479 + ], + "score": 0.84, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 469, + 483, + 483 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 423, + 506, + 483 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 486, + 343, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 344, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 344, + 499 + ], + "score": 1.0, + "content": "The objective for the cost-augmented inference network is:", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 486, + 344, + 499 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 169, + 501, + 441, + 522 + ], + "lines": [ + { + "bbox": [ + 169, + 501, + 441, + 522 + ], + "spans": [ + { + "bbox": [ + 169, + 501, + 441, + 522 + ], + "score": 0.91, + "content": "\\hat { \\Phi } \\underset { \\Phi } { \\mathrm { a r g m a x } } [ \\bigtriangleup ( \\mathbf { A } _ { \\Phi } ( \\boldsymbol { x } _ { i } ) , \\pmb { y } _ { i } ) - E _ { \\Theta } ( \\boldsymbol { x } _ { i } , \\mathbf { A } _ { \\Phi } ( \\boldsymbol { x } ) _ { i } ) + E _ { \\Theta } ( \\boldsymbol { x } _ { i } , \\pmb { y } _ { i } ) ] _ { + }", + "type": "interline_equation", + "image_path": "81b3451fee569d726e87a3eefef544d0a83f9edbf6416c0fbd76d4ca9b61468c.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 169, + 501, + 441, + 522 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 525, + 506, + 581 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 506, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 180, + 539 + ], + "score": 1.0, + "content": "That is, we update", + "type": "text" + }, + { + "bbox": [ + 180, + 526, + 189, + 536 + ], + "score": 0.84, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 525, + 218, + 539 + ], + "score": 1.0, + "content": "so that", + "type": "text" + }, + { + "bbox": [ + 218, + 526, + 234, + 537 + ], + "score": 0.91, + "content": "\\mathbf { A } _ { \\Phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 525, + 506, + 539 + ], + "score": 1.0, + "content": "yields an output that has low energy and high cost, in order to mimic", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 309, + 549 + ], + "score": 1.0, + "content": "cost-augmented inference. 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(4). However, training only", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 507, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 285, + 734 + ], + "score": 1.0, + "content": "gives us a cost-augmented inference network", + "type": "text" + }, + { + "bbox": [ + 286, + 721, + 302, + 732 + ], + "score": 0.9, + "content": "\\mathbf { A } _ { \\Phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 720, + 330, + 734 + ], + "score": 1.0, + "content": ". 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Similar", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 504, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 518 + ], + "score": 1.0, + "content": "observations have been noted about other alternative optimization settings, especially those underlying", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "generative adversarial networks (Salimans et al., 2016). Below we describe several techniques we", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 528, + 467, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 467, + 540 + ], + "score": 1.0, + "content": "found to help stabilize training, which are optional terms added to the objective in Eq. 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We only experiment with this loss for sequence labeling.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 701, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 699, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 119, + 699, + 505, + 714 + ], + "score": 1.0, + "content": "1For MLC, encouraging lower entropy distributions worked better, while for sequence labeling, higher", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 712, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 712, + 505, + 723 + ], + "score": 1.0, + "content": "entropy was better, similar to the effect found by Pereyra et al. (2017). Further research is required to gain", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 438, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 438, + 733 + ], + "score": 1.0, + "content": "understanding of the role of entropy regularization in such alternating optimization settings.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 127 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 82, + 506, + 128 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 108, + 140, + 276, + 151 + ], + "lines": [ + { + "bbox": [ + 105, + 138, + 278, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 278, + 153 + ], + "score": 1.0, + "content": "4.2 VARIATIONS AND SPECIAL CASES", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 160, + 506, + 216 + ], + "lines": [ + { + "bbox": [ + 106, + 161, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 506, + 173 + ], + "score": 1.0, + "content": "This approach also permits us to use large-margin structured prediction with slack rescaling (Tsochan-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 171, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 506, + 185 + ], + "score": 1.0, + "content": "taridis et al., 2005). Slack rescaling can yield higher accuracies than margin rescaling, but requires", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 181, + 507, + 195 + ], + "spans": [ + { + "bbox": [ + 104, + 181, + 507, + 195 + ], + "score": 1.0, + "content": "“cost-scaled” inference during training which is intractable for many classes of output structures.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 192, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 506, + 206 + ], + "score": 1.0, + "content": "However, we can use our notion of inference networks to circumvent this tractability issue and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "approximately optimize the slack-rescaled hinge loss, yielding the following optimization problem:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 104, + 161, + 507, + 217 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 218, + 455, + 247 + ], + "lines": [ + { + "bbox": [ + 155, + 218, + 455, + 247 + ], + "spans": [ + { + "bbox": [ + 155, + 218, + 455, + 247 + ], + "score": 0.93, + "content": "\\operatorname* { m i n } _ { \\Theta } \\operatorname* { m a x } _ { \\Phi } \\sum _ { \\langle \\mathbf { x } _ { i } , \\mathbf { y } _ { i } \\rangle \\in \\mathcal { D } } \\bigtriangleup ( \\mathbf { A } _ { \\Phi } ( \\mathbf { x } _ { i } ) , \\mathbf { y } _ { i } ) [ 1 - E _ { \\Theta } ( \\mathbf { x } _ { i } , \\mathbf { A } _ { \\Phi } ( \\mathbf { x } _ { i } ) ) + E _ { \\Theta } ( \\mathbf { x } _ { i } , \\mathbf { y } _ { i } ) ] _ { + }", + "type": "interline_equation", + "image_path": "c38a7125f1caf5dee8eb0dcc6338c0a0ca5a1baa5f9921e518f50e97c7405cd5.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 155, + 218, + 455, + 227.66666666666666 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 155, + 227.66666666666666, + 455, + 237.33333333333331 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 155, + 237.33333333333331, + 455, + 246.99999999999997 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 109, + 250, + 505, + 262 + ], + "lines": [ + { + "bbox": [ + 107, + 249, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 107, + 249, + 467, + 263 + ], + "score": 1.0, + "content": "Using the same argument as above, we can also break this into alternating optimization of", + "type": "text" + }, + { + "bbox": [ + 467, + 250, + 475, + 260 + ], + "score": 0.84, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 249, + 493, + 263 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 494, + 250, + 502, + 260 + ], + "score": 0.82, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 249, + 506, + 263 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 107, + 249, + 506, + 263 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 266, + 505, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 266, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 280 + ], + "score": 1.0, + "content": "We can optimize a structured perceptron (Collins, 2002) version by using the margin-rescaled hinge", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 206, + 291 + ], + "score": 1.0, + "content": "loss (Eq. 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Similar", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 504, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 518 + ], + "score": 1.0, + "content": "observations have been noted about other alternative optimization settings, especially those underlying", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "generative adversarial networks (Salimans et al., 2016). Below we describe several techniques we", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 528, + 467, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 467, + 540 + ], + "score": 1.0, + "content": "found to help stabilize training, which are optional terms added to the objective in Eq. (8).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 494, + 506, + 540 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 543, + 493, + 556 + ], + "lines": [ + { + "bbox": [ + 107, + 542, + 492, + 558 + ], + "spans": [ + { + "bbox": [ + 107, + 545, + 119, + 555 + ], + "score": 0.83, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 542, + 221, + 558 + ], + "score": 1.0, + "content": "Regularization: We use", + "type": "text" + }, + { + "bbox": [ + 221, + 545, + 233, + 555 + ], + "score": 0.88, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 542, + 392, + 558 + ], + "score": 1.0, + "content": "regularization, adding the penalty term", + "type": "text" + }, + { + "bbox": [ + 392, + 543, + 415, + 556 + ], + "score": 0.92, + "content": "\\| \\Phi \\| _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 542, + 480, + 558 + ], + "score": 1.0, + "content": "with coefficient", + "type": "text" + }, + { + "bbox": [ + 480, + 544, + 492, + 555 + ], + "score": 0.87, + "content": "\\lambda _ { 1 }", + "type": "inline_equation" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 107, + 542, + 492, + 558 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 560, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 361, + 574 + ], + "score": 1.0, + "content": "Entropy Regularization: We add an entropy-based regularizer", + "type": "text" + }, + { + "bbox": [ + 361, + 560, + 421, + 573 + ], + "score": 0.92, + "content": "\\mathrm { l o s s } _ { \\mathrm { H } } ( \\mathbf { A } _ { \\Phi } ( \\pmb { x } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "defined for the prob-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 303, + 585 + ], + "score": 1.0, + "content": "lem under consideration. For MLC, the output of", + "type": "text" + }, + { + "bbox": [ + 304, + 572, + 334, + 584 + ], + "score": 0.93, + "content": "\\mathbf { A } _ { \\Phi } ( \\pmb { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "is a vector of scalars in [0, 1], one for each", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 437, + 596 + ], + "score": 1.0, + "content": "label, where the scalar is interpreted as a label probability. The entropy regularizer", + "type": "text" + }, + { + "bbox": [ + 438, + 584, + 461, + 594 + ], + "score": 0.37, + "content": "\\mathrm { l o s s } _ { \\mathrm { H } }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "is the sum", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 594, + 504, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 496, + 607 + ], + "score": 1.0, + "content": "of the entropies over these label binary distributions. For sequence labeling, where the length of", + "type": "text" + }, + { + "bbox": [ + 496, + 596, + 504, + 604 + ], + "score": 0.76, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 604, + 504, + 618 + ], + "spans": [ + { + "bbox": [ + 104, + 604, + 115, + 618 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 116, + 605, + 126, + 615 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 604, + 208, + 618 + ], + "score": 1.0, + "content": "and where there are", + "type": "text" + }, + { + "bbox": [ + 209, + 605, + 217, + 615 + ], + "score": 0.79, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 604, + 329, + 618 + ], + "score": 1.0, + "content": "unique labels, the output of", + "type": "text" + }, + { + "bbox": [ + 330, + 605, + 360, + 617 + ], + "score": 0.93, + "content": "{ \\bf A } _ { \\Phi } ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 604, + 406, + 618 + ], + "score": 1.0, + "content": "is a length-", + "type": "text" + }, + { + "bbox": [ + 406, + 605, + 416, + 615 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 604, + 496, + 618 + ], + "score": 1.0, + "content": "sequence of length-", + "type": "text" + }, + { + "bbox": [ + 496, + 606, + 504, + 615 + ], + "score": 0.8, + "content": "L", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 615, + 504, + 630 + ], + "spans": [ + { + "bbox": [ + 104, + 615, + 336, + 630 + ], + "score": 1.0, + "content": "vectors, each of which represents the distribution over the", + "type": "text" + }, + { + "bbox": [ + 337, + 617, + 345, + 626 + ], + "score": 0.79, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 615, + 443, + 630 + ], + "score": 1.0, + "content": "labels at that position in", + "type": "text" + }, + { + "bbox": [ + 443, + 618, + 451, + 626 + ], + "score": 0.66, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 615, + 480, + 630 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + }, + { + "bbox": [ + 480, + 617, + 504, + 627 + ], + "score": 0.85, + "content": "\\mathrm { l o s s } _ { \\mathrm { H } }", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 627, + 439, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 439, + 640 + ], + "score": 1.0, + "content": "is the sum of entropies of these label distributions across positions in the sequence.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38, + "bbox_fs": [ + 104, + 560, + 506, + 640 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 503, + 666 + ], + "lines": [ + { + "bbox": [ + 106, + 643, + 504, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 221, + 656 + ], + "score": 1.0, + "content": "When tuning the coefficient", + "type": "text" + }, + { + "bbox": [ + 222, + 644, + 233, + 655 + ], + "score": 0.87, + "content": "\\lambda _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 643, + 504, + 656 + ], + "score": 1.0, + "content": "for this regularizer, we consider both positive and negative values,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 654, + 437, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 437, + 667 + ], + "score": 1.0, + "content": "permitting us to favor either low- or high-entropy distributions as the task prefers.1", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 643, + 504, + 667 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 672, + 504, + 694 + ], + "lines": [ + { + "bbox": [ + 105, + 670, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 422, + 686 + ], + "score": 1.0, + "content": "Local Cross Entropy Loss: We add a local (non-structured) cross entropy", + "type": "text" + }, + { + "bbox": [ + 422, + 671, + 505, + 684 + ], + "score": 0.9, + "content": "\\mathrm { l o s s } _ { \\mathrm { C E } } ( \\mathbf { A } _ { \\Phi } ( \\pmb { x } _ { i } ) , \\pmb { y } _ { i } )", + "type": "inline_equation" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 680, + 507, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 507, + 698 + ], + "score": 1.0, + "content": "defined for the problem under consideration. We only experiment with this loss for sequence labeling.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 670, + 507, + 698 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "It is the sum of the label cross entropy losses over all positions in the sequence. This loss provides", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "more explicit feedback to the inference network, helping the optimization procedure to find a solution", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "that minimizes the energy function while also correctly classifying individual labels. It can also be", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 321, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 321, + 127 + ], + "score": 1.0, + "content": "viewed as a multi-task loss for the inference network.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 105, + 132, + 504, + 155 + ], + "lines": [ + { + "bbox": [ + 106, + 131, + 504, + 146 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 418, + 146 + ], + "score": 1.0, + "content": "Regularization Toward Pretrained Inference Network: We add the penalty", + "type": "text" + }, + { + "bbox": [ + 419, + 132, + 464, + 144 + ], + "score": 0.94, + "content": "\\lVert \\Phi - \\Phi _ { 0 } \\rVert _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 131, + 491, + 146 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 491, + 132, + 504, + 144 + ], + "score": 0.88, + "content": "\\Phi _ { 0 }", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 479, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 468, + 157 + ], + "score": 1.0, + "content": "is a pretrained network, e.g., a local classifier trained to independently predict each part of", + "type": "text" + }, + { + "bbox": [ + 468, + 146, + 475, + 155 + ], + "score": 0.81, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 142, + 479, + 157 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 160, + 413, + 172 + ], + "lines": [ + { + "bbox": [ + 105, + 158, + 413, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 413, + 174 + ], + "score": 1.0, + "content": "Each additional term has its own tunable hyperparameter. Finally we obtain:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 197, + 466, + 237 + ], + "lines": [ + { + "bbox": [ + 145, + 197, + 466, + 237 + ], + "spans": [ + { + "bbox": [ + 145, + 197, + 466, + 237 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\hat { \\Phi } \\underset { \\Phi } { \\operatorname { a r g m a x } } \\ [ \\triangle ( \\mathbf { A } _ { \\Phi } ( \\boldsymbol { x } _ { i } ) , \\boldsymbol { y } _ { i } ) - E _ { \\Theta } ( \\boldsymbol { x } _ { i } , \\mathbf { A } _ { \\Phi } ( \\boldsymbol { x } _ { i } ) ) + E _ { \\Theta } ( \\boldsymbol { x } _ { i } , \\boldsymbol { y } _ { i } ) ] _ { + } - \\lambda _ { 1 } \\| \\Phi \\| _ { 2 } ^ { 2 } } \\\\ { + \\lambda _ { 2 } \\mathrm { l o s s } _ { \\mathrm { H } } ( \\mathbf { A } _ { \\Phi } ( \\boldsymbol { x } _ { i } ) ) - \\lambda _ { 3 } \\mathrm { l o s s } _ { \\mathrm { C E } } ( \\mathbf { A } _ { \\Phi } ( \\boldsymbol { x } _ { i } ) , \\boldsymbol { y } _ { i } ) - \\lambda _ { 4 } \\| \\Phi - \\Phi _ { 0 } \\| _ { 2 } ^ { 2 } } \\end{array}", + "type": "interline_equation", + "image_path": "fc64f56ebe399567e7f1d7d37808748ed8b7c86a56715673a34127347f2f0f18.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 145, + 197, + 466, + 210.33333333333334 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 145, + 210.33333333333334, + 466, + 223.66666666666669 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 145, + 223.66666666666669, + 466, + 237.00000000000003 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 273, + 211, + 286 + ], + "lines": [ + { + "bbox": [ + 105, + 272, + 213, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 213, + 288 + ], + "score": 1.0, + "content": "6 RELATED WORK", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 313, + 505, + 390 + ], + "lines": [ + { + "bbox": [ + 105, + 312, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 326 + ], + "score": 1.0, + "content": "Our methods are reminiscent of other alternating optimization problems like that underlying generative", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "score": 1.0, + "content": "adversarial networks (GANs; Goodfellow et al. 2014). GANs are based on a minimax game and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 334, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 316, + 348 + ], + "score": 1.0, + "content": "have a value function that one agent (a discriminator", + "type": "text" + }, + { + "bbox": [ + 317, + 335, + 326, + 345 + ], + "score": 0.75, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 334, + 506, + 348 + ], + "score": 1.0, + "content": ") seeks to maximize and another (a generator", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 345, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 116, + 356 + ], + "score": 0.56, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 345, + 505, + 359 + ], + "score": 1.0, + "content": ") seeks to minimize. By their analysis, a log loss discriminator converges to a degenerate uniform", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 369 + ], + "score": 1.0, + "content": "solution. When using hinge loss, we can get a non-degenerate discriminator while matching the data", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "score": 1.0, + "content": "distribution (Dai et al., 2017; Zhao et al., 2016). Our formulation is closer to this hinge loss version", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 380, + 158, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 158, + 390 + ], + "score": 1.0, + "content": "of the GAN.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 395, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 106, + 396, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 506, + 408 + ], + "score": 1.0, + "content": "Our approach is also related to knowledge distillation (Ba & Caruana, 2014; Hinton et al., 2015),", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 407, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 419 + ], + "score": 1.0, + "content": "which refers to strategies in which one model (a “student”) is trained to mimic another (a “teacher”).", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 418, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 506, + 430 + ], + "score": 1.0, + "content": "Typically, the teacher is a larger, more accurate model but which is too computationally expensive to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 429, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 441 + ], + "score": 1.0, + "content": "use at test time. Urban et al. (2016) train shallow networks using image classification data labeled", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "score": 1.0, + "content": "by an ensemble of deep teacher nets. Geras et al. (2016) train a convolutional network to mimic an", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 450, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 464 + ], + "score": 1.0, + "content": "LSTM for speech recognition. Others have explored knowledge distillation for sequence-to-sequence", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 462, + 368, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 368, + 474 + ], + "score": 1.0, + "content": "learning (Kim & Rush, 2016) and parsing (Kuncoro et al., 2016).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 478, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "Since we train a single inference network for an entire dataset, our approach is also related to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 489, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 501 + ], + "score": 1.0, + "content": "“amortized inference” (Srikumar et al., 2012; Gershman & Goodman, 2014; Paige & Wood, 2016;", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "Chang et al., 2015). Such methods precompute or save solutions to subproblems for faster overall", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "computation. Our inference networks likely devote more modeling capacity to the most frequent", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "substructures in the data. A kind of inference network is used in variational autoencoders (Kingma &", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 532, + 395, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 395, + 546 + ], + "score": 1.0, + "content": "Welling, 2013) to approximate posterior inference in generative models.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 106, + 550, + 506, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "Our methods are also related to work in structured prediction that seeks to approximate structured", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "models with factorized ones, e.g., mean-field approximations in graphical models (Koller & Friedman,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "2009; Krähenbühl & Koltun, 2011). Like our use of inference networks, there have been efforts in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 582, + 507, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 507, + 596 + ], + "score": 1.0, + "content": "designing differentiable approximations of combinatorial search procedures (Martins & Kreutzer,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 593, + 507, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 507, + 607 + ], + "score": 1.0, + "content": "2017; Goyal et al., 2018) and structured losses for training with them (Wiseman & Rush, 2016).", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "Since we relax discrete output variables to be continuous, there is also a connection to recent work", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "that focuses on structured prediction with continuous valued output variables (Wang et al., 2016).", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "score": 1.0, + "content": "They also propose a formulation that yields an alternating optimization problem, but it is based on", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 639, + 183, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 183, + 649 + ], + "score": 1.0, + "content": "proximal methods.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "There are other settings in which gradient descent is used for inference, e.g., image generation", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "applications like DeepDream (Mordvintsev et al., 2015) and neural style transfer (Gatys et al., 2015),", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "as well as machine translation (Hoang et al., 2017). In these and related settings, gradient descent", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "has started to be replaced by inference networks, especially for image transformation tasks (Johnson", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "et al., 2016; Li & Wand, 2016). Our results below provide more evidence for making this transition.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "An alternative to what we pursue here would be to obtain an easier convex optimization problem for", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 721, + 362, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 362, + 732 + ], + "score": 1.0, + "content": "inference via input convex neural networks (Amos et al., 2017).", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "It is the sum of the label cross entropy losses over all positions in the sequence. This loss provides", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "more explicit feedback to the inference network, helping the optimization procedure to find a solution", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "that minimizes the energy function while also correctly classifying individual labels. It can also be", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 321, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 321, + 127 + ], + "score": 1.0, + "content": "viewed as a multi-task loss for the inference network.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 83, + 506, + 127 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 132, + 504, + 155 + ], + "lines": [ + { + "bbox": [ + 106, + 131, + 504, + 146 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 418, + 146 + ], + "score": 1.0, + "content": "Regularization Toward Pretrained Inference Network: We add the penalty", + "type": "text" + }, + { + "bbox": [ + 419, + 132, + 464, + 144 + ], + "score": 0.94, + "content": "\\lVert \\Phi - \\Phi _ { 0 } \\rVert _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 131, + 491, + 146 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 491, + 132, + 504, + 144 + ], + "score": 0.88, + "content": "\\Phi _ { 0 }", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 479, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 468, + 157 + ], + "score": 1.0, + "content": "is a pretrained network, e.g., a local classifier trained to independently predict each part of", + "type": "text" + }, + { + "bbox": [ + 468, + 146, + 475, + 155 + ], + "score": 0.81, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 142, + 479, + 157 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 131, + 504, + 157 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 160, + 413, + 172 + ], + "lines": [ + { + "bbox": [ + 105, + 158, + 413, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 413, + 174 + ], + "score": 1.0, + "content": "Each additional term has its own tunable hyperparameter. Finally we obtain:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 158, + 413, + 174 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 197, + 466, + 237 + ], + "lines": [ + { + "bbox": [ + 145, + 197, + 466, + 237 + ], + "spans": [ + { + "bbox": [ + 145, + 197, + 466, + 237 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\hat { \\Phi } \\underset { \\Phi } { \\operatorname { a r g m a x } } \\ [ \\triangle ( \\mathbf { A } _ { \\Phi } ( \\boldsymbol { x } _ { i } ) , \\boldsymbol { y } _ { i } ) - E _ { \\Theta } ( \\boldsymbol { x } _ { i } , \\mathbf { A } _ { \\Phi } ( \\boldsymbol { x } _ { i } ) ) + E _ { \\Theta } ( \\boldsymbol { x } _ { i } , \\boldsymbol { y } _ { i } ) ] _ { + } - \\lambda _ { 1 } \\| \\Phi \\| _ { 2 } ^ { 2 } } \\\\ { + \\lambda _ { 2 } \\mathrm { l o s s } _ { \\mathrm { H } } ( \\mathbf { A } _ { \\Phi } ( \\boldsymbol { x } _ { i } ) ) - \\lambda _ { 3 } \\mathrm { l o s s } _ { \\mathrm { C E } } ( \\mathbf { A } _ { \\Phi } ( \\boldsymbol { x } _ { i } ) , \\boldsymbol { y } _ { i } ) - \\lambda _ { 4 } \\| \\Phi - \\Phi _ { 0 } \\| _ { 2 } ^ { 2 } } \\end{array}", + "type": "interline_equation", + "image_path": "fc64f56ebe399567e7f1d7d37808748ed8b7c86a56715673a34127347f2f0f18.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 145, + 197, + 466, + 210.33333333333334 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 145, + 210.33333333333334, + 466, + 223.66666666666669 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 145, + 223.66666666666669, + 466, + 237.00000000000003 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 273, + 211, + 286 + ], + "lines": [ + { + "bbox": [ + 105, + 272, + 213, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 213, + 288 + ], + "score": 1.0, + "content": "6 RELATED WORK", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 313, + 505, + 390 + ], + "lines": [ + { + "bbox": [ + 105, + 312, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 326 + ], + "score": 1.0, + "content": "Our methods are reminiscent of other alternating optimization problems like that underlying generative", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "score": 1.0, + "content": "adversarial networks (GANs; Goodfellow et al. 2014). GANs are based on a minimax game and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 334, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 316, + 348 + ], + "score": 1.0, + "content": "have a value function that one agent (a discriminator", + "type": "text" + }, + { + "bbox": [ + 317, + 335, + 326, + 345 + ], + "score": 0.75, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 334, + 506, + 348 + ], + "score": 1.0, + "content": ") seeks to maximize and another (a generator", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 345, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 116, + 356 + ], + "score": 0.56, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 345, + 505, + 359 + ], + "score": 1.0, + "content": ") seeks to minimize. By their analysis, a log loss discriminator converges to a degenerate uniform", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 369 + ], + "score": 1.0, + "content": "solution. When using hinge loss, we can get a non-degenerate discriminator while matching the data", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "score": 1.0, + "content": "distribution (Dai et al., 2017; Zhao et al., 2016). Our formulation is closer to this hinge loss version", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 380, + 158, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 158, + 390 + ], + "score": 1.0, + "content": "of the GAN.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 312, + 506, + 390 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 395, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 106, + 396, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 506, + 408 + ], + "score": 1.0, + "content": "Our approach is also related to knowledge distillation (Ba & Caruana, 2014; Hinton et al., 2015),", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 407, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 419 + ], + "score": 1.0, + "content": "which refers to strategies in which one model (a “student”) is trained to mimic another (a “teacher”).", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 418, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 506, + 430 + ], + "score": 1.0, + "content": "Typically, the teacher is a larger, more accurate model but which is too computationally expensive to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 429, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 441 + ], + "score": 1.0, + "content": "use at test time. Urban et al. (2016) train shallow networks using image classification data labeled", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "score": 1.0, + "content": "by an ensemble of deep teacher nets. Geras et al. (2016) train a convolutional network to mimic an", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 450, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 464 + ], + "score": 1.0, + "content": "LSTM for speech recognition. Others have explored knowledge distillation for sequence-to-sequence", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 462, + 368, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 368, + 474 + ], + "score": 1.0, + "content": "learning (Kim & Rush, 2016) and parsing (Kuncoro et al., 2016).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 396, + 506, + 474 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 478, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "Since we train a single inference network for an entire dataset, our approach is also related to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 489, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 501 + ], + "score": 1.0, + "content": "“amortized inference” (Srikumar et al., 2012; Gershman & Goodman, 2014; Paige & Wood, 2016;", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "Chang et al., 2015). Such methods precompute or save solutions to subproblems for faster overall", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "computation. Our inference networks likely devote more modeling capacity to the most frequent", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "substructures in the data. A kind of inference network is used in variational autoencoders (Kingma &", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 532, + 395, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 395, + 546 + ], + "score": 1.0, + "content": "Welling, 2013) to approximate posterior inference in generative models.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 478, + 506, + 546 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 550, + 506, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "Our methods are also related to work in structured prediction that seeks to approximate structured", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "models with factorized ones, e.g., mean-field approximations in graphical models (Koller & Friedman,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "2009; Krähenbühl & Koltun, 2011). Like our use of inference networks, there have been efforts in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 582, + 507, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 507, + 596 + ], + "score": 1.0, + "content": "designing differentiable approximations of combinatorial search procedures (Martins & Kreutzer,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 593, + 507, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 507, + 607 + ], + "score": 1.0, + "content": "2017; Goyal et al., 2018) and structured losses for training with them (Wiseman & Rush, 2016).", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "Since we relax discrete output variables to be continuous, there is also a connection to recent work", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "that focuses on structured prediction with continuous valued output variables (Wang et al., 2016).", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "score": 1.0, + "content": "They also propose a formulation that yields an alternating optimization problem, but it is based on", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 639, + 183, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 183, + 649 + ], + "score": 1.0, + "content": "proximal methods.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 550, + 507, + 649 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "There are other settings in which gradient descent is used for inference, e.g., image generation", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "applications like DeepDream (Mordvintsev et al., 2015) and neural style transfer (Gatys et al., 2015),", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "as well as machine translation (Hoang et al., 2017). In these and related settings, gradient descent", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "has started to be replaced by inference networks, especially for image transformation tasks (Johnson", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "et al., 2016; Li & Wand, 2016). Our results below provide more evidence for making this transition.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "An alternative to what we pursue here would be to obtain an easier convex optimization problem for", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 721, + 362, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 362, + 732 + ], + "score": 1.0, + "content": "inference via input convex neural networks (Amos et al., 2017).", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 654, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 190, + 104, + 421, + 158 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 142, + 89, + 467, + 101 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 142, + 88, + 469, + 102 + ], + "spans": [ + { + "bbox": [ + 142, + 88, + 469, + 102 + ], + "score": 1.0, + "content": "Table 1: Test F1 when comparing methods on multi-label classification datasets.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 190, + 104, + 421, + 158 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 190, + 104, + 421, + 158 + ], + "spans": [ + { + "bbox": [ + 190, + 104, + 421, + 158 + ], + "score": 0.927, + "html": "
BibtexBookmarksDeliciousavg.
MLP38.933.837.836.8
SPEN (BM16)42.234.437.538.0
SPEN (E2E)38.133.934.435.5
SPEN (InfNet)42.237.637.539.1
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We compare", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "score": 1.0, + "content": "accuracy and speed, finding our approach to outperform prior work. We then perform experiments", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 230, + 269, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 269, + 241 + ], + "score": 1.0, + "content": "with sequence labeling tasks in Sec. 7.2.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 108, + 255, + 267, + 267 + ], + "lines": [ + { + "bbox": [ + 105, + 254, + 269, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 269, + 268 + ], + "score": 1.0, + "content": "7.1 MULTI-LABEL CLASSIFICATION", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 276, + 506, + 343 + ], + "lines": [ + { + "bbox": [ + 106, + 277, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 506, + 288 + ], + "score": 1.0, + "content": "We use the MLC datasets used by Belanger & McCallum (2016): Bibtex, Delicious, and Bookmarks.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 286, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 300 + ], + "score": 1.0, + "content": "Dataset statistics are shown in Table 7 in the Appendix. For Bibtex and Delicious, we follow Belanger", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 299, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 505, + 310 + ], + "score": 1.0, + "content": "and McCallum and tune the hyperparameters using a different sampling of train and test data, then use", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 310, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 506, + 321 + ], + "score": 1.0, + "content": "the standard train/test split for final experimentation using the tuned hyperparameters. For Bookmarks,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "we use the same train/dev/test split as (Belanger & McCallum, 2016). For evaluation, we report the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 331, + 304, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 304, + 344 + ], + "score": 1.0, + "content": "example averaged (macro averaged) F1 measure.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 348, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 347, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 506, + 361 + ], + "score": 1.0, + "content": "We use the SPEN for MLC described in Section 2 and also used by Belanger & McCallum (2016). For", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 358, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 241, + 372 + ], + "score": 1.0, + "content": "the feature representation network", + "type": "text" + }, + { + "bbox": [ + 242, + 359, + 264, + 371 + ], + "score": 0.91, + "content": "F ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 358, + 506, + 372 + ], + "score": 1.0, + "content": ", we use feed-forward networks with two hidden layers, using", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "score": 1.0, + "content": "their same layer widths: 150 for Bibtex/Bookmarks and 250 for Delicious. We pretrain the feature", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 380, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 104, + 380, + 145, + 394 + ], + "score": 1.0, + "content": "networks", + "type": "text" + }, + { + "bbox": [ + 146, + 381, + 168, + 393 + ], + "score": 0.92, + "content": "F ( { \\dot { \\mathbf { x } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 380, + 506, + 394 + ], + "score": 1.0, + "content": "by minimizing independent-label cross entropy for 10 epochs using Adam (Kingma", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "& Ba, 2014) with learning rate 0.001. While training SPENs, we only update the parameters of the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 173, + 416 + ], + "score": 1.0, + "content": "energy function", + "type": "text" + }, + { + "bbox": [ + 174, + 403, + 189, + 414 + ], + "score": 0.69, + "content": "( \\Theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 403, + 299, + 416 + ], + "score": 1.0, + "content": "and the inference network", + "type": "text" + }, + { + "bbox": [ + 299, + 403, + 314, + 414 + ], + "score": 0.73, + "content": "( \\Phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 403, + 482, + 416 + ], + "score": 1.0, + "content": ", keeping the feature network parameters", + "type": "text" + }, + { + "bbox": [ + 482, + 403, + 505, + 415 + ], + "score": 0.92, + "content": "F ( { \\pmb x } )", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 414, + 358, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 318, + 426 + ], + "score": 1.0, + "content": "fixed. We use Adam with learning rate 0.001 to train", + "type": "text" + }, + { + "bbox": [ + 319, + 415, + 328, + 424 + ], + "score": 0.81, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 414, + 345, + 426 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 346, + 415, + 354, + 424 + ], + "score": 0.82, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 414, + 358, + 426 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 431, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 106, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "The inference networks are feed-forward networks with two hidden layers, using the same archi-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 441, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 238, + 455 + ], + "score": 1.0, + "content": "tectures as the feature networks", + "type": "text" + }, + { + "bbox": [ + 238, + 442, + 261, + 453 + ], + "score": 0.92, + "content": "F ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 441, + 505, + 455 + ], + "score": 1.0, + "content": ". This permits us to initialize inference network parameters", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 452, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 115, + 463 + ], + "score": 0.79, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 452, + 505, + 466 + ], + "score": 1.0, + "content": "using pretrained feature network parameters. For the output, we use an affine transformation", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 426, + 477 + ], + "score": 1.0, + "content": "layer with a sigmoid nonlinearity function, so the output values are in the range", + "type": "text" + }, + { + "bbox": [ + 426, + 464, + 448, + 475 + ], + "score": 0.89, + "content": "( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 464, + 505, + 477 + ], + "score": 1.0, + "content": ". We interpret", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 474, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 488 + ], + "score": 1.0, + "content": "each value as the probability of predicting the corresponding label. We obtain discrete predictions", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 234, + 498 + ], + "score": 1.0, + "content": "by thresholding at a threshold", + "type": "text" + }, + { + "bbox": [ + 234, + 487, + 241, + 496 + ], + "score": 0.73, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "tuned to maximize F1 on the development data. We add three", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 496, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 337, + 510 + ], + "score": 1.0, + "content": "terms to the inference network objective from Section 5:", + "type": "text" + }, + { + "bbox": [ + 337, + 497, + 350, + 507 + ], + "score": 0.85, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 496, + 506, + 510 + ], + "score": 1.0, + "content": "regularization, entropy regularization,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 508, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 506, + 521 + ], + "score": 1.0, + "content": "and regularization toward the pretrained feature network. Margin rescaling and slack rescaling use", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 518, + 455, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 140, + 532 + ], + "score": 1.0, + "content": "squared", + "type": "text" + }, + { + "bbox": [ + 140, + 519, + 152, + 529 + ], + "score": 0.88, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 518, + 203, + 532 + ], + "score": 1.0, + "content": "distance for", + "type": "text" + }, + { + "bbox": [ + 203, + 519, + 213, + 528 + ], + "score": 0.8, + "content": "\\triangle", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 518, + 455, + 532 + ], + "score": 1.0, + "content": ". Additional details are provided in Sec. 9.1 in the appendix.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 535, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "score": 1.0, + "content": "Comparison to Prior Work. Table 1 shows results comparing to prior work. The MLP and “SPEN", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "score": 1.0, + "content": "(BM16)” baseline results are taken from (Belanger & McCallum, 2016). We obtained the “SPEN", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 557, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 506, + 570 + ], + "score": 1.0, + "content": "(E2E)” (Belanger et al., 2017) results by running the code available from the authors on these datasets.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "This method constructs a recurrent neural network that performs gradient-based minimization of the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 579, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 196, + 592 + ], + "score": 1.0, + "content": "energy with respect to", + "type": "text" + }, + { + "bbox": [ + 196, + 581, + 204, + 591 + ], + "score": 0.75, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 579, + 506, + 592 + ], + "score": 1.0, + "content": ". They noted in their software release that, while this method is more stable,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 590, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 104, + 590, + 506, + 602 + ], + "score": 1.0, + "content": "it is prone to overfitting and actually performs worse than the original SPEN. We indeed find this to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 601, + 426, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 426, + 614 + ], + "score": 1.0, + "content": "be the case, as SPEN (E2E) underperforms SPEN (BM16) on all three datasets.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 618, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 617, + 507, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 507, + 631 + ], + "score": 1.0, + "content": "Our method (“SPEN (InfNet)”) achieves the best average performance across the three datasets.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "It performs especially well on Bookmarks, which is the largest of the three. Our results use the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 639, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 653 + ], + "score": 1.0, + "content": "contrastive hinge loss and retune the inference network on the development data after the energy is", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "trained; these decisions were made based on the tuning described in Sec. 9.1, but all four hinge losses", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 663, + 227, + 675 + ], + "spans": [ + { + "bbox": [ + 106, + 663, + 227, + 675 + ], + "score": 1.0, + "content": "led to similarly strong results.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 108, + 679, + 504, + 701 + ], + "lines": [ + { + "bbox": [ + 105, + 678, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 692 + ], + "score": 1.0, + "content": "Speed Comparison. Table 2 compares training and test-time inference speed among the different", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "methods. We only report speeds of methods that we ran.2 The SPEN (E2E) times were obtained", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 712, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 709, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 118, + 709, + 506, + 724 + ], + "score": 1.0, + "content": "2The MLP F1 scores above were taken from Belanger & McCallum (2016), but the MLP timing results", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 721, + 388, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 388, + 732 + ], + "score": 1.0, + "content": "reported in Table 2 are from our own experimental replication of their results.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 190, + 104, + 421, + 158 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 142, + 89, + 467, + 101 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 142, + 88, + 469, + 102 + ], + "spans": [ + { + "bbox": [ + 142, + 88, + 469, + 102 + ], + "score": 1.0, + "content": "Table 1: Test F1 when comparing methods on multi-label classification datasets.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 190, + 104, + 421, + 158 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 190, + 104, + 421, + 158 + ], + "spans": [ + { + "bbox": [ + 190, + 104, + 421, + 158 + ], + "score": 0.927, + "html": "
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We compare", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "score": 1.0, + "content": "accuracy and speed, finding our approach to outperform prior work. We then perform experiments", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 230, + 269, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 269, + 241 + ], + "score": 1.0, + "content": "with sequence labeling tasks in Sec. 7.2.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 207, + 505, + 241 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 255, + 267, + 267 + ], + "lines": [ + { + "bbox": [ + 105, + 254, + 269, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 269, + 268 + ], + "score": 1.0, + "content": "7.1 MULTI-LABEL CLASSIFICATION", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 276, + 506, + 343 + ], + "lines": [ + { + "bbox": [ + 106, + 277, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 506, + 288 + ], + "score": 1.0, + "content": "We use the MLC datasets used by Belanger & McCallum (2016): Bibtex, Delicious, and Bookmarks.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 286, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 300 + ], + "score": 1.0, + "content": "Dataset statistics are shown in Table 7 in the Appendix. For Bibtex and Delicious, we follow Belanger", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 299, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 505, + 310 + ], + "score": 1.0, + "content": "and McCallum and tune the hyperparameters using a different sampling of train and test data, then use", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 310, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 506, + 321 + ], + "score": 1.0, + "content": "the standard train/test split for final experimentation using the tuned hyperparameters. For Bookmarks,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "we use the same train/dev/test split as (Belanger & McCallum, 2016). For evaluation, we report the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 331, + 304, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 304, + 344 + ], + "score": 1.0, + "content": "example averaged (macro averaged) F1 measure.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 277, + 506, + 344 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 348, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 347, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 506, + 361 + ], + "score": 1.0, + "content": "We use the SPEN for MLC described in Section 2 and also used by Belanger & McCallum (2016). For", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 358, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 241, + 372 + ], + "score": 1.0, + "content": "the feature representation network", + "type": "text" + }, + { + "bbox": [ + 242, + 359, + 264, + 371 + ], + "score": 0.91, + "content": "F ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 358, + 506, + 372 + ], + "score": 1.0, + "content": ", we use feed-forward networks with two hidden layers, using", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "score": 1.0, + "content": "their same layer widths: 150 for Bibtex/Bookmarks and 250 for Delicious. We pretrain the feature", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 380, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 104, + 380, + 145, + 394 + ], + "score": 1.0, + "content": "networks", + "type": "text" + }, + { + "bbox": [ + 146, + 381, + 168, + 393 + ], + "score": 0.92, + "content": "F ( { \\dot { \\mathbf { x } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 380, + 506, + 394 + ], + "score": 1.0, + "content": "by minimizing independent-label cross entropy for 10 epochs using Adam (Kingma", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "& Ba, 2014) with learning rate 0.001. While training SPENs, we only update the parameters of the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 173, + 416 + ], + "score": 1.0, + "content": "energy function", + "type": "text" + }, + { + "bbox": [ + 174, + 403, + 189, + 414 + ], + "score": 0.69, + "content": "( \\Theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 403, + 299, + 416 + ], + "score": 1.0, + "content": "and the inference network", + "type": "text" + }, + { + "bbox": [ + 299, + 403, + 314, + 414 + ], + "score": 0.73, + "content": "( \\Phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 403, + 482, + 416 + ], + "score": 1.0, + "content": ", keeping the feature network parameters", + "type": "text" + }, + { + "bbox": [ + 482, + 403, + 505, + 415 + ], + "score": 0.92, + "content": "F ( { \\pmb x } )", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 414, + 358, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 318, + 426 + ], + "score": 1.0, + "content": "fixed. We use Adam with learning rate 0.001 to train", + "type": "text" + }, + { + "bbox": [ + 319, + 415, + 328, + 424 + ], + "score": 0.81, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 414, + 345, + 426 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 346, + 415, + 354, + 424 + ], + "score": 0.82, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 414, + 358, + 426 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19, + "bbox_fs": [ + 104, + 347, + 506, + 426 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 431, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 106, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "The inference networks are feed-forward networks with two hidden layers, using the same archi-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 441, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 238, + 455 + ], + "score": 1.0, + "content": "tectures as the feature networks", + "type": "text" + }, + { + "bbox": [ + 238, + 442, + 261, + 453 + ], + "score": 0.92, + "content": "F ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 441, + 505, + 455 + ], + "score": 1.0, + "content": ". This permits us to initialize inference network parameters", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 452, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 115, + 463 + ], + "score": 0.79, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 452, + 505, + 466 + ], + "score": 1.0, + "content": "using pretrained feature network parameters. For the output, we use an affine transformation", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 426, + 477 + ], + "score": 1.0, + "content": "layer with a sigmoid nonlinearity function, so the output values are in the range", + "type": "text" + }, + { + "bbox": [ + 426, + 464, + 448, + 475 + ], + "score": 0.89, + "content": "( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 464, + 505, + 477 + ], + "score": 1.0, + "content": ". We interpret", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 474, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 488 + ], + "score": 1.0, + "content": "each value as the probability of predicting the corresponding label. We obtain discrete predictions", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 234, + 498 + ], + "score": 1.0, + "content": "by thresholding at a threshold", + "type": "text" + }, + { + "bbox": [ + 234, + 487, + 241, + 496 + ], + "score": 0.73, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "tuned to maximize F1 on the development data. We add three", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 496, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 337, + 510 + ], + "score": 1.0, + "content": "terms to the inference network objective from Section 5:", + "type": "text" + }, + { + "bbox": [ + 337, + 497, + 350, + 507 + ], + "score": 0.85, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 496, + 506, + 510 + ], + "score": 1.0, + "content": "regularization, entropy regularization,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 508, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 506, + 521 + ], + "score": 1.0, + "content": "and regularization toward the pretrained feature network. Margin rescaling and slack rescaling use", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 518, + 455, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 140, + 532 + ], + "score": 1.0, + "content": "squared", + "type": "text" + }, + { + "bbox": [ + 140, + 519, + 152, + 529 + ], + "score": 0.88, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 518, + 203, + 532 + ], + "score": 1.0, + "content": "distance for", + "type": "text" + }, + { + "bbox": [ + 203, + 519, + 213, + 528 + ], + "score": 0.8, + "content": "\\triangle", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 518, + 455, + 532 + ], + "score": 1.0, + "content": ". Additional details are provided in Sec. 9.1 in the appendix.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 430, + 506, + 532 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 535, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "score": 1.0, + "content": "Comparison to Prior Work. Table 1 shows results comparing to prior work. The MLP and “SPEN", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "score": 1.0, + "content": "(BM16)” baseline results are taken from (Belanger & McCallum, 2016). We obtained the “SPEN", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 557, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 506, + 570 + ], + "score": 1.0, + "content": "(E2E)” (Belanger et al., 2017) results by running the code available from the authors on these datasets.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "This method constructs a recurrent neural network that performs gradient-based minimization of the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 579, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 196, + 592 + ], + "score": 1.0, + "content": "energy with respect to", + "type": "text" + }, + { + "bbox": [ + 196, + 581, + 204, + 591 + ], + "score": 0.75, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 579, + 506, + 592 + ], + "score": 1.0, + "content": ". They noted in their software release that, while this method is more stable,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 590, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 104, + 590, + 506, + 602 + ], + "score": 1.0, + "content": "it is prone to overfitting and actually performs worse than the original SPEN. We indeed find this to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 601, + 426, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 426, + 614 + ], + "score": 1.0, + "content": "be the case, as SPEN (E2E) underperforms SPEN (BM16) on all three datasets.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35, + "bbox_fs": [ + 104, + 534, + 506, + 614 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 618, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 617, + 507, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 507, + 631 + ], + "score": 1.0, + "content": "Our method (“SPEN (InfNet)”) achieves the best average performance across the three datasets.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "It performs especially well on Bookmarks, which is the largest of the three. Our results use the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 639, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 653 + ], + "score": 1.0, + "content": "contrastive hinge loss and retune the inference network on the development data after the energy is", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "trained; these decisions were made based on the tuning described in Sec. 9.1, but all four hinge losses", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 663, + 227, + 675 + ], + "spans": [ + { + "bbox": [ + 106, + 663, + 227, + 675 + ], + "score": 1.0, + "content": "led to similarly strong results.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 617, + 507, + 675 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 679, + 504, + 701 + ], + "lines": [ + { + "bbox": [ + 105, + 678, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 692 + ], + "score": 1.0, + "content": "Speed Comparison. Table 2 compares training and test-time inference speed among the different", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "methods. We only report speeds of methods that we ran.2 The SPEN (E2E) times were obtained", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 182, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 505, + 194 + ], + "score": 1.0, + "content": "using code obtained from Belanger and McCallum. We suspect that SPEN (BM16) training would be", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 192, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 505, + 205 + ], + "score": 1.0, + "content": "comparable to or slower than SPEN (E2E). 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Training Speed (examples/sec)Testing Speed (examples/sec)
BibtexBookmarksDeliciousBibtexBookmarksDelicious
MLP2167019591261589070692307113750
SPEN (E2E)55155938314201401832
SPEN (InfNet)5533546746679419488888112148
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perceptron (MR,△= 0)89.289.6
88.889.4
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SPEN hinge lossvalidation accuracy (%)
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perceptron (MR,△= 0)89.289.6
88.889.4
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We found that the cross entropy stabilization term worked well for this setting; details and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 484, + 327, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 327, + 495 + ], + "score": 1.0, + "content": "an empirical comparison are provided in Section 9.2.1.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 406, + 506, + 495 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 556 + ], + "lines": [ + { + "bbox": [ + 106, + 500, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "We compare to standard BLSTM and CRF baselines. 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Further details", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 545, + 224, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 224, + 556 + ], + "score": 1.0, + "content": "are provided in the appendix.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 500, + 505, + 556 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 567, + 179, + 579 + ], + "lines": [ + { + "bbox": [ + 105, + 566, + 180, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 180, + 581 + ], + "score": 1.0, + "content": "7.2.3 RESULTS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 586, + 505, + 664 + ], + "lines": [ + { + "bbox": [ + 106, + 587, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 505, + 599 + ], + "score": 1.0, + "content": "Loss Function Comparison. Table 3 shows results when comparing SPEN training objectives. We", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 599, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 505, + 610 + ], + "score": 1.0, + "content": "see a larger difference among losses here than for MLC tasks. When using the perceptron loss, there", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 609, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 621 + ], + "score": 1.0, + "content": "is no margin, which leads to overfitting: 89.4 on validation, 88.6 on test (not shown in the table).", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 619, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 506, + 632 + ], + "score": 1.0, + "content": "The contrastive loss, which strives to achieve a margin of 1, does better on test (89.0). We also see", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 631, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 505, + 642 + ], + "score": 1.0, + "content": "here that margin rescaling and slack rescaling both outperform the contrastive hinge, unlike the MLC", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 642, + 506, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 506, + 654 + ], + "score": 1.0, + "content": "tasks. We suspect that in the case in which each input/output has a different length, using a cost that", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 653, + 246, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 246, + 666 + ], + "score": 1.0, + "content": "captures length is more important.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 587, + 506, + 666 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 669, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 669, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 505, + 682 + ], + "score": 1.0, + "content": "Comparison to Standard Baselines. Table 4 compares our final tuned SPEN configuration to two", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 681, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 506, + 693 + ], + "score": 1.0, + "content": "standard baselines: a BLSTM tagger and a CRF. The SPEN achieves higher validation and test", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 691, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 505, + 705 + ], + "score": 1.0, + "content": "accuracies with faster test-time inference. While our method is slower than the baselines during", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 173, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 505, + 186 + ], + "score": 1.0, + "content": "training, it is faster than the CRF at test time, operating at essentially the same speed as the BLSTM", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 184, + 252, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 184, + 252, + 196 + ], + "score": 1.0, + "content": "baseline while being more accurate.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 669, + 506, + 705 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 212, + 115, + 398, + 148 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 89, + 503, + 112 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 88, + 504, + 102 + ], + "spans": [ + { + "bbox": [ + 105, + 88, + 504, + 102 + ], + "score": 1.0, + "content": "Table 5: Twitter POS validation/test accuracies when adding tag language model (TLM) energy term", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 99, + 294, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 294, + 113 + ], + "score": 1.0, + "content": "to a SPEN trained with margin-rescaled hinge.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 212, + 115, + 398, + 148 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 212, + 115, + 398, + 148 + ], + "spans": [ + { + "bbox": [ + 212, + 115, + 398, + 148 + ], + "score": 0.959, + "html": "
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(13). We note that the SPEN outperforms the CRF, despite using the same", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "form for the energy. There are two factors that can explain this. First, the losses are different. The", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 235, + 504, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 504, + 245 + ], + "score": 1.0, + "content": "CRF uses conditional log-likelihood while the SPEN results here use slack-rescaled hinge, which", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 244, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 505, + 258 + ], + "score": 1.0, + "content": "outperforms the other hinge loss variants (Table 3). Second, the stabilization terms used when training", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 255, + 506, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 506, + 269 + ], + "score": 1.0, + "content": "the inference network may be providing a regularizing effect for the model. Our motivation for", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 266, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 281 + ], + "score": 1.0, + "content": "these experiments was to show the impact of these differences while keeping the form of the energy", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 277, + 295, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 295, + 290 + ], + "score": 1.0, + "content": "function fixed. We now turn to richer energies.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 109, + 301, + 504, + 313 + ], + "lines": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "score": 1.0, + "content": "7.2.4 TOWARDS GLOBAL ENERGIES: TAG LANGUAGE MODELS FOR TWITTER POS TAGGING", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 320, + 505, + 431 + ], + "lines": [ + { + "bbox": [ + 105, + 319, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 505, + 333 + ], + "score": 1.0, + "content": "The above results only use the pairwise energy; no results used the tag language model (TLM). To", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "score": 1.0, + "content": "compute the TLM energy term, we first automatically tag unlabeled tweets, then train an LSTM", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 342, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 505, + 355 + ], + "score": 1.0, + "content": "language model on the automatic tag sequences. When doing so, we define the input tag embeddings", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 127, + 366 + ], + "score": 1.0, + "content": "to be", + "type": "text" + }, + { + "bbox": [ + 128, + 354, + 136, + 363 + ], + "score": 0.77, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 353, + 506, + 366 + ], + "score": 1.0, + "content": "-dimensional one-hot vectors specifying the tags in the training sequences. This is nonstandard", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 364, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 506, + 377 + ], + "score": 1.0, + "content": "compared to standard language modeling. In standard language modeling, we train on observed", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 376, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 388 + ], + "score": 1.0, + "content": "sequences and compute likelihoods of other fully-observed sequences. However, in our case, we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 387, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 399 + ], + "score": 1.0, + "content": "train on tag sequences but we want to use the same model on sequences of tag distributions produced", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "by an inference network. We train the TLM on sequences of one-hot vectors and then use it to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "compute likelihoods of sequences of tag distributions. Further details about training are provided in", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 419, + 228, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 228, + 432 + ], + "score": 1.0, + "content": "Section 9.2.2 in the appendix.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 435, + 505, + 469 + ], + "lines": [ + { + "bbox": [ + 104, + 433, + 504, + 451 + ], + "spans": [ + { + "bbox": [ + 104, + 433, + 259, + 451 + ], + "score": 1.0, + "content": "We define an additional energy term", + "type": "text" + }, + { + "bbox": [ + 260, + 435, + 300, + 448 + ], + "score": 0.92, + "content": "E ^ { \\mathrm { T L M } } ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 433, + 496, + 451 + ], + "score": 1.0, + "content": "based on the pretrained TLM. If the argument", + "type": "text" + }, + { + "bbox": [ + 496, + 438, + 504, + 448 + ], + "score": 0.69, + "content": "\\textbf { { y } }", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "consisted of one-hot vectors, we could simply compute its likelihood. However, to support relaxed", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 458, + 297, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 114, + 470 + ], + "score": 0.67, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 458, + 297, + 470 + ], + "score": 1.0, + "content": "’s, we need to define a more general function:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 474, + 408, + 510 + ], + "lines": [ + { + "bbox": [ + 203, + 474, + 408, + 510 + ], + "spans": [ + { + "bbox": [ + 203, + 474, + 408, + 510 + ], + "score": 0.94, + "content": "E ^ { \\mathrm { T L M } } ( \\pmb { y } ) = - \\sum _ { t = 1 } ^ { | \\pmb { y } | + 1 } \\log ( \\pmb { y } _ { t } ^ { \\top } \\mathrm { T L M } ( \\langle \\pmb { y } _ { 0 } , . . . , \\pmb { y } _ { t - 1 } \\rangle ) )", + "type": "interline_equation", + "image_path": "5eb397eae23ba8116a03e4aa804b4d4ce2322a93afe9110101cb66e890e1379c.jpg" + } + ] + } + ], + "index": 28.5, + "virtual_lines": [ + { + "bbox": [ + 203, + 474, + 408, + 492.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 203, + 492.0, + 408, + 510.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 514, + 505, + 560 + ], + "lines": [ + { + "bbox": [ + 104, + 513, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 104, + 513, + 138, + 529 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 138, + 517, + 149, + 526 + ], + "score": 0.79, + "content": "y _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 513, + 306, + 529 + ], + "score": 1.0, + "content": "is the start-of-sequence symbol,", + "type": "text" + }, + { + "bbox": [ + 306, + 516, + 333, + 528 + ], + "score": 0.89, + "content": "y _ { \\vert \\pmb { y } \\vert + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 513, + 506, + 529 + ], + "score": 1.0, + "content": "is the end-of-sequence symbol, and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 525, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 190, + 538 + ], + "score": 0.88, + "content": "\\mathrm { T L M } \\big ( \\langle y _ { 0 } , . . . , y _ { t - 1 } \\rangle \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 525, + 410, + 540 + ], + "score": 1.0, + "content": "returns the softmax distribution over tags at position", + "type": "text" + }, + { + "bbox": [ + 410, + 527, + 415, + 536 + ], + "score": 0.67, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 525, + 506, + 540 + ], + "score": 1.0, + "content": "(under the pretrained", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 537, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 368, + 551 + ], + "score": 1.0, + "content": "tag language model) given the preceding tag vectors. When each", + "type": "text" + }, + { + "bbox": [ + 369, + 539, + 379, + 549 + ], + "score": 0.85, + "content": "y _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 537, + 505, + 551 + ], + "score": 1.0, + "content": "is a one-hot vector, this energy", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 547, + 400, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 388, + 562 + ], + "score": 1.0, + "content": "reduces to the negative log-likelihood of the tag sequence specified by", + "type": "text" + }, + { + "bbox": [ + 388, + 550, + 396, + 560 + ], + "score": 0.77, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 547, + 400, + 562 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 106, + 565, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 564, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 579 + ], + "score": 1.0, + "content": "We define the new joint energy as the sum of the energy function in Eq. (13) and the TLM energy", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 575, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 506, + 589 + ], + "score": 1.0, + "content": "function in Eq. (14). During learning, we keep the TLM parameters fixed to their pretrained values,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 586, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 329, + 600 + ], + "score": 1.0, + "content": "but we tune the weight of the TLM energy (over the set", + "type": "text" + }, + { + "bbox": [ + 329, + 587, + 389, + 599 + ], + "score": 0.8, + "content": "\\{ 0 . 1 , 0 . 2 , 0 . 5 \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 586, + 506, + 600 + ], + "score": 1.0, + "content": "in the joint energy. We train", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 598, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 506, + 611 + ], + "score": 1.0, + "content": "SPENs with the new joint energy using the margin-rescaled hinge, training the inference network", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 610, + 220, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 220, + 622 + ], + "score": 1.0, + "content": "with the cross entropy term.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 626, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 104, + 625, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 104, + 625, + 505, + 640 + ], + "score": 1.0, + "content": "Table 5 shows results.4 Adding the TLM energy leads to a gain of 0.6 on the test set. Other settings", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 636, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 505, + 650 + ], + "score": 1.0, + "content": "showed more variance; when using slack-rescaled hinge, we found a small drop on test, while when", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 649, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 506, + 661 + ], + "score": 1.0, + "content": "simply training inference networks for a fixed, pretrained joint energy with tuned mixture coefficient,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 659, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 672 + ], + "score": 1.0, + "content": "we found a gain of 0.3 on test when adding the TLM energy. We investigated the improvements and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 669, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 506, + 683 + ], + "score": 1.0, + "content": "found some to involve corrections that seemingly stem from handling non-local dependencies better.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 680, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 104, + 680, + 506, + 694 + ], + "score": 1.0, + "content": "Table 10 in the appendix shows examples in which the model with the TLM appears to be better at", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 691, + 506, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 506, + 704 + ], + "score": 1.0, + "content": "using the broader context when making tagging decisions. 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To", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "score": 1.0, + "content": "compute the TLM energy term, we first automatically tag unlabeled tweets, then train an LSTM", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 342, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 505, + 355 + ], + "score": 1.0, + "content": "language model on the automatic tag sequences. 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We train", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 598, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 506, + 611 + ], + "score": 1.0, + "content": "SPENs with the new joint energy using the margin-rescaled hinge, training the inference network", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 610, + 220, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 220, + 622 + ], + "score": 1.0, + "content": "with the cross entropy term.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 564, + 506, + 622 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 626, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 104, + 625, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 104, + 625, + 505, + 640 + ], + "score": 1.0, + "content": "Table 5 shows results.4 Adding the TLM energy leads to a gain of 0.6 on the test set. Other settings", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 636, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 505, + 650 + ], + "score": 1.0, + "content": "showed more variance; when using slack-rescaled hinge, we found a small drop on test, while when", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 649, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 506, + 661 + ], + "score": 1.0, + "content": "simply training inference networks for a fixed, pretrained joint energy with tuned mixture coefficient,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 659, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 672 + ], + "score": 1.0, + "content": "we found a gain of 0.3 on test when adding the TLM energy. 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Future work includes exploring the space of network architectures for inference", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "networks to balance accuracy and efficiency, experimenting with additional global terms in structured", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 560, + 473, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 473, + 573 + ], + "score": 1.0, + "content": "energy functions, and exploring richer structured output spaces such as trees and sentences.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5 + }, + { + "type": "title", + "bbox": [ + 108, + 586, + 200, + 596 + ], + "lines": [ + { + "bbox": [ + 107, + 587, + 200, + 597 + ], + "spans": [ + { + "bbox": [ + 107, + 587, + 200, + 597 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 605, + 504, + 628 + ], + "lines": [ + { + "bbox": [ + 106, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "We thank the anonymous reviewers, David Belanger, Weiran Wang and Zheng Cai. We also thank", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 616, + 358, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 358, + 629 + ], + "score": 1.0, + "content": "NVIDIA Corporation for donating GPUs used in this research.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "title", + "bbox": [ + 108, + 646, + 176, + 658 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 177, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 177, + 660 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 665, + 504, + 677 + ], + "lines": [ + { + "bbox": [ + 106, + 664, + 504, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 504, + 678 + ], + "score": 1.0, + "content": "Brandon Amos, Lei Xu, and J. Zico Kolter. Input convex neural networks. In Proc. of ICML, 2017.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 490, + 699 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 490, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 490, + 701 + ], + "score": 1.0, + "content": "Jimmy Ba and Rich Caruana. Do deep nets really need to be deep? In Advances in NIPS, 2014.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 722 + ], + "score": 1.0, + "content": "David Belanger and Andrew McCallum. Structured prediction energy networks. In Proc. of ICML,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 114, + 719, + 143, + 733 + ], + "spans": [ + { + "bbox": [ + 114, + 719, + 143, + 733 + ], + "score": 1.0, + "content": "2016.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "10", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 107, + 126, + 504, + 180 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 89, + 505, + 123 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 88, + 505, + 102 + ], + "spans": [ + { + "bbox": [ + 105, + 88, + 505, + 102 + ], + "score": 1.0, + "content": "Table 6: Comparison of test-time inference algorithms for a trained CRF (Twitter POS tagging). 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When using the squared", + "type": "text" + }, + { + "bbox": [ + 243, + 379, + 256, + 389 + ], + "score": 0.88, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 378, + 506, + 390 + ], + "score": 1.0, + "content": "distance term (which regularizes the inference network toward", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 478, + 401 + ], + "score": 1.0, + "content": "the pretrained BLSTM), the accuracy reduces to be closer to that of the BLSTM, which reaches", + "type": "text" + }, + { + "bbox": [ + 478, + 389, + 505, + 400 + ], + "score": 0.84, + "content": "8 8 . 6 \\%", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 399, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 414 + ], + "score": 1.0, + "content": "on validation (see Table 4). When using no stabilization terms for the inference network, learning", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 167, + 424 + ], + "score": 1.0, + "content": "fails, reaching", + "type": "text" + }, + { + "bbox": [ + 167, + 411, + 194, + 422 + ], + "score": 0.85, + "content": "1 3 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "on the development set, showing the importance of using some stabilization", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 422, + 277, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 277, + 434 + ], + "score": 1.0, + "content": "term while training the inference network.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 366, + 506, + 434 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 439, + 504, + 461 + ], + "lines": [ + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "score": 1.0, + "content": "These results show promise for training inference networks to speed up combinatorial algorithms for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 450, + 270, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 270, + 461 + ], + "score": 1.0, + "content": "structured prediction and other domains.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 438, + 506, + 461 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 479, + 309, + 492 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 310, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 310, + 494 + ], + "score": 1.0, + "content": "8 CONCLUSIONS AND FUTURE WORK", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 505, + 505, + 571 + ], + "lines": [ + { + "bbox": [ + 106, + 505, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 506, + 518 + ], + "score": 1.0, + "content": "We presented ways to jointly train structured energy functions and inference networks using large-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "margin objectives. The energy function captures arbitrary dependencies among the labels, while the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "inference networks learns to capture the properties of the energy in an efficient manner, yielding fast", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 538, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 506, + 551 + ], + "score": 1.0, + "content": "test-time inference. Future work includes exploring the space of network architectures for inference", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "networks to balance accuracy and efficiency, experimenting with additional global terms in structured", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 560, + 473, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 473, + 573 + ], + "score": 1.0, + "content": "energy functions, and exploring richer structured output spaces such as trees and sentences.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 505, + 506, + 573 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 586, + 200, + 596 + ], + "lines": [ + { + "bbox": [ + 107, + 587, + 200, + 597 + ], + "spans": [ + { + "bbox": [ + 107, + 587, + 200, + 597 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 605, + 504, + 628 + ], + "lines": [ + { + "bbox": [ + 106, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "We thank the anonymous reviewers, David Belanger, Weiran Wang and Zheng Cai. 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#labels# features#train#dev#test
Bibtex15918364836-2515
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hinge loss-retuning+retuning
margin rescaling38.5138.68
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This suggests that by the end of training, the cost-augmented", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 517, + 474, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 474, + 531 + ], + "score": 1.0, + "content": "network may be approaching the argmin and that there may not be much need for retuning.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 505, + 568 + ], + "lines": [ + { + "bbox": [ + 106, + 534, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 156, + 547 + ], + "score": 1.0, + "content": "When using", + "type": "text" + }, + { + "bbox": [ + 156, + 535, + 185, + 545 + ], + "score": 0.9, + "content": "\\triangle = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 534, + 506, + 547 + ], + "score": 1.0, + "content": "or 1, retuning leads to the same small gain as when using the margin-rescaled or", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 544, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 558 + ], + "score": 1.0, + "content": "slack-rescaled losses. Here the gain is presumably from adjusting the inference network for other", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 556, + 479, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 479, + 569 + ], + "score": 1.0, + "content": "inputs rather than from converting it from a cost-augmented to a test-time inference network.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 107, + 584, + 236, + 596 + ], + "lines": [ + { + "bbox": [ + 105, + 584, + 237, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 237, + 597 + ], + "score": 1.0, + "content": "9.2 TWITTER POS TAGGING", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 108, + 606, + 259, + 618 + ], + "lines": [ + { + "bbox": [ + 106, + 605, + 261, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 261, + 619 + ], + "score": 1.0, + "content": "9.2.1 HYPERPARAMETER TUNING", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 626, + 506, + 694 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 507, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 507, + 639 + ], + "score": 1.0, + "content": "When training inference networks and SPENs for Twitter POS tagging, we use the following hyper-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 637, + 507, + 651 + ], + "spans": [ + { + "bbox": [ + 104, + 637, + 353, + 651 + ], + "score": 1.0, + "content": "parameter tuning. 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#labels# features#train#dev#test
Bibtex15918364836-2515
Bookmarks2082151480001200027856
Delicious98250112896-3185
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hinge loss-retuning+retuning
margin rescaling38.5138.68
slack rescaling38.5738.62
perceptron (MR,△= 0)38.5538.70
contrastive (△= 1)38.8038.88
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Table 8 shows", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 435, + 507, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 507, + 448 + ], + "score": 1.0, + "content": "results comparing the four loss functions from Section 4.2 on the development set for Bookmarks,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "the largest of the three datasets. We find performance to be highly similar across the losses, with the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 457, + 334, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 334, + 469 + ], + "score": 1.0, + "content": "contrastive loss appearing slightly better than the others.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 423, + 507, + 469 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 473, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 106, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "After training, we “retune” the inference network as specified by Eq. (5) on the development set for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "20 epochs using a smaller learning rate of 0.00001. Table 8 shows slightly higher F1 for all losses", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 496, + 504, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 504, + 508 + ], + "score": 1.0, + "content": "with retuning. We were surprised to see that the final cost-augmented inference network performs", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 506, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 520 + ], + "score": 1.0, + "content": "well as a test-time inference network. This suggests that by the end of training, the cost-augmented", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 517, + 474, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 474, + 531 + ], + "score": 1.0, + "content": "network may be approaching the argmin and that there may not be much need for retuning.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 473, + 505, + 531 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 505, + 568 + ], + "lines": [ + { + "bbox": [ + 106, + 534, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 156, + 547 + ], + "score": 1.0, + "content": "When using", + "type": "text" + }, + { + "bbox": [ + 156, + 535, + 185, + 545 + ], + "score": 0.9, + "content": "\\triangle = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 534, + 506, + 547 + ], + "score": 1.0, + "content": "or 1, retuning leads to the same small gain as when using the margin-rescaled or", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 544, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 558 + ], + "score": 1.0, + "content": "slack-rescaled losses. Here the gain is presumably from adjusting the inference network for other", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 556, + 479, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 479, + 569 + ], + "score": 1.0, + "content": "inputs rather than from converting it from a cost-augmented to a test-time inference network.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 534, + 506, + 569 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 584, + 236, + 596 + ], + "lines": [ + { + "bbox": [ + 105, + 584, + 237, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 237, + 597 + ], + "score": 1.0, + "content": "9.2 TWITTER POS TAGGING", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 108, + 606, + 259, + 618 + ], + "lines": [ + { + "bbox": [ + 106, + 605, + 261, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 261, + 619 + ], + "score": 1.0, + "content": "9.2.1 HYPERPARAMETER TUNING", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 626, + 506, + 694 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 507, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 507, + 639 + ], + "score": 1.0, + "content": "When training inference networks and SPENs for Twitter POS tagging, we use the following hyper-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 637, + 507, + 651 + ], + "spans": [ + { + "bbox": [ + 104, + 637, + 353, + 651 + ], + "score": 1.0, + "content": "parameter tuning. We tune the inference network learning rate", + "type": "text" + }, + { + "bbox": [ + 354, + 638, + 501, + 650 + ], + "score": 0.76, + "content": "( \\{ 0 . 1 , 0 . 0 5 , 0 . 0 2 , 0 . 0 1 , 0 . 0 0 5 , 0 . 0 0 1 \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 637, + 507, + 651 + ], + "score": 1.0, + "content": "),", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 648, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 107, + 649, + 119, + 660 + ], + "score": 0.86, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 648, + 181, + 661 + ], + "score": 1.0, + "content": "regularization", + "type": "text" + }, + { + "bbox": [ + 181, + 649, + 369, + 660 + ], + "score": 0.79, + "content": "( \\lambda _ { 1 } \\in \\{ 0 , 1 { \\mathrm { e } } - 3 , 1 { \\mathrm { e } } - 4 , 1 { \\mathrm { e } } - 5 , 1 { \\mathrm { e } } - { \\bar { 6 } } , 1 { \\mathrm { e } } - { \\bar { 7 } } \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 648, + 506, + 661 + ], + "score": 1.0, + "content": "), the entropy regularization term", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 108, + 660, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 108, + 660, + 218, + 672 + ], + "score": 0.89, + "content": "( \\lambda _ { 2 } \\in \\{ 0 . 1 , 0 . 5 , 1 , 2 , 5 , 1 0 \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 660, + 374, + 673 + ], + "score": 1.0, + "content": ", the cross entropy regularization term", + "type": "text" + }, + { + "bbox": [ + 374, + 660, + 483, + 672 + ], + "score": 0.89, + "content": "( \\lambda _ { 3 } \\in \\{ 0 . 1 , 0 . 5 , 1 , 2 , 5 , 1 0 \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 660, + 506, + 673 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 671, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 205, + 684 + ], + "score": 1.0, + "content": "the squared L2 distance", + "type": "text" + }, + { + "bbox": [ + 206, + 671, + 331, + 682 + ], + "score": 0.88, + "content": "( \\bar { \\lambda } _ { 4 } \\in \\{ 0 , 0 . 1 , 0 . 2 , \\bar { 0 . 5 } , 1 , 2 , 1 0 \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 671, + 506, + 684 + ], + "score": 1.0, + "content": "). We train the energy functions with Adam", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 681, + 497, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 238, + 694 + ], + "score": 1.0, + "content": "with a learning rate of 0.001 and", + "type": "text" + }, + { + "bbox": [ + 239, + 682, + 251, + 693 + ], + "score": 0.88, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 681, + 311, + 694 + ], + "score": 1.0, + "content": "regularization", + "type": "text" + }, + { + "bbox": [ + 311, + 682, + 493, + 694 + ], + "score": 0.88, + "content": "( \\lambda _ { 1 } \\in \\{ 0 , 1 \\mathrm { { e } - 3 , 1 \\mathrm { { e } - 4 , 1 \\mathrm { { e } - 5 } , 1 \\mathrm { { e } - 6 } , 1 \\mathrm { { e } - 7 } \\} ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 681, + 497, + 694 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 626, + 507, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "Table 9 compares the use of the cross entropy and entropy stabilization terms when training inference", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "networks for a SPEN with margin-rescaled hinge. Cross entropy works better than entropy in this", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 721, + 423, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 423, + 734 + ], + "score": 1.0, + "content": "setting, though retuning permits the latter to bridge the gap more than halfway.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 698, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 183, + 116, + 427, + 158 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 89, + 502, + 112 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 89, + 504, + 102 + ], + "spans": [ + { + "bbox": [ + 106, + 89, + 504, + 102 + ], + "score": 1.0, + "content": "Table 9: Comparison of inference network stabilization terms and showing impact of retuning when", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 100, + 429, + 113 + ], + "spans": [ + { + "bbox": [ + 106, + 100, + 429, + 113 + ], + "score": 1.0, + "content": "training SPENs with margin-rescaled hinge (Twitter POS validation accuracies).", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 183, + 116, + 427, + 158 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 183, + 116, + 427, + 158 + ], + "spans": [ + { + "bbox": [ + 183, + 116, + 427, + 158 + ], + "score": 0.966, + "html": "
validation accuracy (%)
inference network stabilization terms-retuning+retuning
cross entropy89.189.3
entropy84.286.8
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predicted tags
#tweet (target word in bold)-TLM+TLM
1... that's a t-17, technically . does that count as top-25 ?determinerpronoun
2... lol you know im down like 4 flats on a cadillac ... lol...adjectivepreposition
3... them who he is : he wants her to like him for his pers..prepositionverb
4I wonder when Nic Cage is going to film " Another Somethingnounverb
5Something Las Vegas " . Cut my hair, gag and bore menounverb
6 7... they had their fun,we hd ours !;) lmaooo " Logic will get you from A to B . Imagination will take youproper noun verbverb
everywhere ." - Albert Einstein .noun
8lmao I'm not a sheep who listens to it cos everyone else does ..verbpreposition
9Noo its not cuss you have swag andd you wont look dumb !..nouncoord. conj.
", + "type": "table", + "image_path": "6825bbc3e98b08c9213a525503588606eea7ad0c4d07b08a4ef13e3a4ee9ddcc.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 107, + 210, + 505, + 258.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 107, + 258.6666666666667, + 505, + 307.33333333333337 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 107, + 307.33333333333337, + 505, + 356.00000000000006 + ], + "spans": [], + "index": 9 + } + ] + } + ], + "index": 6.75 + }, + { + "type": "text", + "bbox": [ + 107, + 380, + 505, + 414 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 506, + 393 + ], + "score": 1.0, + "content": "When training CRFs, we use SGD with momentum. We tune the learning rate (over", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 391, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 107, + 392, + 253, + 403 + ], + "score": 0.81, + "content": "\\{ 0 . 1 , 0 . 0 5 , 0 . 0 2 , 0 . 0 1 , 0 . 0 0 5 , 0 . 0 0 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 391, + 273, + 404 + ], + "score": 1.0, + "content": ") and", + "type": "text" + }, + { + "bbox": [ + 273, + 392, + 285, + 403 + ], + "score": 0.88, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 391, + 415, + 404 + ], + "score": 1.0, + "content": "regularization coefficient (over", + "type": "text" + }, + { + "bbox": [ + 415, + 392, + 505, + 403 + ], + "score": 0.9, + "content": "\\{ 0 , 1 \\mathrm { e } - 3 , 1 \\mathrm { e } - 4 , 1 \\mathrm { e } -", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 402, + 457, + 416 + ], + "spans": [ + { + "bbox": [ + 107, + 403, + 173, + 415 + ], + "score": 0.78, + "content": "5 , 1 \\mathrm { e } - 6 , 1 \\mathrm { e } - 7 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 402, + 457, + 416 + ], + "score": 1.0, + "content": "). For all methods, we use early stopping based on validation accuracy.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 106, + 426, + 352, + 437 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 353, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 353, + 439 + ], + "score": 1.0, + "content": "9.2.2 TAG LANGUAGE MODEL DETAILS AND ANALYSIS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 446, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 459 + ], + "score": 1.0, + "content": "To obtain training data for training the tag language model, we run the Twitter POS tagger from", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 455, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 505, + 470 + ], + "score": 1.0, + "content": "Owoputi et al. (2013) on a dataset of 303K randomly-sampled English tweets. We train the tag", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 466, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 481 + ], + "score": 1.0, + "content": "language model on 300K tweets and use the remaining 3K for tuning hyperparameters and early", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 479, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 491 + ], + "score": 1.0, + "content": "stopping. We train an LSTM language model on the tag sequences using stochastic gradient descent", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "with momentum and early stopping on the validation set. We used a dropout rate of 0.5 for the LSTM", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 268, + 514 + ], + "score": 1.0, + "content": "hidden layer. We tune the learning rate", + "type": "text" + }, + { + "bbox": [ + 269, + 500, + 348, + 512 + ], + "score": 0.8, + "content": "( \\{ 0 . 1 , 0 . 2 , 0 . 5 , 1 . 0 \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 500, + 470, + 514 + ], + "score": 1.0, + "content": ", the number of LSTM layers", + "type": "text" + }, + { + "bbox": [ + 470, + 500, + 502, + 513 + ], + "score": 0.91, + "content": "( \\{ 1 , 2 \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 500, + 506, + 514 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 511, + 278, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 209, + 524 + ], + "score": 1.0, + "content": "and the hidden layer size", + "type": "text" + }, + { + "bbox": [ + 210, + 511, + 272, + 523 + ], + "score": 0.66, + "content": "( \\{ 5 0 , 1 0 0 , 2 0 0 \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 511, + 278, + 524 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 528, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "Table 10 shows examples in which our SPEN that includes the TLM appears to be using broader", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 540, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 551 + ], + "score": 1.0, + "content": "context when making tagging decisions. These are examples from the test set labeled by two models:", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 289, + 563 + ], + "score": 1.0, + "content": "the SPEN without the TLM (which achieves", + "type": "text" + }, + { + "bbox": [ + 290, + 550, + 317, + 561 + ], + "score": 0.89, + "content": "8 9 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "accuracy, as shown in Table 5) and the SPEN", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 230, + 573 + ], + "score": 1.0, + "content": "with the TLM (which reaches", + "type": "text" + }, + { + "bbox": [ + 231, + 561, + 258, + 572 + ], + "score": 0.86, + "content": "9 0 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "accuracy). In example 1, the token “that” is predicted to be", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "a determiner based on local context, but is correctly labeled a pronoun when using the TLM. This", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "example is difficult because of the noun/verb tag ambiguity of the next word (“count”) and its impact", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "score": 1.0, + "content": "on the tag for “that”. Examples 2 and 3 show two corrections for the token “like”, which is a highly", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "ambiguous word in Twitter POS tagging. The broader context makes it much clearer which tag is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 615, + 146, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 146, + 628 + ], + "score": 1.0, + "content": "intended.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "The next two examples (4 and 5) are cases of noun/verb ambiguity that are resolvable with larger", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "context. The last four examples show improvements for nonstandard word forms. The shortened form", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "of “had” (example 6) is difficult to tag due to its collision with “HD” (high-definition), but the model", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "with the TLM is able to tag it correctly. In example 7, the ambiguous token “b” is frequently used as", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "a short form of “be” on Twitter, and since it comes after “to” in this context, the verb interpretation is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 688, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 699 + ], + "score": 1.0, + "content": "encouraged. However, the broader context makes it clear that it is not a verb and the TLM-enriched", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "model tags it correctly. The words in the last two examples are nonstandard word forms that were not", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 707, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 506, + 724 + ], + "score": 1.0, + "content": "observed in the training data, which is likely the reason for their erroneous predictions. When using", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 720, + 417, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 417, + 732 + ], + "score": 1.0, + "content": "the TLM, we can better handle these rare forms based on the broader context.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 183, + 116, + 427, + 158 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 89, + 502, + 112 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 89, + 504, + 102 + ], + "spans": [ + { + "bbox": [ + 106, + 89, + 504, + 102 + ], + "score": 1.0, + "content": "Table 9: Comparison of inference network stabilization terms and showing impact of retuning when", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 100, + 429, + 113 + ], + "spans": [ + { + "bbox": [ + 106, + 100, + 429, + 113 + ], + "score": 1.0, + "content": "training SPENs with margin-rescaled hinge (Twitter POS validation accuracies).", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 183, + 116, + 427, + 158 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 183, + 116, + 427, + 158 + ], + "spans": [ + { + "bbox": [ + 183, + 116, + 427, + 158 + ], + "score": 0.966, + "html": "
validation accuracy (%)
inference network stabilization terms-retuning+retuning
cross entropy89.189.3
entropy84.286.8
", + "type": "table", + "image_path": "52e56f1f7963ce24dd015d0b30495b629bbf3b1fbb7952dcbcd1238e1cc48616.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 183, + 116, + 427, + 130.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 183, + 130.0, + 427, + 144.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 183, + 144.0, + 427, + 158.0 + ], + "spans": [], + "index": 4 + } + ] + } + ], + "index": 1.75 + }, + { + "type": "table", + "bbox": [ + 107, + 210, + 505, + 356 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 103, + 182, + 506, + 205 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "score": 1.0, + "content": "Table 10: Examples of improvements in Twitter POS tagging when using tag language model (TLM).", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 193, + 470, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 470, + 205 + ], + "score": 1.0, + "content": "In all of these examples, the predicted tag when using the TLM matches the gold standard.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "table_body", + "bbox": [ + 107, + 210, + 505, + 356 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 107, + 210, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 107, + 210, + 505, + 356 + ], + "score": 0.983, + "html": "
predicted tags
#tweet (target word in bold)-TLM+TLM
1... that's a t-17, technically . does that count as top-25 ?determinerpronoun
2... lol you know im down like 4 flats on a cadillac ... lol...adjectivepreposition
3... them who he is : he wants her to like him for his pers..prepositionverb
4I wonder when Nic Cage is going to film " Another Somethingnounverb
5Something Las Vegas " . Cut my hair, gag and bore menounverb
6 7... they had their fun,we hd ours !;) lmaooo " Logic will get you from A to B . Imagination will take youproper noun verbverb
everywhere ." - Albert Einstein .noun
8lmao I'm not a sheep who listens to it cos everyone else does ..verbpreposition
9Noo its not cuss you have swag andd you wont look dumb !..nouncoord. conj.
", + "type": "table", + "image_path": "6825bbc3e98b08c9213a525503588606eea7ad0c4d07b08a4ef13e3a4ee9ddcc.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 107, + 210, + 505, + 258.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 107, + 258.6666666666667, + 505, + 307.33333333333337 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 107, + 307.33333333333337, + 505, + 356.00000000000006 + ], + "spans": [], + "index": 9 + } + ] + } + ], + "index": 6.75 + }, + { + "type": "text", + "bbox": [ + 107, + 380, + 505, + 414 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 506, + 393 + ], + "score": 1.0, + "content": "When training CRFs, we use SGD with momentum. We tune the learning rate (over", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 391, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 107, + 392, + 253, + 403 + ], + "score": 0.81, + "content": "\\{ 0 . 1 , 0 . 0 5 , 0 . 0 2 , 0 . 0 1 , 0 . 0 0 5 , 0 . 0 0 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 391, + 273, + 404 + ], + "score": 1.0, + "content": ") and", + "type": "text" + }, + { + "bbox": [ + 273, + 392, + 285, + 403 + ], + "score": 0.88, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 391, + 415, + 404 + ], + "score": 1.0, + "content": "regularization coefficient (over", + "type": "text" + }, + { + "bbox": [ + 415, + 392, + 505, + 403 + ], + "score": 0.9, + "content": "\\{ 0 , 1 \\mathrm { e } - 3 , 1 \\mathrm { e } - 4 , 1 \\mathrm { e } -", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 402, + 457, + 416 + ], + "spans": [ + { + "bbox": [ + 107, + 403, + 173, + 415 + ], + "score": 0.78, + "content": "5 , 1 \\mathrm { e } - 6 , 1 \\mathrm { e } - 7 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 402, + 457, + 416 + ], + "score": 1.0, + "content": "). For all methods, we use early stopping based on validation accuracy.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 380, + 506, + 416 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 426, + 352, + 437 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 353, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 353, + 439 + ], + "score": 1.0, + "content": "9.2.2 TAG LANGUAGE MODEL DETAILS AND ANALYSIS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 446, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 459 + ], + "score": 1.0, + "content": "To obtain training data for training the tag language model, we run the Twitter POS tagger from", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 455, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 505, + 470 + ], + "score": 1.0, + "content": "Owoputi et al. (2013) on a dataset of 303K randomly-sampled English tweets. We train the tag", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 466, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 481 + ], + "score": 1.0, + "content": "language model on 300K tweets and use the remaining 3K for tuning hyperparameters and early", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 479, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 491 + ], + "score": 1.0, + "content": "stopping. We train an LSTM language model on the tag sequences using stochastic gradient descent", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "with momentum and early stopping on the validation set. We used a dropout rate of 0.5 for the LSTM", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 268, + 514 + ], + "score": 1.0, + "content": "hidden layer. We tune the learning rate", + "type": "text" + }, + { + "bbox": [ + 269, + 500, + 348, + 512 + ], + "score": 0.8, + "content": "( \\{ 0 . 1 , 0 . 2 , 0 . 5 , 1 . 0 \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 500, + 470, + 514 + ], + "score": 1.0, + "content": ", the number of LSTM layers", + "type": "text" + }, + { + "bbox": [ + 470, + 500, + 502, + 513 + ], + "score": 0.91, + "content": "( \\{ 1 , 2 \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 500, + 506, + 514 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 511, + 278, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 209, + 524 + ], + "score": 1.0, + "content": "and the hidden layer size", + "type": "text" + }, + { + "bbox": [ + 210, + 511, + 272, + 523 + ], + "score": 0.66, + "content": "( \\{ 5 0 , 1 0 0 , 2 0 0 \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 511, + 278, + 524 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 444, + 506, + 524 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 528, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "Table 10 shows examples in which our SPEN that includes the TLM appears to be using broader", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 540, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 551 + ], + "score": 1.0, + "content": "context when making tagging decisions. These are examples from the test set labeled by two models:", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 289, + 563 + ], + "score": 1.0, + "content": "the SPEN without the TLM (which achieves", + "type": "text" + }, + { + "bbox": [ + 290, + 550, + 317, + 561 + ], + "score": 0.89, + "content": "8 9 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "accuracy, as shown in Table 5) and the SPEN", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 230, + 573 + ], + "score": 1.0, + "content": "with the TLM (which reaches", + "type": "text" + }, + { + "bbox": [ + 231, + 561, + 258, + 572 + ], + "score": 0.86, + "content": "9 0 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "accuracy). In example 1, the token “that” is predicted to be", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "a determiner based on local context, but is correctly labeled a pronoun when using the TLM. This", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "example is difficult because of the noun/verb tag ambiguity of the next word (“count”) and its impact", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "score": 1.0, + "content": "on the tag for “that”. Examples 2 and 3 show two corrections for the token “like”, which is a highly", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "ambiguous word in Twitter POS tagging. The broader context makes it much clearer which tag is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 615, + 146, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 146, + 628 + ], + "score": 1.0, + "content": "intended.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 528, + 506, + 628 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "The next two examples (4 and 5) are cases of noun/verb ambiguity that are resolvable with larger", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "context. The last four examples show improvements for nonstandard word forms. The shortened form", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "of “had” (example 6) is difficult to tag due to its collision with “HD” (high-definition), but the model", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "with the TLM is able to tag it correctly. In example 7, the ambiguous token “b” is frequently used as", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "a short form of “be” on Twitter, and since it comes after “to” in this context, the verb interpretation is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 688, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 699 + ], + "score": 1.0, + "content": "encouraged. However, the broader context makes it clear that it is not a verb and the TLM-enriched", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "model tags it correctly. The words in the last two examples are nonstandard word forms that were not", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 707, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 506, + 724 + ], + "score": 1.0, + "content": "observed in the training data, which is likely the reason for their erroneous predictions. When using", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 720, + 417, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 417, + 732 + ], + "score": 1.0, + "content": "the TLM, we can better handle these rare forms based on the broader context.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 632, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 166, + 135, + 444, + 189 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 88, + 504, + 133 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 89, + 506, + 102 + ], + "spans": [ + { + "bbox": [ + 106, + 89, + 506, + 102 + ], + "score": 1.0, + "content": "Table 11: Named entity recognition F1 of BLSTM, CRF, and SPEN (InfNet) with slack-rescaled", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 100, + 505, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 505, + 113 + ], + "score": 1.0, + "content": "hinge where inference networks used cross entropy stabilization term. Though slowest to train, the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 110, + 506, + 125 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 506, + 125 + ], + "score": 1.0, + "content": "SPEN matches the test-time speed of the BLSTM while improving F1 by 2 points, though it lags", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 123, + 173, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 123, + 173, + 133 + ], + "score": 1.0, + "content": "behind the CRF.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "table_body", + "bbox": [ + 166, + 135, + 444, + 189 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 166, + 135, + 444, + 189 + ], + "spans": [ + { + "bbox": [ + 166, + 135, + 444, + 189 + ], + "score": 0.972, + "html": "
validation F1test F1training speed (examples/sec)testing speed (examples/sec)
BLSTM88.3083.023851042
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Training Speed (examples/sec)Testing Speed (examples/sec)
BibtexBookmarksDeliciousBibtexBookmarksDelicious
MLP2167019591261589070692307113750
SPEN (E2E)55155938314201401832
SPEN (InfNet)5533546746679419488888112148
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validation accuracy (%)test accuracy (%)training speed (examples/sec)testing speed (examples/sec)
BLSTM88.688.83851250
CRF89.189.2250500
SPEN (InfNet)89.689.81251250
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SPEN hinge lossvalidation accuracy (%)
-retuning 89.1+retuning
margin rescaling slack rescaling89.489.3
perceptron (MR,△= 0)89.289.6
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test-time inference algorithmval. accuracy (%)test accuracy (%)speed (examples/sec)
Viterbialgorithm89.189.2500
Inference network + cross entropy89.789.51250
Inference network+ entropy89.6
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Bookmarks2082151480001200027856
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predicted tags
#tweet (target word in bold)-TLM+TLM
1... that's a t-17, technically . does that count as top-25 ?determinerpronoun
2... lol you know im down like 4 flats on a cadillac ... lol...adjectivepreposition
3... them who he is : he wants her to like him for his pers..prepositionverb
4I wonder when Nic Cage is going to film " Another Somethingnounverb
5Something Las Vegas " . Cut my hair, gag and bore menounverb
6 7... they had their fun,we hd ours !;) lmaooo " Logic will get you from A to B . Imagination will take youproper noun verbverb
everywhere ." - Albert Einstein .noun
8lmao I'm not a sheep who listens to it cos everyone else does ..verbpreposition
9Noo its not cuss you have swag andd you wont look dumb !..nouncoord. conj.
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validation accuracy (%)
inference network stabilization terms-retuning+retuning
cross entropy89.189.3
entropy84.286.8
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validation F1test F1training speed (examples/sec)testing speed (examples/sec)
BLSTM88.3083.023851042
CRF91.3187.15222454
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0000000000000000000000000000000000000000..7432e45290bd821ed9086699fa259aaf539cffa6 --- /dev/null +++ b/parse/train/HkxJpnVtPr/HkxJpnVtPr.md @@ -0,0 +1,853 @@ +# A STOCHASTIC TRUST REGION METHOD FOR NONCONVEX MINIMIZATION + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +We target the problem of finding a local minimum in non-convex finite-sum minimization. Towards this goal, we first prove that the trust region method with inexact gradient and Hessian estimation can achieve a convergence rate of order $\mathcal { O } ( 1 / k ^ { 2 / 3 } )$ as long as those differential estimations are sufficiently accurate. Combining such result with a novel Hessian estimator, we propose a sample-efficient√ stochastic trust region (STR) algorithm which finds an $( \epsilon , \sqrt { \epsilon } )$ -approximate local minimum within $\tilde { \mathcal { O } } ( \sqrt { n } / \epsilon ^ { 1 . 5 } )$ stochastic Hessian oracle queries. This improves the state-of-the-art result by a factor of $\mathcal { O } ( n ^ { 1 / 6 } )$ . Finally, we also develop Hessian-free STR algorithms which achieve the lowest runtime complexity. Experiments verify theoretical conclusions and the efficiency of the proposed algorithms. + +# 1 INTRODUCTION + +We consider the following finite-sum non-convex minimization problem + +$$ +\operatorname* { m i n } _ { \mathbf { x } \in \mathbb { R } ^ { d } } F ( \mathbf { x } ) = { \frac { 1 } { n } } \sum _ { i = 1 } ^ { n } f _ { i } ( \mathbf { x } ) , +$$ + +where each (non-convex) component function $f _ { i } : \mathbb { R } ^ { d } \mathbb { R }$ is assumed to have $L _ { 1 }$ -Lipschitz continuous gradient and $L _ { 2 }$ -Lipschitz continuous Hessian. Since first-order stationary points could be saddle points with inferior generalization performance (Dauphin et al., 2014), in this work we are√ particularly interested in computing $( \epsilon , \sqrt { \epsilon } )$ -approximate second-order stationary points, $\epsilon$ -SOSP: + +$$ +\begin{array} { r } { \| \nabla F ( { \bf x } _ { \epsilon } ) \| \le \epsilon \quad \mathrm { a n d } \quad \nabla ^ { 2 } F ( { \bf x } _ { \epsilon } ) \succcurlyeq - \sqrt { L _ { 2 } \epsilon } { \bf I } . } \end{array} +$$ + +To find a local minimum of problem (1), the cubic regularization approach (Nesterov & Polyak, 2006) and the trust region algorithm (Conn et al., 2000; Curtis et al., 2017) are two classical methods. Specifically, cubic regularization forms a cubic surrogate function for the objective $F ( \mathbf { x } )$ by adding a third-order regularization term to the second-order Taylor expansion, and minimizes it iteratively. Such a method is proved to achieve an $\mathcal { O } ( 1 / k ^ { 2 / 3 } )$ global convergence rate and thus needs $\mathcal { O } ( n / \epsilon ^ { 1 . 5 } )$ stochastic first- and second-order oracle queries, namely the evaluation number of stochastic gradient and Hessian, to achieve a point that satisfies (2). On the other hand, trust region algorithms estimate the objective with its second-order Taylor expansion but minimize it only within a local region. Recently, Curtis et al. (2017) proposes a trust region variant to achieve the same convergence rate as the cubic regularization approach. But both methods require computing full gradients and Hessians of $F ( \mathbf { x } )$ and thus suffer from high computational cost in large-scale problems. + +To avoid costly exact differential evaluations, many works explore the finite-sum structure of problem (1) and develop stochastic cubic regularization approaches. Both Kohler & Lucchi (2017b) and Xu et al. (2017) propose to directly subsample the gradient and Hessian in the cubic surrogate function, and achieve $\tilde { \mathcal { O } } ( 1 / \epsilon ^ { 3 . 5 } )$ and $\tilde { \mathcal { O } } ( 1 / \epsilon ^ { 2 . 5 } )$ stochastic first- and second-order oracle complexities respectively. By plugging a stochastic variance reduced estimator (Johnson & Zhang, 2013) and the Hessian tracking technique (Gower et al., 2018) into the gradient and Hessian estimation, the approach in (Zhou et al., 2018a) improves both the stochastic first- and second-order oracle complexities to $\tilde { \mathcal { O } } ( n ^ { 0 . 8 } / \epsilon ^ { 1 . 5 } )$ . Recently, Zhang et al. (2018) and Zhou et al. (2018b) develop more efficient stochastic cubic regularization variants, which further reduce the stochastic second-order oracle complexity to $\tilde { \mathcal { O } } ( n ^ { 2 / 3 } \bar { / } \epsilon ^ { 1 . 5 } )$ at the cost of increasing the stochastic first-order oracle complexity to ${ \tilde { \mathcal { O } } } ( n ^ { 2 \bar { / } 3 } / \epsilon ^ { 2 . \bar { 5 } } )$ . + +Table 1: Stochastic first- and second-order oracle complexities, SFO and SSO for short respectively, of the proposed STR approaches and other state-of-the-art methods. When SSO is prioritized, our $\mathrm { { S T R } _ { 1 } }$ has strictly better complexity than both SCR and Lite-SVRC. When SFO and SSO are treated equally, $\mathrm { { S T R _ { 2 } } }$ improves the existing result in SVRC. + +
AlgorithmSFOSSO
TR (Conn et al., 2000; Curtis et al., 2017)CR (Nesterov & Polyak, 2006)SCR (Kohler & Lucchi, 2017a)SVRC (Zhou et al., 2018c)Lite-SVRC (Zhou et al., 2018b)0(共n01.5013.52.5O n共0(1.50（1（.5）6国1.5
STRi (this paper)STR2 (this paper)O(min{n1.5，1.5O(min{1，1.5
+ +Contributions: In this paper we propose and exploit a formulation in which we make explicit control of the step size in the trust region method. This idea is leveraged to develop two efficient stochastic trust region (STR) approaches. We tailor our methods to achieve state-of-the-art oracle complexities under the following two measurements: (i) the stochastic second-order oracle complexity is prioritized; (ii) the stochastic first- and second-order oracle complexities are treated equally. Specifically, in Setting (i), our method $\mathrm { { S T R } _ { 1 } }$ employs a newly proposed estimator to approximate the Hessian and adopts the estimator in (Fang et al., 2018) for gradient approximation. Our novel Hessian estimator maintains an accurate second-order differential approximation with lower amortized oracle complexity. In this way, $\operatorname { S T R } _ { 1 }$ achieves $\tilde { \mathcal { O } } ( \operatorname* { m i n } \{ 1 / \epsilon ^ { 2 } , \sqrt { n } / \epsilon ^ { 1 . 5 } \} )$ stochastic second-order oracle complexity. This is lower than existing results for solving problem (1). In Setting (ii), our method $\mathrm { { S T R } _ { 2 } }$ substitutes the gradient estimator in $\mathrm { { S T R } _ { 1 } }$ with one that integrates stochastic gradient and Hessian together to maintain an accurate gradient approximation. As a result, $\mathrm { { S T R _ { 2 } } }$ achieves convergence in ${ \tilde { \mathcal { O } } } ( n ^ { 3 / 4 } / \epsilon ^ { 1 . 5 } )$ overall stochastic first- and second-order oracle queries. Finally, based on STR, we further develop Hessian-free STR algorithms, namely $\mathrm { { S T R } _ { \mathrm { { f r e e } } } }$ and ${ \mathrm { S T R } } _ { \mathrm { f r e e } } +$ , which outperform existing Hessian-free algorithms theoretically. + +# 1.1 RELATED WORK + +Computing local minimum to a non-convex optimization problem is gaining considerable amount of attentions in recent years. Both cubic regularization (CR) approaches (Nesterov & Polyak, 2006) and trust region (TR) algorithms (Conn et al., 2000; Curtis et al., 2017) can escape saddle points and find a local minimum by iterating the variable along the direction related to the eigenvector of the Hessian with the most negative eigenvalue. As the CR heavily depends on the regularization parameter for the cubic term, Cartis et al. (2011) propose an adaptive cubic regularization (ARC) approach to boost the efficiency by adaptively tuning the regularization parameter according to the current objective decrease. Noting the high cost of full gradient and Hessian computation in ARC, sub-sampled cubic regularization (SCR) (Kohler & Lucchi, 2017a) is developed for sampling partial data points to estimate the full gradient and Hessian. Recently, by exploring the finite-sum structure of the target problem, many works incorporate variance reduction technique (Johnson & Zhang, 2013) into CR and propose stochastic variance-reduced methods. For example, Zhou et al. (2018c) propose stochastic variance-reduced cubic (SVRC) in which they integrate the stochastic variance-reduced gradient estimator (Johnson & Zhang, 2013) and the Hessian tracking technique (Gower et al., 2018) with CR. Such a method is proved to be at least $\mathcal { O } ( n ^ { 1 / 5 } )$ faster than CR and TR. Then Zhou et al. (2018b) use adaptive gradient batch size and constant Hessian batch size, and develop Lite-SVRC to further reduce the stochastic second-order oracle ${ \tilde { \mathcal { O } } } ( n ^ { 4 / 5 } / \epsilon ^ { 1 . 5 } )$ of SVRC to $\tilde { \mathcal { O } } ( n ^ { 2 / 3 } / \epsilon ^ { 1 . 5 } )$ at the cost of higher gradient computation cost. Similarly, except turning the gradient batch size, Zhang et al. (2018) further adaptively sample a certain number of data points to estimate the Hessian and prove the proposed method to have the same stochastic second-order oracle complexity as Lite-SVRC. + +# 2 PRELIMINARY + +Notation. We use $\| \mathbf { v } \|$ to denote the Euclidean norm of vector $\mathbf { v }$ and use $\| \mathbf { A } \|$ to denote the spectral norm of matrix A. Let $s$ be the set of component indices. We define the minibatch average of component functions by $\begin{array} { r } { f ( \mathbf { x } ; \mathcal { S } ) \ \stackrel { \mathrm { d e f } } { = } \ \frac { 1 } { | \mathcal { S } | } \sum _ { i \in \mathcal { S } } f _ { i } ( \mathbf { x } ) } \end{array}$ . Then we specify the assumptions that are necessary to the analysis of our methods. + +# MetaAlgorithm 1 Inexact Trust Region Method + +Input: initial point $\mathbf { x } ^ { 0 }$ , step size $r$ , number of iterations $K$ , construction of differential estimators $\mathbf { g } ^ { k }$ and $\mathbf { H } ^ { k }$ +1: for $k = 0$ to $K - 1$ do +2: Compute $\mathbf { h } ^ { k }$ and $\lambda ^ { k }$ by solving (8); +3: $\mathbf { x } ^ { k + 1 } : = \mathbf { x } ^ { k } + \mathbf { h } ^ { k }$ ; +4: if $\lambda ^ { k } \le 3 \sqrt { \epsilon / L _ { 2 } }$ then +5: Output ${ \bf x } _ { \epsilon } = { \bf x } ^ { k + 1 }$ ; +6: end if +7: end for + +Assumption 2.1. $F$ is bounded from below and its global optimal is achieved at $\mathbf { x } ^ { * }$ . We denote $\Delta = \bar { F ( \mathbf { x } ^ { 0 } ) } - F ( \mathbf { x } ^ { * } )$ . + +Assumption 2.2. Each $f _ { i } : \mathbb { R } ^ { d } \mathbb { R }$ has $L _ { 1 }$ -Lipschitz continuous gradient: for any $\mathbf { x } , \mathbf { y } \in \mathbb { R } ^ { d }$ + +$$ +\| \nabla f _ { i } ( \mathbf { x } ) - \nabla f _ { i } ( \mathbf { y } ) \| \leq L _ { 1 } \| \mathbf { x } - \mathbf { y } \| . +$$ + +Assumption 2.3. Each $f _ { i } : \mathbb { R } ^ { d } \mathbb { R }$ has $L _ { 2 }$ -Lipschitz continuous Hessian: for any $\mathbf { x } , \mathbf { y } \in \mathbb { R } ^ { d }$ + +$$ +\| \nabla ^ { 2 } f _ { i } ( \mathbf { x } ) - \nabla ^ { 2 } f _ { i } ( \mathbf { y } ) \| \leq L _ { 2 } \| \mathbf { x } - \mathbf { y } \| . +$$ + +# 2.1 TRUST REGION METHOD + +Here we briefly introduce the trust region method (Conn et al., 2000). In each step, it first solves the Quadratic Constraint Quadratic Program (QCQP) defined as + +$$ +\mathbf { h } ^ { k } : = \operatorname * { a r g m i n } _ { \mathbf { h } \in \mathbb { R } ^ { d } , \| \mathbf { h } \| \leq r } \langle \nabla F ( \mathbf { x } ^ { k } ) , \mathbf { h } \rangle + \frac { 1 } { 2 } \langle \nabla ^ { 2 } F ( \mathbf { x } ^ { k } ) \mathbf { h } , \mathbf { h } \rangle , +$$ + +where $r$ is the trust-region radius. Then it updates the new variable as + +$$ +\mathbf { x } ^ { k + 1 } : = \mathbf { x } ^ { k } + \mathbf { h } ^ { k } . +$$ + +Since $\nabla ^ { 2 } F ( \mathbf { x } ^ { k } )$ is indefinite, the trust-region subproblem (5) is non-convex. But its global optimizer can be characterized by the following lemma (Corollary 7.2.2 in (Conn et al., 2000)). + +Lemma 2.1. Any global minimizer of problem (5) satisfies the equation + +$$ +\left( \nabla ^ { 2 } F ( \mathbf { x } ^ { k } ) + \lambda \mathbf { I } \right) \mathbf { h } ^ { k } = - \nabla F ( \mathbf { x } ^ { k } ) , +$$ + +where the dual variable $\lambda \geq 0$ should satisfy $\nabla ^ { 2 } F ( \mathbf { x } ^ { k } ) + \lambda \mathbf { I } \succcurlyeq 0$ and $\lambda ( \| \mathbf { h } ^ { k } \| - r ) = 0 .$ . + +In particular, the standard QCQP solver returns both the minimizer $\mathbf { h } ^ { k }$ as well as the corresponding dual variable $\lambda$ of subproblem (5). In the following section, we first prove that the deterministic trust-region update (5) and (6) converges at the rate of √ $\mathcal { O } ( 1 / k ^ { 2 / 3 } )$ , much sharper than existing provable convergence rate $\mathcal { O } ( 1 / \sqrt { k } )$ (Conn et al., 2000), and then develop a more efficient stochastic trust-region approach. + +# 3 METHODOLOGY + +Here we first introduce a general inexact trust region method which is summarized in MetaAlgorithm 1. It accepts inexact gradient estimation $\mathbf { g } ^ { k }$ and Hessian estimation $\mathbf { H } ^ { k }$ as input to the QCQP subproblem + +$$ +\mathbf { h } ^ { k } : = \operatorname * { a r g m i n } _ { \mathbf { h } \in \mathbb { R } ^ { d } , \| \mathbf { h } \| \leq r } \langle \mathbf { g } ^ { k } , \mathbf { h } \rangle + \frac { 1 } { 2 } \langle \mathbf { H } ^ { k } \mathbf { h } , \mathbf { h } \rangle . +$$ + +Similar to (5), Lemma 2.1 characterizes the global optimum to problem (8) which can be efficiently solved by Lanczos method (Gould et al., 1999). Assume the dual variable of the minimizer $\mathbf { h } ^ { k }$ is $\lambda ^ { \dot { k } }$ + +We prove that such inexact trust region method achieves the optimal $\mathcal { O } ( 1 / k ^ { 2 / 3 } )$ convergence rate when the estimation $\mathbf { g } ^ { k }$ and $\mathbf { H } ^ { k }$ at each iteration are sufficiently close to their full (exact) counterparts $\nabla F ( \mathbf { x } ^ { k } )$ and $\nabla ^ { 2 } F ( \bar { \mathbf { x } } ^ { k } )$ respectively: + +$$ +\| \mathbf { g } ^ { k } - \nabla F ( \mathbf { x } ^ { k } ) \| \leq \frac { \epsilon } { 6 } , \| \mathbf { H } ^ { k } - \nabla ^ { 2 } F ( \mathbf { x } ^ { k } ) \| \leq \frac { \sqrt { \epsilon L _ { 2 } } } { 3 } . +$$ + +# Algorithm 2 STR1 + +
1:for k= 1 to K do 2: 3: Compute hk and Xk by solving (8);Input:initial point x°,step size r,number of iterations K Construct gradient estimator gk by Estimator 4; Construct Hessian estimator Hk by Estimator 3;
+ +Such result allows us to derive stochastic trust-region variants with novel differential estimators that are tailored to ensure the optimal convergence rate. We state our formal results in Theorem 3.1, whose proof is deferred to Appendix B.1 due to the space limit. + +Theorem 3.1 (Main Result). Consider problem (1) under Assumption 2.1-2.3. If the differential estimators $\mathbf { g } ^ { k }$ and $\mathbf { H } ^ { k }$ satisfy Eqn. (9) for all $k$ , MetaAlgorithm 1 finds an $\mathcal { O } ( \epsilon )$ -SOSP in less than $K = \mathcal { O } ( \sqrt { L _ { 2 } } \Delta / \epsilon ^ { 1 . 5 } )$ iterations by setting the trust-region radius as $r = \sqrt { \epsilon / L _ { 2 } }$ . + +Remark 3.1. We emphasize that MetaAlgorithm 1 degenerates to the exact trust region method by taking $\mathbf { g } ^ { k } = \nabla F ( \mathbf { x } ^ { k } )$ and $\mathbf { H } ^ { k } = \nabla ^ { 2 } F ( \mathbf { \bar { x } } ^ { k } )$ . Such result is of its own interest because this is the first proof to show that the vanilla trust region method has the optimal $\mathcal { O } ( 1 / k ^ { 2 / 3 } )$ convergence rate. Similar rate is achieved by Curtis et al. (2017) but with a much more complicated trust region variant. + +Theorem 3.1 shows the explicit step size control of the trust region method: Since the dual variable√ satisfies $\lambda ^ { k } > 3 \epsilon ^ { 0 . 5 } / \sqrt { L _ { 2 } } ^ { \bullet } > 0$ for all but the last iteration, we always find the solution to the trustregion subproblem (8) in the boundary, i.e. $\| \mathbf { h } ^ { k } \| = r$ , according to the complementary condition (15) in Appendix B.1. Such exact step size control property is missing in the cubic-regularization method where the step size is implicitly decided by the cubic regularization parameter. + +More importantly, we emphasize that such explicit step size control is crucial to the sample efficiency of our variance reduced differential estimators. The essence of variance reduction is to exploit the correlations between the differentials in consecutive iterations. Intuitively, when two neighboring iterates are close, so are their differentials due to the Lipschitz continuity, and hence a smaller number of samples suffice to maintain the accuracy of the estimators. On the other hand, smaller step size reduces the per-iteration objective decrease which harms the convergence rate of the algorithm (see proof of Theorem 3.1). Therefore, the explicit step size control in trust region method allows us to well trade-off the per-iteration sample complexity and convergence rate, from which we can derive stochastic trust region approaches with the state-of-the-art sample efficiency. In contrast, existing trust region methods change the step size at every iteration according to progress made, which requires loss evaluations that can be as expensive as gradient computations (e.g. the non-convex linear model in Section 7) and is thus prohibitive for large-scale problems. + +# 4 STOCHASTIC TRUST REGION METHOD: TYPE I + +Having the inexact trust region method as prototype, we now present our first sample-efficient stochastic trust region method, namely $\mathrm { { S T R } _ { 1 } }$ , in Algorithm 2 which emphasizes cheaper stochastic second-order oracle complexity. As Theorem 3.1 already guarantees the optimal convergence rate of MetaAlgorithm 1 when the gradient estimator $\mathbf { g } ^ { k }$ and the Hessian estimator $\mathbf { H } ^ { k }$ meet the requirement (9), here we focus on constructing such novel differential estimators. Specifically, we first present our Hessian estimator in Estimator 3 and our first gradient estimator in Estimator 4, both of which exploit the trust region radius $r = \sqrt { \epsilon / L _ { 2 } }$ to reduce their variances. + +# 4.1 HESSIAN ESTIMATOR + +Our epoch-wise Hessian estimator $\mathbf { H } ^ { k }$ is given in Estimator 3, where $p _ { 2 }$ controls the epoch length and $s _ { 2 }$ (and optionally $s _ { 2 } ^ { \prime }$ ) controls the minibatch size. At the beginning of each epoch, Estimator 3 has two options, designed for different target accuracy: Option I is preferable for the high accuracy + +# Estimator 3 Hessian Estimator + +Input: Epoch length $p _ { 2 }$ , sample size $s _ { 2 }$ , $s _ { 2 } ^ { \prime }$ (optional) +1: if $\mod ( k , p _ { 2 } ) = 0$ then +2: Option I: $\diamond$ high accuracy case (small $\epsilon$ ) +3: $\mathbf { H } ^ { k } : = \nabla ^ { 2 } F ( \mathbf { x } ^ { k } )$ ; +4: Option II: $\diamond$ low accuracy case (moderate $\epsilon$ ) +5: Draw $s _ { 2 } ^ { \prime }$ samples indexed by $\mathcal { H } ^ { \prime }$ and let $\mathbf { H } ^ { k } : = \nabla ^ { 2 } f ( \mathbf { x } ^ { k } ; \mathcal { H } ^ { \prime } )$ ; +6: else +7: Draw $s _ { 2 }$ samples indexed by $\mathcal { H }$ and let $\mathbf { H } ^ { k } : = \nabla ^ { 2 } f ( \mathbf { x } ^ { k } ; \mathcal { H } ) - \nabla ^ { 2 } f ( \mathbf { x } ^ { k - 1 } ; \mathcal { H } ) + \mathbf { H } ^ { k - 1 }$ ; +8: end if + +# Estimator 4 Gradient Estimator: Case (1) + +1: if $\mod ( k , p _ { 1 } ) = 0$ then +2: $\mathbf { g } ^ { k } : = \nabla F ( \mathbf { x } ^ { k } )$ ; +3: else +4: Draw $s _ { 1 }$ samples indexed by $\mathcal { G }$ and $\mathbf { g } ^ { k } = \nabla f ( \mathbf { x } ^ { k } ; \mathcal { G } ) - \nabla f ( \mathbf { x } ^ { k - 1 } ; \mathcal { G } ) + \mathbf { g } ^ { k - 1 } ;$ ; +5: end if + +case $( \epsilon < \mathcal { O } ( 1 / n ) )$ where we compute the full Hessian to avoid approximation error, and Option II is designed for the moderate accuracy case $( \epsilon > \mathcal { O } ( 1 / n ) )$ where we only need an approximate Hessian estimator. Then, $p _ { 2 }$ iterations follow with $\mathbf { H } ^ { k }$ defined in a recurrent manner. These recurrent estimators exist for the first-order case (Nguyen et al., 2017; Fang et al., 2018), but their bound only holds under the vector $\ell _ { 2 }$ norm. Here we generalize them into Hessian estimation with matrix spectral norm bound. + +The following lemma analyzes the amortized stochastic second-order oracle (Hessian) complexity for Algorithm 3 to meet the requirement in Theorem 3.1. As we need an error bound under the spectral norm, we will appeal to the matrix Azuma’s inequality (Tropp, 2012). The proof is deferred to Appendix B.2. + +Lemma 4.1. Assume Algorithm 2 takes the trust region radius $r = \sqrt { \epsilon / L _ { 2 } }$ as in Theorem 3.1. For any $k \geq 0$ , Estimator 3 produces estimator $\mathbf { H } ^ { k }$ for the second order differential $\nabla ^ { 2 } F ( \mathbf { x } ^ { k } )$ such that $\| \mathbf { H } ^ { k } - \nabla ^ { 2 } F ( \mathbf { x } ^ { k } ) \| \leq \sqrt { \epsilon L _ { 2 } } / 3$ with probability at least $1 - \delta / K _ { 0 }$ if we set (1) $\boldsymbol { p } _ { 2 } = \sqrt { n }$ and $s _ { 2 } = 3 2 \sqrt { n } \log ( d K _ { 0 } / \delta )$ in option $I ,$ or (2) $p _ { 2 } = L _ { 1 } / ( 2 \sqrt { \epsilon L _ { 2 } } )$ , $s _ { 2 } ^ { \prime } = 1 6 L _ { 1 } ^ { 2 } / ( \epsilon L _ { 2 } ) \log ( d K _ { 0 } / \delta )$ , and $s _ { 2 } = 3 2 L _ { 1 } / ( \sqrt { \epsilon L _ { 2 } } ) \log ( d K _ { 0 } / \delta )$ in option $\boldsymbol { { I I } }$ . Here $K _ { 0 }$ is a constant to be determined later. Consequently the amortized per-iteration stochastic second-order oracle complexity to construct is no more than $\begin{array} { r } { 2 s _ { 2 } = \operatorname* { m i n } \{ 6 4 \sqrt { n } \log \frac { d K _ { 0 } } { \delta } } \end{array}$ , $\begin{array} { r } { \frac { 6 4 L _ { 1 } } { \sqrt { \epsilon L _ { 2 } } } \log \frac { d K _ { 0 } } { \delta } \biggr \} } \end{array}$ $\mathbf { H } ^ { k }$ + +# 4.2 GRADIENT ESTIMATOR: CASE (1) + +When the stochastic second-order oracle complexity is prioritized, we directly employ the SPIDER gradient estimator to construct $\mathbf { g } ^ { k }$ (Fang et al., 2018). Similar to the construction for $\bar { \mathbf { H } } ^ { k }$ , the estimator $\bar { \mathbf { g } } ^ { k }$ is also constructed in an epoch-wise manner as presented in Estimator 4, where $p _ { 1 }$ controls the epoch length and $s _ { 1 }$ controls the minibatch size. + +We now analyze the stochastic first-order oracle complexity to meet the requirement in Theorem 3.1. Lemma 4.2. Assume Algorithm 2 takes the trust region radius $r = \sqrt { \epsilon / L _ { 2 } }$ . Estimator $\cdot$ produces estimator $\mathbf { g } ^ { k }$ of the first order differential $\nabla F ( \mathbf { x } ^ { k } )$ such that $\| \mathbf { g } ^ { k } - \nabla F ( \mathbf { x } ^ { k } ) \| \leq \epsilon / 6$ with probability at least $1 - \delta / K _ { 0 }$ for any $k \geq 0 ,$ , if we set $\begin{array} { r } { p _ { 1 } = \operatorname* { m a x } \lbrace 1 , \sqrt { n \epsilon L _ { 2 } / ( c L _ { 1 } ^ { 2 } \log \frac { K _ { 0 } } { \delta } ) } \rbrace } \end{array}$ and $s _ { 1 } = \operatorname* { m i n } \{ n , \sqrt { c n L _ { 1 } ^ { 2 } \log ( K _ { 0 } / \delta ) / ( \epsilon L _ { 2 } ) } \}$ , where the constant $c = 1 1 5 2$ and $K _ { 0 }$ is a constant to be determined later. Consequently, the amortized per-iteration stochastic first-order oracle complexity to construct $\mathbf { g } ^ { k } i s \operatorname* { m i n } \{ n , \sqrt { 4 c n L _ { 1 } ^ { 2 } \log { ( K _ { 0 } / \delta ) } / ( \epsilon L _ { 2 } ) } \}$ . + +The proof of Lemma 4.2 is similar to the one of Lemma 4.1 and is deferred to Appendix B.3. These two lemmas only guarantee that the differential estimators satisfy the requirement (9) in a single iteration and can be extended to hold for all $k$ by using the union bound with $K _ { 0 } = 2 K$ , where $K$ denotes the number of iterations. Combining such lifted result with Theorem 3.1, we can establish the computational complexity bound as follows. + +# Algorithm 5 STR2 + +Input: initial point $\mathbf { x } ^ { 0 }$ , step size $r$ , number of iterations $K$ + +1: for $k = 1$ to $K$ do +2: Construct gradient estimator $\mathbf { g } ^ { k }$ by Estimator 6; +3: Construct Hessian estimator $\bar { \mathbf { H } } ^ { k }$ by Estimator 3; +4: Compute $\mathbf { h } ^ { k }$ and $\lambda ^ { k }$ by solving (8); +5: $\mathbf { x } ^ { k + 1 } : = \mathbf { x } ^ { k } + \mathbf { h } ^ { k }$ ; +6: if $\lambda ^ { k } \le 3 \sqrt { \epsilon / L _ { 2 } }$ then +7: Output ${ \bf x } _ { \epsilon } = { \bf x } ^ { k + 1 }$ ; +8: end if +9: end for + +# Estimator 6 Gradient Estimator: Case (2) + +1: if $\mod ( k , p _ { 1 } ) = 0$ then +2: Let $\widetilde \mathbf { x } : = \mathbf { x } ^ { k }$ , $\mathbf { g } ^ { k } : = \nabla F ( \tilde { \mathbf { x } } )$ +3: else +4: Draw $s _ { 1 }$ samples indexed by $\mathcal { G }$ ; +5: $\mathbf { g } ^ { k } = \bar { \nabla } f ( \mathbf { x } ^ { k ^ { \prime } } ; \mathcal { G } ) - \nabla f ( \mathbf { x } ^ { k ^ { \prime } - 1 } ; \mathcal { G } ) + \mathbf { g } ^ { k - 1 } + [ \nabla ^ { 2 } F ( \tilde { \mathbf { x } } ) - \nabla ^ { 2 } f ( \tilde { \mathbf { x } } ; \mathcal { G } ) ] ( \mathbf { x } ^ { k } - \mathbf { x } ^ { k - 1 } ) ;$ +6: end if + +Corollary 4.1 (Computational Complexity of $\mathrm { S T R _ { 1 } }$ ). Assume Algorithm 2 will use Estimator 4 to construct the first-order differential estimator $\mathbf { g } ^ { k }$ and use Estimator $^ 3$ to construct the second-order differential estimator $\mathbf { H } ^ { k }$ . To find a $1 2 \epsilon$ -SOSP with probability at least $1 - \delta$ , the overall stochastic first-order oracle complexity is O(min{ n L2∆1.5 , $\begin{array} { r l } { \mathcal { O } ( \operatorname* { m i n } \{ \frac { n \sqrt { L _ { 2 } } \Delta } { \epsilon ^ { 1 . 5 } } , \frac { \sqrt { n } L _ { 1 } } { \epsilon ^ { 2 } } \log ( \frac { L _ { 2 } \Delta } { \delta \epsilon ^ { 1 . 5 } } ) \} ) } \end{array}$ and the overall stochastic secondorder oracle complexity is $\begin{array} { r l } { \mathcal { O } ( \operatorname* { m i n } \{ \frac { \sqrt { n L _ { 2 } } \Delta } { \epsilon ^ { 1 . 5 } } , \frac { L _ { 1 } \Delta } { \epsilon ^ { 2 } } \} \log ( \frac { d \sqrt { L _ { 2 } } \Delta } { \delta \epsilon ^ { 1 . 5 } } ) ) } \end{array}$ + +From Corollary 4.1 we see that $\tilde { \mathcal { O } } ( \operatorname* { m i n } \{ \sqrt { n } / \epsilon ^ { 1 . 5 } , 1 / \epsilon ^ { 2 } \} )$ stochastic second-order oracle queries are sufficient for $\operatorname { S T R } _ { 1 }$ to find an $\epsilon$ -SOSP which is significantly better than both the subsampled cubic regularization method $\tilde { \mathcal { O } } ( 1 / \epsilon ^ { 2 . 5 } )$ (Kohler & Lucchi, 2017a) and the variance reduction based ones $\tilde { \mathcal { O } } \bar { ( } n ^ { 2 / 3 } / \epsilon ^ { 1 . 5 } )$ (Zhou et al., 2018b; Zhang et al., 2018). Recently, Zhou & Gu (2019) developed a√ stochastic recursive variance-reduced cubic regularization (SRVRC) method which finds an $( \epsilon , \sqrt { \epsilon } )$ - approximate local minimum with $\tilde { \mathcal { O } } ( n / \epsilon ^ { 1 . 5 } , 1 / \epsilon ^ { 3 } )$ SFO and $\tilde { \mathcal { O } } ( \sqrt { n } / \epsilon ^ { 1 . 5 } , 1 / \epsilon ^ { 2 } )$ SSO. But the result of SRVRC needs to assume stochastic gradient to be bounded, i.e., $\| \nabla f _ { i } ( { \mathbf { x } } ) - \nabla F ( { \mathbf { x } } ) \| \leq \sigma$ . With this extra assumption, STR1 enjoys $\tilde { \mathcal { O } } ( n / \epsilon ^ { 1 . 5 } , n / \epsilon ^ { 2 } , 1 / \epsilon ^ { 3 } )$ SFO and $\tilde { \mathcal { O } } ( \sqrt { n } / \epsilon ^ { 1 . 5 } , 1 / \epsilon ^ { 2 } )$ SSO. Thus, if $1 / { \epsilon } \le n \le 1 / \bar { \epsilon } ^ { 2 }$ , STR1 outperforms SRVRC; otherwise they have the same complexities. + +# 5 STOCHASTIC TRUST REGION METHOD: TYPE II + +In the above section, we focus on the setting where the stochastic second-order oracle complexity is prioritized over the stochastic first-order oracle complexity. In this setting, $\mathrm { { S T R } _ { 1 } }$ achieves the stateof-the-art efficiency. In this section, we consider a different complexity measure where first-order and second-order oracle complexities are treated equally and our goal is to minimize the maximum of them. We note that, currently the best result is $\tilde { \mathcal { O } } ( n ^ { 4 / 5 } / \epsilon ^ { 1 . 5 } )$ of the SVRC method (Zhou et al., 2018c). + +Since the Hessian estimator $\mathbf { H } ^ { k }$ of $\operatorname { S T R } _ { 1 }$ already delivers the superior $\tilde { \mathcal { O } } ( \sqrt { n } / \epsilon ^ { 1 . 5 } )$ stochastic Hessian complexity, in $\mathrm { { S T R _ { 2 } } }$ (see Algorithm 5), we retain Estimator 3 for second-order differential estimation and use Estimator 6 to further reduce the stochastic gradient complexity. + +# 5.1 GRADIENT ESTIMATOR: CASE (2) + +When stochastic gradient and Hessian complexities are equally important, we use Hessian to improve the gradient estimation. Denote $\mathbf { x } ( a ) = a \bar { \mathbf { x } } ^ { t } + ( 1 - a ) \tilde { \mathbf { x } }$ . From Assumption 2.3, we have + +$$ +\nabla f _ { i } ( \mathbf { x } ^ { t } ) - \nabla f _ { i } ( \widetilde { \mathbf { x } } ) - \nabla ^ { 2 } f _ { i } ( \widetilde { \mathbf { x } } ) ( \mathbf { x } ^ { t } - \widetilde { \mathbf { x } } ) \| = \| \int _ { 0 } ^ { 1 } [ \nabla ^ { 2 } f _ { i } ( \mathbf { x } ( a ) ) - \nabla ^ { 2 } f _ { i } ( \widetilde { \mathbf { x } } ) ] ( \mathbf { x } ^ { t } - \widetilde { \mathbf { x } } ) \mathbf { d } a \| \leq \frac { L _ { 2 } } { 2 } \| \mathbf { x } ^ { t } - \widetilde { \mathbf { x } } \| ^ { 2 } +$$ + +Such property can be used to improve Lemma 4.2 of Estimator 4. Specifically, define the correction + +$$ +\mathbf { c } ^ { k } = [ \nabla ^ { 2 } F ( \tilde { \mathbf { x } } ) - \nabla ^ { 2 } f ( \tilde { \mathbf { x } } ; \mathcal { G } ) ] ( \mathbf { x } ^ { k } - \mathbf { x } ^ { k - 1 } ) , +$$ + +where $\tilde { \bf x }$ is some reference point updated in an epoch-wise manner. Estimator 6 adds $\mathbf { c } ^ { k }$ to the estimator in Estimator 4. Note that in Estimator 6, the first- and second-order oracle complexities are the same. We now analyze the first-order (and second-order) oracle complexity to meet requirement (9). + +Lemma 5.1. Assume Algorithm $\cdot$ takes the trust region radius $r = \sqrt { \epsilon / L _ { 2 } }$ as in Theorem 3.1. For any $k \geq 0$ , Estimator 6 produces estimator $\mathbf { g } ^ { k }$ for the first order differential $\nabla F ( \mathbf { x } ^ { k } )$ such that $\| \mathbf { g } ^ { k } - \mathbf { \mu }$ $\nabla F ( \mathbf { x } ^ { k } ) \| \leq \epsilon / 6$ with probability at least $1 - \delta / K _ { 0 }$ , if we set $p _ { 1 } = n ^ { 0 . 2 5 }$ and $s _ { 1 } \doteq \dot { n } ^ { 0 . 7 5 } c \log ( \ddot { K _ { 0 } } / \delta )$ where $c = 1 1 5 2$ and $K _ { 0 }$ is a constant to be determined. Consequently, the amortized per-iteration stochastic first-order oracle complexity to construct $\mathbf { g } ^ { k }$ is $2 s _ { 1 } = 2 n ^ { 0 . \dot { 7 } 5 } c \log ( K _ { 0 } / \delta )$ . + +The proof of Lemma 5.1 is similar to the one of Lemma 4.1 and is deferred to Appendix B.4. Similar to the previous section, Lemma 5.1 only guarantees that the gradient estimator satisfies the requirement (9) in a single iteration. Such result can be extended to hold for all $k$ by using the union bound with $K _ { 0 } = 2 K$ , which together with Theorem 3.1 gives the following corollary. + +Corollary 5.1 (Computational Complexity of $\mathrm { { S T R } _ { 2 } }$ ). Algorithm 5 finds a $1 2 \epsilon$ -SOSP with probability at least $1 - \delta$ , within $\begin{array} { r } { \mathcal { O } ( \frac { n ^ { 0 . 7 5 } \sqrt { L _ { 2 } } \Delta } { \epsilon ^ { 1 . 5 } } \log ( \frac { \sqrt { L _ { 2 } } \Delta } { \delta \epsilon ^ { 1 . 5 } } ) ) } \end{array}$ ) overall stochastic first-order oracle queries and $\begin{array} { r } { \mathcal { O } ( \frac { n ^ { 0 . 7 5 } \sqrt { L _ { 2 } } \Delta } { \epsilon ^ { 1 . 5 } } \log ( \frac { d \sqrt { L _ { 2 } } \Delta } { \delta \epsilon ^ { 1 . 5 } } ) ) } \end{array}$ 1.5 δ1.5 overall stochastic second-order oracle queries. + +Corollary 5.1 shows that to find an $\epsilon$ -SOSP, both SFO and SSO of $\mathrm { { S T R _ { 2 } } }$ are $\tilde { \mathcal { O } } ( n ^ { 3 / 4 } / \epsilon ^ { 1 . 5 } )$ which surpasses the best existing one ${ \tilde { \mathcal { O } } } ( n ^ { 4 / 5 } / \epsilon ^ { 1 . 5 } )$ in (Zhou et al., 2018c). + +# 6 PRACTICAL STOCHASTIC TRUST REGION VARIANTS + +# 6.1 HANDLING INEXACT QCQP SOLUTIONS + +One drawback of MetaAlgorithm 1 is that it requires the exact solution to the QCQP subproblem (8) and uses the dual variable as stopping criterion. We address this problem by developing a practical variant, MetaAlgorithm 7, which admits inexact QCQP solutions without access to the dual variable. This algorithm repeatedly invokes a procedure called INEXACTTR $\mathbf { W E A K }$ , which, as we shall see, outputs an $\mathcal { O } ( \epsilon )$ -SOSP with a constant probability of $2 / 3$ in $\mathcal { O } ( 1 / \epsilon ^ { 1 . 5 } )$ iterations. By repeatedly invoking INEXACTTR $\mathbf { \partial } \cdot \mathbf { W E A K }$ for $\Theta ( \log ( 1 / \bar { \delta } ) )$ times, MetaAlgorithm 7 boosts the probability to $( 1 - \delta )$ for any desired $\delta$ . This repeating technique has been studied by, e.g., (Allen-Zhu & Li, 2018; AllenZhu, 2018b). To test whether the $t$ -th run outputs an $\mathcal { O } ( \epsilon )$ -SOSP, we need to compute $\| \nabla F ( \mathbf { x } ^ { t } ) \|$ and the smallest eigenvalue of $\nabla ^ { 2 } F ( \mathbf { x } ^ { t } )$ . The latter one can be approximated by solving the QCQP + +$$ +\mathbf { v } ^ { t } : = \operatorname * { a r g m i n } _ { \| \mathbf { v } \| \leq 1 } \psi _ { t } ( \mathbf { v } ) = \langle \mathbf { H } ^ { t } \mathbf { v } , \mathbf { v } \rangle , +$$ + +where $\mathbf { H } ^ { t }$ is the full Hessian $\nabla ^ { 2 } F ( \mathbf { x } ^ { t } )$ or its estimation. One can show that MetaAlgorithm 7 finds an $\mathcal { O } ( \epsilon )$ -SOSP w.p. at least $( 1 - \delta )$ in $\tilde { \mathcal { O } } ( 1 / \epsilon ^ { 1 . 5 } )$ iterations. We defer the detailed analysis to Appendix C. + +# 6.2 HESSIAN-FREE IMPLEMENTATION + +Based on MetaAlgorithm 7, we propose a Hessian-free method named ${ \mathrm { { S T R } } } _ { \mathrm { { f r e e } } }$ , which is summarized in Algorithm 8. $\mathrm { S T R } _ { \mathrm { f r e e } }$ leverages the full/stochastic Hessian and Estimator 4 to construct $\mathbf { H } ^ { k }$ and $\mathbf { g } ^ { k }$ , respectively. Besides, it uses Lanczos method (Gould et al., 1999; Carmon & Duchi, 2018) as the QCQP solver, which can be implemented in a Hessian-free manner (i.e., using only Hessian-vector products without explicit Hessian matrix evaluations). Thus, $\mathbf { H } ^ { k }$ is only accessed through Hessianvector products and is never explicitly constructed. Since Hessian-vector products can be computed in linear time (in terms of the dimension $d$ ) for many machine learning problems (Allen-Zhu, 2018b; Agarwal et al., 2017), Hessian-free methods are usually more practical than Hessian based ones for high dimensional problems. The following theorem, whose proof can be found in Appendix D, establishes the runtime complexity (i.e., the total complexity of stochastic gradient and Hessian-vector product evaluations (Zhou & Gu, 2019)) of ${ \bf S T R } _ { \mathrm { f r e e } }$ . + +# MetaAlgorithm 7 Inexact Trust Region Method II + +Input: initial point $\mathbf { x } ^ { 0 }$ , step size $r$ , number of inner iterations $K$ , constants $\delta , \zeta \in ( 0 , 1 )$ , c1, c2, +number of outer iterations $T = \Theta ( \log ( 1 / \delta ) )$ , sample size $s$ (optional) +1: for $t = 1$ to $T$ do +2: $\mathbf { x } ^ { t } \gets \mathrm { I N E X A C T T R } _ { \mathrm { W E A K } } ( \mathbf { x } ^ { 0 } , r , K , \zeta )$ ; +3: Option I: $\diamond$ high accuracy case (small $\epsilon$ ) +4: $\mathbf { H } ^ { t } : = \nabla ^ { 2 } F ( \mathbf { x } ^ { t } )$ ; +5: Option II: $\diamond$ low accuracy case (moderate $\epsilon$ ) +6: Draw $s$ samples indexed by $\mathcal { H }$ and let $\mathbf { H } ^ { t } : = \nabla ^ { 2 } f ( \mathbf { x } ^ { t } ; \mathcal { H } )$ ; +7: Compute $\tilde { \mathbf { v } } ^ { t }$ by solving (10) up to accuracy $\sqrt { \epsilon L _ { 2 } }$ with probability $1 - \delta / 4$ ; +8: if $\| \nabla F ( \mathbf { x } ^ { t } ) \| \leq c _ { 1 } \epsilon _ { . }$ and $\psi _ { t } ( \tilde { \mathbf { v } } ^ { t } ) \geq - \sqrt { c _ { 2 } \epsilon L _ { 2 } }$ then +9: return $\mathbf { x } _ { \epsilon } : = \mathbf { x } ^ { t }$ ; +10: end if +11: end for +12: procedure INEXACTTR $\mathbf { w } _ { \mathrm { E A K } } ( \mathbf { x } ^ { 0 } , r , K , \zeta )$ +13: for $k = 0$ to $K - 1$ do +14: Compute $\mathbf { g } ^ { k }$ and $\mathbf { H } ^ { k }$ such that (9) holds with probability $\textstyle { 1 - \frac { \zeta } { 4 K } }$ +15: Compute $\tilde { \mathbf { h } } ^ { k }$ by solving (8) up to accuracy $\epsilon ^ { 1 . 5 } / \sqrt { L _ { 2 } }$ 4Kwith probability $\textstyle 1 - { \frac { \zeta } { 4 K } }$ ; +16: $\mathbf { x } ^ { k + 1 } : = \mathbf { x } ^ { k } + \tilde { \mathbf { h } } ^ { k }$ ; +17: end for +18: Randomly select $\bar { k }$ from $\{ 0 , \ldots , K - 1 \}$ ; +19: return xk¯+1; +20: end procedure + +# Algorithm 8 STRfree + +1: In the same setting as MetaAlgorithm 7, +2: construct gradient estimator $\mathbf { g } ^ { k }$ by Estimator 4; +3: construct Hessian estimator $\bar { \mathbf { H } } ^ { k }$ by +4: Option I: $\mathbf { H } ^ { k } : = \nabla ^ { 2 } F ( \mathbf { x } ^ { k } )$ ; +5: Option II: Draw $s$ samples indexed by $\mathcal { H }$ and let $\mathbf { H } ^ { k } : = \nabla ^ { 2 } f ( \mathbf { x } ^ { k } ; \mathcal { H } )$ ; +6: use Lanczos method to solve QCQP subproblems. + +Theorem 6.1. Consider Algorithm 8 for solving problem (1). Let $\zeta = 1 / 3$ , $r = \sqrt { \epsilon / L _ { 2 } }$ , $K =$ $4 \sqrt { L _ { 2 } } \Delta / \epsilon ^ { 1 . 5 }$ , $\begin{array} { r } { T = \frac { 3 } { 2 } \mathrm { { l o g } ( 2 / \delta ) } } \end{array}$ , $c _ { 1 } = 6 0 0$ , $c _ { 2 } = 5 0 0 \mathrm { \Omega }$ , and 32L21L2 log(4d/δ). The hyper-parameters in Estimator $^ { 4 }$ are set to the same values as those in Lemma 4.2. The number of iterations of Lanczos method is set to $\tilde { \mathcal { O } } ( 1 / ( L _ { 2 } \epsilon ) ^ { 0 . 2 5 } )$ . To find an $\mathcal { O } ( \epsilon )$ -SOSP w.p. at least $1 - \delta$ , the runtime complexity is $\tilde { \mathcal { O } } ( d \operatorname* { m i n } \{ n / \epsilon ^ { 1 . 7 5 } , 1 / \epsilon ^ { 2 . 7 5 } + \sqrt { n } / \epsilon ^ { 2 } \} \log ( 1 / \delta ) )$ . + +To solve the QCQP more efficiently, here we develop a faster solver which is based on the AppxPCA method (Allen-Zhu & Li, 2016) and KatyushaXW (Allen-Zhu, 2018a). See details in Appendix E. By replacing Lanczos method with this solver in ${ \bf S T R } _ { \mathrm { f r e e } }$ , we further improve the runtime complexity to $\tilde { \mathcal { O } } ( d \operatorname* { m i n } \{ n / \epsilon ^ { 1 . 5 } + n ^ { 0 . 7 5 } / \epsilon ^ { 1 . 7 5 } , 1 / \epsilon ^ { 2 . 5 } + \sqrt { n } / \epsilon ^ { 2 } \}$ ). We call this new algorithm ${ \mathrm { S T R } } _ { \mathrm { f r e e } } +$ whose details can be found in Appendix E. Table 2 shows that for the runtime complexities, both $\mathrm { { S T R } _ { \mathrm { { f r e e } } } }$ and ${ \mathrm { S T R } } _ { \mathrm { f r e e } } +$ outperform existing methods. See more comparison and discussion in Appendix E.4. + +# 7 EXPERIMENTS + +Here we compare the proposed STR with several state-of-the-art (stochastic) cubic regularized algorithms and trust region approaches, including trust region (TR) algorithm (Conn et al., 2000), adaptive cubic regularization (ARC) (Cartis et al., 2011), sub-sampled cubic regularization (SCR) (Kohler & Lucchi, 2017a), stochastic variance-reduced cubic (SVRC) (Zhou et al., 2018c), Lite-SVRC (Zhou et al., 2018b), and SRVRC (Zhou & Gu, 2019). For STR, we estimate the gradient as the way in case (1). This is because such a method enjoys lower Hessian computational complexity over the way in case (2) and for most problems, computing their Hessian matrices is much more time-consuming than computing their gradients. For the subproblems in these compared methods, we use Lanczos method (Gould et al., 1999; Kohler & Lucchi, 2017a) to solve the subproblem approximately in a Hessian-related Krylov subspace. We run simulations on seven datasets from LibSVM (a9a, ijcnn, codrna, phishing, w8a, epsilon and mnist). We run our algorithm for 40 epochs and use the output as the optimal value $f ^ { * }$ for sub-optimality estimation. Note the output has very small gradient already verified by Figure 2 and 4 in appendix. For all the considered algorithms, we set their initializations as zeros and tune their hyper-parameters optimally. For more experimental settings, e.g. details of testing datasets and algorithm parameter settings, please refer to Appendix F. + +Table 2: Runtime complexities of $\operatorname { S T R } _ { \mathrm { f r e e } }$ , ${ \mathrm { S T R } } _ { \mathrm { f r e e } } +$ , and other state-of-the-art methods. + +
AlgorithmRuntime
Hessian-free Cubic (Carmon & Duchi, 2016)Fast-Cubic (Agarwal et al., 2017)Stochastic Cubic (Tripuraneni et al., 2018)SRVRCfree (Zhou & Gu, 2019)（）0dn dn0.75+151.756d *35O(min{ d}）*
STRfree (this paper)STRfree+ (this paper)O(min{dn d dn0.5e1.75，2.752O(min{ dn0.75dn d dn0.51.5+e1.75，2.5+20)
+ +These entries rely on an additional assumption: $\| \nabla f _ { i } ( { \mathbf { x } } ) - \nabla F ( { \mathbf { x } } ) \| \leq \sigma$ a.s. + +![](images/8013d004ab6213afb97a494ee9b3163ce86d5b5c7e6f77d3cb61238a0eee4a13.jpg) +Figure 1: Comparison on the logistic regression with non-convex regularizer. + +Two evaluation non-convex problems. Following (Kohler & Lucchi, 2017a; Zhou et al., 2018c), we evaluate all considered algorithms on two learning tasks: the logistic regression with nonconvex regularizer and the nonlinear least square. Given $n$ data points $( { \pmb x } _ { i } , y _ { i } )$ where $\pmb { x } _ { i } \in \mathbb { R } ^ { d }$ is the sample vector and $y _ { i } ~ \in ~ \{ - 1 , 1 \}$ is the label, logistic regression with non-convex regularizer aims at distinguishing these two kinds of samples by solving the following problem $\begin{array} { r } { \operatorname* { m i n } _ { \pmb { w } } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \log ( 1 + \mathrm { { e x p } } ( - y _ { i } \pmb { w } ^ { T } \pmb { x } _ { i } ) ) + \lambda R ( \pmb { w } ; \alpha ) } \end{array}$ , where the non-convex regularizer $R ( { \boldsymbol { w } } ; \alpha )$ is defined as $\begin{array} { r } { R ( \pmb { w } ; \alpha ) = \sum _ { i = 1 } ^ { d } \alpha \pmb { w } _ { i } ^ { 2 } / ( 1 + \alpha \pmb { w } _ { i } ^ { 2 } ) } \end{array}$ . The nonlinear least square problem fits the nonlinear data by minimizing $\begin{array} { r } { \operatorname* { m i n } _ { \pmb { w } } \frac { 1 } { 2 n } \sum _ { i = 1 } ^ { n } \left[ y _ { i } - \phi ( \pmb { w } ^ { T } \pmb { x } _ { i } ) \right] ^ { 2 } + \lambda R ( \pmb { w } , \alpha ) } \end{array}$ . For these two kinds of problems, we set the parameters $\lambda = 1 0 ^ { - 3 }$ and $\alpha = 1 0$ for all testing datasets. + +Comparison of Hessian based algorithms. Figure 1 summarizes testing results on the non-convex logistic regression problem. For each dataset, we report the function value gap v.s. the overall algorithm running time which can reflect the overall computational complexity of an algorithm, and also show the function value gap v.s. Hessian sample complexity which reveals the complexity of Hessian computation. From Figure 1, one can observe that our proposed STR algorithm runs faster than the compared algorithms in terms of the algorithm running time, showing the overall superiority of STR. Furthermore, STR also reveals much sharper convergence curves in terms of the Hessian sample complexity which is consistent with our theory. This is because to achieve an $\epsilon$ -accuracy local minimum, the Hessian sample complexity of the proposed STR is $\tilde { \mathcal { O } } ( n ^ { 0 . 5 } / \epsilon ^ { 1 . 5 } )$ and is superior over the complexity of the compared methods (see the comparison in Sec. 4.2). Indeed, this also explains why our algorithm is also faster in terms of algorithm running time, since for most optimization problems, Hessian matrix is much more computationally expensive than the gradient and thus more efficient Hessian sample complexity means faster overall convergence speed. Note, as all compared methods need to compute the Hessian and gradient, their memory complexity are all $\mathcal { O } ( d ^ { 2 } + \dot { d } )$ . + +![](images/1f795eb6ac39b80b39a1210d8ceb789239034df7ee57a3eb2a229984d19c4413.jpg) +Figure 2: Comparison on the nonlinear least square problem. + +Figure 2 displays results of the compared algorithms on the nonlinear least square problem. STR shows very similar behaviors as those in Figure 1. Specifically, STR achieves fastest convergence rate in terms of both algorithm running time and Hessian sample complexity. On the codrna dataset we further plot the gradient norm versus running time and Hessian sample complexity. One can obverse that the gradient in STR vanishes significantly faster than other algorithms which means that STR can find the stationary point with high efficiency. See Figure 4 in Appendix F.2 for more experimental results on gradient norm comparison. All these results confirm the superiority of the proposed STR. + +Comparison of Hessian-free algorithms. Here we compare our proposed Hessian-free STR, namely $\mathrm { { S T R } _ { \mathrm { { f r e e } } } }$ , with other state-ofthe-art Hessian-free algorithms on the two high-dimensional datasets, including epsilon and mnist (see details in Appendix F). Here we do not compare ${ \mathrm { S T R } } _ { \mathrm { f r e e } } +$ , as it is based on AppxPCA method (Allen-Zhu & Li, 2016) and KatyushaXW (Allen-Zhu, 2018a) which require tuning a lot of hyper-parameters. From the results in Figure 3, one can observe that compared with other algorithms, our ${ \mathrm { { S T R } } } _ { \mathrm { { f r e e } } }$ + +![](images/6ae2d6fe5712438b857d2a33bd755443d66d2c7c1e1911c6e91971831b384a67.jpg) +Figure 3: Comparison of Hessian-free algorithms on non-convex logistic and soft-max regressions. + +achieves the best convergence speed which demonstrates its high efficiency in realistic applications. Besides, one also can find that $\operatorname { S T R } _ { \mathrm { f r e e } }$ is much faster than Hessian based STR since computing full Hessian is actually much computationally expensive than the computation of the Hessian vector. + +# 8 CONCLUSION + +We proposed two stochastic trust region variants. Under two settings (whether stochastic first- and second-order oracle complexities are treated equally), the proposed methods achieve state-of-theart oracle complexities. We also propose Hessian-free variants with lowest runtime complexity. 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In ICML, 2018c. 2, 6, 7, 8, 9, 22 + +# A APPENDIX + +In this appendix, Sec. B first provides the proofs for the results in the manuscript. Then, we analyze MetaAlgorithm 7 and $\mathrm { { S T R } _ { \mathrm { { f r e e } } } }$ in Sec. C and Sec. D, respectively. Next, in Sec. E, we develop a fast QCQP solver to further improve the computational complexity of $\operatorname { S T R } _ { \mathrm { f r e e } }$ . Finally, more experimental details and results are presented in Sec. F. + +# B DEFERRED PROOFS + +# B.1 PROOF OF THEOREM 3.1 + +Proof. For simplicity of notation, we denote + +$$ +\nabla _ { k } \overset { \mathrm { d e f } } { = } \nabla F ( { \mathbf x } ^ { k } ) - { \mathbf g } ^ { k } \mathrm { a n d } \nabla _ { k } ^ { 2 } \overset { \mathrm { d e f } } { = } \nabla ^ { 2 } F ( { \mathbf x } ^ { k } ) - { \mathbf H } ^ { k } . +$$ + +From Assumption 2.3 we have + +$$ +\begin{array} { r } { F ( { \mathbf x } ^ { k + 1 } ) \leq F ( { \mathbf x } ^ { k } ) + \langle \nabla F ( { \mathbf x } ^ { k } ) , \mathbf { h } ^ { k } \rangle + \displaystyle \frac { 1 } { 2 } \langle \nabla ^ { 2 } F ( { \mathbf x } ^ { k } ) { \mathbf h } ^ { k } , \mathbf { h } ^ { k } \rangle + \displaystyle \frac { L _ { 2 } } { 6 } \| { \mathbf h } ^ { k } \| ^ { 3 } } \\ { = F ( { \mathbf x } ^ { k } ) + \langle \nabla _ { k } + \mathbf { g } ^ { k } , \mathbf { h } ^ { k } \rangle + \displaystyle \frac { 1 } { 2 } \langle [ \nabla _ { k } ^ { 2 } + \mathbf { H } ^ { k } ] { \mathbf h } ^ { k } , \mathbf { h } ^ { k } \rangle + \displaystyle \frac { L _ { 2 } } { 6 } \| { \mathbf h } ^ { k } \| ^ { 3 } . } \end{array} +$$ + +Use the CauchySchwarz inequality to obtain + +$$ +F ( { \mathbf x } ^ { k + 1 } ) \leq F ( { \mathbf x } ^ { k } ) + \langle { \mathbf g } ^ { k } , { \mathbf h } ^ { k } \rangle + \frac { 1 } { 2 } \langle \mathbf H ^ { k } { \mathbf h } ^ { k } , { \mathbf h } ^ { k } \rangle + \frac { L _ { 2 } } { 6 } \| { \mathbf h } ^ { k } \| ^ { 3 } + \| \nabla _ { k } \| \| { \mathbf h } ^ { k } \| + \frac { 1 } { 2 } \| \nabla _ { k } ^ { 2 } \| \| { \mathbf h } ^ { k } \| ^ { 2 } . +$$ + +The requirement (9) together with the trust region radius $\| \mathbf { h } \| \le r = \sqrt { \epsilon / L _ { 2 } }$ allows us to bound + +$$ +\| \nabla _ { k } \| \| \mathbf h ^ { k } \| + \frac { 1 } { 2 } \| \nabla _ { k } ^ { 2 } \| \| \mathbf h ^ { k } \| ^ { 2 } \leq \frac { 1 } { 3 } \cdot \frac { \epsilon ^ { 1 . 5 } } { \sqrt { L _ { 2 } } } . +$$ + +The optimality of (5) indicates that there exists a dual variable $\lambda ^ { k } \geq 0$ so that (Corollary 7.2.2 in (Conn et al., 2000)) + +$$ +\begin{array} { c } { { \displaystyle { \mathrm { F i r s t ~ O r d e r : ~ } } { \bf { g } } ^ { k } + { \bf H } ^ { k } { \bf h } ^ { k } + \frac { \lambda ^ { k } L _ { 2 } } { 2 } { \bf h } ^ { k } = 0 , } } \\ { { \displaystyle \mathrm { S e c o n d ~ O r d e r : ~ } { \bf H } ^ { k } + \frac { \lambda ^ { k } L _ { 2 } } { 2 } \cdot { \bf I } \approx 0 , } } \\ { { \displaystyle \mathrm { C o m p l e m e n t a r y : ~ } \lambda ^ { k } \cdot ( \| { \bf h } ^ { k } \| - r ) = 0 . } } \end{array} +$$ + +Multiplying (13) by $\mathbf { h } ^ { k }$ , we have + +$$ +\langle \mathbf { g } ^ { k } + \mathbf { H } ^ { k } \mathbf { h } ^ { k } + \frac { \lambda ^ { k } L _ { 2 } } { 2 } \mathbf { h } ^ { k } , \mathbf { h } ^ { k } \rangle = 0 . +$$ + +Additionally, using (14) we have + +$$ +\langle ( \mathbf { H } ^ { k } + \frac { \lambda ^ { k } L _ { 2 } } { 2 } \mathbf { I } ) \mathbf { h } ^ { k } , \mathbf { h } ^ { k } \rangle \geq 0 , +$$ + +which together with (16) gives + +$$ +\langle \mathbf { g } ^ { k } , \mathbf { h } ^ { k } \rangle \leq 0 . +$$ + +Moreover, the complementary property (15) indicates $\| \mathbf { h } ^ { k } \| = \sqrt { \epsilon / L _ { 2 } }$ as we have $\lambda ^ { k } > 3 \sqrt { \epsilon / L _ { 2 } } > 0$ before MetaAlgorithm 1 terminates. Plug (12), (16), and (17) into (11) and use $\| \mathbf { h } ^ { k } \| = \sqrt { \epsilon / L _ { 2 } }$ : + +$$ +F ( \mathbf { x } ^ { k + 1 } ) \leq F ( \mathbf { x } ^ { k } ) - \frac { L _ { 2 } \lambda ^ { k } } { 4 } \cdot \frac { \epsilon } { L _ { 2 } } + \frac { 1 } { 2 } \cdot \frac { \epsilon ^ { 1 . 5 } } { \sqrt { L _ { 2 } } } . +$$ + +Therefore, if we have $\lambda ^ { k } > 3 \epsilon ^ { 0 . 5 } / \sqrt { L _ { 2 } }$ , then + +$$ +F ( \mathbf { x } ^ { k + 1 } ) \leq F ( \mathbf { x } ^ { k } ) - { \frac { 1 } { 4 { \sqrt { L _ { 2 } } } } } \cdot \epsilon ^ { 1 . 5 } . +$$ + +Using Assumption 2.1, we find $\lambda ^ { k } \le 3 \epsilon ^ { 0 . 5 } / \sqrt { L _ { 2 } }$ in no more than $4 \sqrt { L _ { 2 } } \cdot ( F ( \mathbf { x } ^ { 0 } ) - F ( \mathbf { x } ^ { * } ) ) / \epsilon ^ { 1 . 5 }$ iterations. + +We now show that once $\lambda ^ { k } \le 3 \epsilon ^ { 0 . 5 } / \sqrt { L _ { 2 } }$ , then $\mathbf { x } ^ { k + 1 }$ is already an $\mathcal { O } ( \epsilon )$ -SOSP: From (13), we have + +$$ +\| \mathbf { g } ^ { k } + \mathbf { H } ^ { k } \mathbf { h } ^ { k } \| = \frac { L _ { 2 } \lambda ^ { k } } { 2 } \cdot \| \mathbf { h } ^ { k } \| \leq 2 \epsilon . +$$ + +The assumptions $\lVert \nabla _ { k } \rVert \leq \epsilon / 6$ and $\| \nabla _ { k } ^ { 2 } \| \le \sqrt { \epsilon L _ { 2 } } / 3$ together with the trust region radius $\| \mathbf { h } \| \leq$ $\sqrt { \epsilon / L _ { 2 } }$ imply + +$$ +\| \nabla F ( { \mathbf x } ^ { k } ) + \nabla ^ { 2 } F ( { \mathbf x } ^ { k } ) { \mathbf h } ^ { k } \| \leq \| { \mathbf g } ^ { k } + { \mathbf H } ^ { k } { \mathbf h } ^ { k } \| + \| \nabla _ { k } \| + \| \nabla _ { k } ^ { 2 } \cdot { \mathbf h } ^ { k } \| \leq 2 . 5 \epsilon . +$$ + +On the other hand use Assumption 2.3 to bound + +$$ +\| \nabla F ( { \mathbf { x } } ^ { k + 1 } ) - \nabla F ( { \mathbf { x } } ^ { k } ) - \nabla ^ { 2 } F ( { \mathbf { x } } ^ { k } ) { \mathbf { h } } ^ { k } \| \leq \frac { L _ { 2 } } { 2 } \| { \mathbf { h } } ^ { k } \| ^ { 2 } \leq \frac { \epsilon } { 2 } . +$$ + +Combining these two results gives $\| \nabla F ( \mathbf { x } ^ { k + 1 } ) \| \leq 3 \epsilon$ + +Besides, using Assumption 2.3, $\| \nabla _ { k } ^ { 2 } \| \le \sqrt { \epsilon L _ { 2 } } / 3$ , and (14), we derive the Hessian lower bound + +$$ +\nabla ^ { 2 } F ( \mathbf { x } ^ { k + 1 } ) \approx \nabla ^ { 2 } F ( \mathbf { x } ^ { k } ) - L _ { 2 } \cdot \Vert \mathbf { h } ^ { k } \Vert \mathbf { I } \approx \mathbf { H } ^ { k } - \sqrt { \epsilon L _ { 2 } } / 3 \mathbf { I } - L _ { 2 } \Vert \mathbf { h } ^ { k } \Vert \mathbf { I } \approx - \sqrt { 1 2 \epsilon L _ { 2 } } \mathbf { I } . +$$ + +Hence $\mathbf { x } ^ { k + 1 }$ is a $1 2 \epsilon$ -stationary point. Additionally, we have $\| \mathbf { h } ^ { k } \| = r$ according to the complementary condition (15) for all but the last iteration. □ + +# B.2 PROOF OF LEMMA 4.1 + +Proof. Without loss of generality, we analyze the case $0 \leq k < q _ { 2 }$ for ease of notation. We first focus on Option II. The proof for Option I follows the similar argument. + +Option II: Define for $k = 0$ and $i \in [ s _ { 2 } ^ { \prime } ]$ + +$$ +\mathbf { B } _ { i } ^ { 0 } \overset { \mathrm { d e f } } { = } \nabla ^ { 2 } f _ { i } ( \mathbf { x } ^ { 0 } ) - \nabla ^ { 2 } F ( \mathbf { x } ^ { 0 } ) , +$$ + +and define for $k \geq 1$ and $i \in [ s _ { 2 } ]$ + +$$ +\mathbf { B } _ { i } ^ { k } \stackrel { \mathrm { d e f } } { = } \nabla ^ { 2 } f _ { i } ( \mathbf { x } ^ { k } ) - \nabla ^ { 2 } f _ { i } ( \mathbf { x } ^ { k - 1 } ) - \big ( \nabla ^ { 2 } F ( \mathbf { x } ^ { k } ) - \nabla ^ { 2 } F ( \mathbf { x } ^ { k - 1 } ) \big ) . +$$ + +$\{ \mathbf { B } _ { i } ^ { k } \}$ is a martingale difference sequence. We have for all $k$ and $i$ + +$$ +\mathbb { E } [ \mathbf { B } _ { i } ^ { k } | \mathbf { x } ^ { k } ] = 0 . +$$ + +Besides, we use Assumption 2.2 for $k = 0$ to bound + +$$ +\| \mathbf { B } _ { i } ^ { 0 } \| \leq \| \nabla ^ { 2 } f _ { i } ( \mathbf { x } ^ { 0 } ) \| + \| \nabla ^ { 2 } F ( \mathbf { x } ^ { 0 } ) \| = 2 L _ { 1 } , +$$ + +and use Assumption 2.3 for $k \geq 1$ to bound + +$$ +\| \mathbf { B } _ { i } ^ { k } \| \leq \| \nabla ^ { 2 } f _ { i } ( \mathbf { x } ^ { k } ) - \nabla ^ { 2 } f _ { i } ( \mathbf { x } ^ { k - 1 } ) \| + \| \nabla ^ { 2 } F ( \mathbf { x } ^ { k } ) - \nabla ^ { 2 } F ( \mathbf { x } ^ { k - 1 } ) \| \leq 2 \sqrt { \epsilon L _ { 2 } } . +$$ + +From the construction of $\mathbf { H } ^ { k }$ , we have + +$$ +\mathbf { H } ^ { k } - \nabla ^ { 2 } F ( \mathbf { x } ^ { k } ) = \sum _ { i = 1 } ^ { s _ { 2 } ^ { \prime } } { \frac { \mathbf { B } _ { i } ^ { 0 } } { s _ { 2 } ^ { \prime } } } + \sum _ { j = 1 } ^ { k } \sum _ { i = 1 } ^ { s _ { 2 } } { \frac { \mathbf { B } _ { i } ^ { j } } { s _ { 2 } } } . +$$ + +Thus using the matrix Azuma’s Inequality in Theorem 7.1 of (Tropp, 2012) and $k \leq p _ { 2 }$ , we have + +$$ +\begin{array} { r l } & { P r \{ \| \mathbf H ^ { k } - \nabla ^ { 2 } F ( { \mathbf x } ^ { k } ) \| \ge t \} \le d \cdot \exp \{ - \frac { t ^ { 2 } / 8 } { \sum _ { i = 1 } ^ { s _ { 2 } ^ { \prime } } 4 L _ { 1 } ^ { 2 } / s _ { 2 } ^ { \prime 2 } + \sum _ { j = 1 } ^ { k } \sum _ { i = 1 } ^ { s _ { 2 } } 4 \epsilon L _ { 2 } / s _ { 2 } ^ { 2 } } \} } \\ & { \qquad \le d \cdot \exp \{ - \frac { t ^ { 2 } / 8 } { 4 L _ { 1 } ^ { 2 } / s _ { 2 } ^ { \prime } + 4 p _ { 2 } \epsilon L _ { 2 } / s _ { 2 } } \} . } \end{array} +$$ + +Consequently, we have + +$$ +P r \{ \| \mathbf { H } ^ { k } - \nabla ^ { 2 } F ( \mathbf { x } ^ { k } ) \| \leq \sqrt { \epsilon L _ { 2 } } \} \geq 1 - \delta / K _ { 0 } . +$$ + +by taking √ $t = \sqrt { \epsilon L _ { 2 } }$ , $s _ { 2 } ^ { \prime } = 1 6 L _ { 1 } ^ { 2 } / ( \epsilon L _ { 2 } ) \log ( d K _ { 0 } / \delta )$ , $s _ { 2 } = 3 2 L _ { 1 } / ( \sqrt { \epsilon L _ { 2 } } ) \log ( d K _ { 0 } / \delta )$ , and $p _ { 2 } =$ $L _ { 1 } / ( 2 \sqrt { \epsilon L _ { 2 } } )$ . + +Option I: The proof is similar to the one of Option $\mathrm { I I }$ except that we replace $\mathbf { B } _ { i } ^ { 0 }$ with zero matrix. In such case, the matrix Azuma’s Inequality implies + +$$ +P r \{ \| \mathbf H ^ { k } - \nabla ^ { 2 } F ( \mathbf x ^ { k } ) \| \geq t \} \leq d \cdot \exp \{ - \frac { t ^ { 2 } / 8 } { \sum _ { j = 1 } ^ { k } \sum _ { i = 1 } ^ { s _ { 2 } } 4 \epsilon L _ { 2 } / s _ { 2 } ^ { 2 } } \} \leq d \cdot \exp \{ - \frac { t ^ { 2 } / 8 } { 4 p _ { 2 } \epsilon L _ { 2 } / s _ { 2 } } \} . +$$ + +Thus by taking $t = \sqrt { \epsilon L _ { 2 } }$ , $s _ { 2 } = 3 2 \sqrt { n } \log ( d K _ { 0 } / \delta )$ , and $p _ { 2 } = { \sqrt { n } }$ , we have the result. + +Amortized Complexity: In option I, the choice of parameters ensures that: $s _ { 2 } ^ { \prime } \leq p _ { 2 } \times s _ { 2 }$ and in option II: $n \leq p _ { 2 } \times s _ { 2 }$ . Consequently, the amortized stochastic second-order oracle complexity is bounded from above by $2 s _ { 2 }$ . □ + +# B.3 PROOF OF LEMMA 4.2 + +Without loss of generality, we analyze the case $0 \leq k < q _ { 1 }$ for ease of notation. Define for $k \geq 1$ and $i \in [ s _ { 1 } ]$ + +$$ +\mathbf { a } _ { i } ^ { k } \overset { \mathrm { d e f } } { = } \nabla f _ { i } ( \mathbf { x } ^ { k } ) - \nabla f _ { i } ( \mathbf { x } ^ { k - 1 } ) - ( \nabla F ( \mathbf { x } ^ { k } ) - \nabla F ( \mathbf { x } ^ { k - 1 } ) ) . +$$ + +$\{ \mathbf { a } _ { i } ^ { k } \}$ is a martingale difference sequence: for all $k$ and $i$ + +$$ +\mathbb { E } [ \mathbf { a } _ { i } ^ { k } | \mathbf { x } ^ { k } ] = 0 . +$$ + +Besides, $\mathbf { a } _ { i } ^ { k }$ has bounded norm: + +$$ +\begin{array} { r l } & { \| \mathbf { a } _ { i } ^ { k } \| \leq \| \nabla f _ { i } ( \mathbf { x } ^ { k } ) - \nabla f _ { i } ( \mathbf { x } ^ { k - 1 } ) \| + \| \nabla F ( \mathbf { x } ^ { k } ) - \nabla F ( \mathbf { x } ^ { k - 1 } ) \| } \\ & { \qquad \leq L _ { 1 } \| \mathbf { x } ^ { k } - \mathbf { x } ^ { k - 1 } \| + L _ { 1 } \| \mathbf { x } ^ { k } - \mathbf { x } ^ { k - 1 } \| } \\ & { \qquad \leq 2 L _ { 1 } \sqrt { \epsilon / L _ { 2 } } . } \end{array} +$$ + +From the construction of $\mathbf { g } ^ { k }$ , we have + +$$ +\mathbf { g } ^ { k } - \nabla F ( \mathbf { x } ^ { k } ) = \sum _ { j = 1 } ^ { k } \sum _ { i = 1 } ^ { s _ { 1 } } \frac { \mathbf { a } _ { i } ^ { j } } { s _ { 1 } } . +$$ + +Recall the Azuma’s Inequality. Using $k \leq p _ { 1 }$ , we have + +$$ +\begin{array} { r l } & { \quad P r \{ | | \mathbf { g } ^ { k } - \nabla F ( \mathbf { x } ^ { k } ) | | \geq t \} } \\ & { \leq \exp \{ - \frac { t ^ { 2 } / 8 } { \sum _ { j = 1 } ^ { k } \sum _ { i = 1 } ^ { s _ { 1 } } \frac { 4 \epsilon L _ { 1 } ^ { 2 } } { L _ { 2 } s _ { 1 } ^ { 2 } } } \} \leq \exp \{ - \frac { t ^ { 2 } / 8 } { 4 \epsilon L _ { 1 } ^ { 2 } p _ { 1 } / \left( s _ { 1 } L _ { 2 } \right) } \} . } \end{array} +$$ + +Take $t = \epsilon / 6$ and denote $c = 1 1 5 2$ . To ensure that + +$$ +\begin{array} { r } { P r \{ \| \mathbf { g } ^ { k } - \nabla F ( \mathbf { x } ^ { k } ) \| \ge \epsilon / 6 \} \le \delta / K _ { 0 } , } \end{array} +$$ + +we need $\begin{array} { r } { \frac { c L _ { 1 } ^ { 2 } } { L _ { 2 } } \log \frac { K _ { 0 } } { \delta } \ \leq \ \frac { \epsilon s _ { 1 } } { p _ { 1 } } } \end{array}$ . The best amortized stochastic first-order oracle complexity can be obtained by solving the following two-dimensional programming: + +$$ +\begin{array} { r l } { \underset { p _ { 1 } \geq 1 , s _ { 1 } \geq 1 } { \operatorname* { m i n } } } & { ( n + s _ { 1 } ( p _ { 1 } - 1 ) ) / p _ { 1 } } \\ { s . t . } & { \frac { c L _ { 1 } ^ { 2 } } { L _ { 2 } } \log \frac { K _ { 0 } } { \delta } \leq \frac { \epsilon s _ { 1 } } { p _ { 1 } } , } \end{array} +$$ + +which has the soluthat when we take $\begin{array} { r } { s _ { 1 } = \operatorname* { m i n } \{ n , \sqrt { \frac { n } { \epsilon } \cdot \frac { c L _ { 1 } ^ { 2 } \log \frac { K _ { 0 } } { \delta } } { L _ { 2 } } } \} } \end{array}$ $\begin{array} { r } { p _ { 1 } = \operatorname* { m a x } \lbrace 1 , \sqrt { n \epsilon \cdot \frac { L _ { 2 } } { c L _ { 1 } ^ { 2 } \log \frac { K _ { 0 } } { \delta } } } \rbrace } \end{array}$ . Note $s _ { 1 } = n$ $\mathbf { g } ^ { k } = \nabla F ( \mathbf { x } ^ { k } )$ + +The amortized stochastic first-order oracle complexity is obtained by plugging in the choice of $s _ { 1 }$ and $p _ { 1 }$ , which completes the proof. + +# B.4 PROOF OF LEMMA 5.1 + +Without loss of generality, we analyze the case $0 \leq k < q _ { 1 }$ for ease of notation. Define for $k \geq 1$ and $i \in [ s _ { 1 } ]$ + +$$ +\begin{array} { r l } { \mathbf { b } _ { i } ^ { k } \stackrel { \mathrm { d e f } } { = } \nabla f _ { i } ( \mathbf { x } ^ { k } ) - \nabla f _ { i } ( \mathbf { x } ^ { k - 1 } ) - \nabla ^ { 2 } f _ { i } ( \tilde { \mathbf { x } } ) ( \mathbf { x } ^ { k } - \mathbf { x } ^ { k - 1 } ) } & { } \\ { - \left[ \nabla F ( \mathbf { x } ^ { k } ) - \nabla F ( \mathbf { x } ^ { k - 1 } ) - \nabla ^ { 2 } F ( \tilde { \mathbf { x } } ) ( \mathbf { x } ^ { k } - \mathbf { x } ^ { k - 1 } ) \right] . } & { } \end{array} +$$ + +$\{ \mathbf { b } _ { i } ^ { k } \}$ is a martingale difference sequence: for all $k$ and $i$ + +$$ +\mathbb { E } [ \mathbf { b } _ { i } ^ { k } | \mathbf { x } ^ { k } ] = 0 . +$$ + +Besides, $\mathbf { b } _ { i } ^ { k }$ has bounded norm: + +$$ +\begin{array} { r l } & { \quad \| \mathbf { b } _ { \star } ^ { k } \| \leq \| \nabla f _ { i } ( \mathbf { x } ^ { k } ) - \nabla f _ { 1 } ( \mathbf { x } ^ { k - 1 } ) - \nabla ^ { 2 } f _ { i } ( \widetilde { \mathbf { x } } ) ( \mathbf { x } ^ { k } - \mathbf { x } ^ { k - 1 } ) \| } \\ & { \qquad + \| \nabla F ( \mathbf { x } ^ { k } ) - \nabla F ( \mathbf { x } ^ { k - 1 } ) - \nabla ^ { 2 } F ( \widetilde { \mathbf { x } } ) ( \mathbf { x } ^ { k } - \mathbf { x } ^ { k - 1 } ) \| . } \\ & { \quad + \| \nabla f _ { i } ( \mathbf { x } ^ { k } ) - \nabla f _ { \widetilde { \mathbf { x } } } ( \mathbf { x } ^ { k - 1 } ) - \nabla ^ { 2 } f _ { i } ( \widetilde { \mathbf { x } } ) ( \mathbf { x } ^ { k } - \mathbf { x } ^ { k - 1 } ) \| \arcsin ^ { k } , } \\ & { \| \nabla f _ { i } ( \mathbf { x } ^ { k } ) - \nabla f _ { i } ( \mathbf { x } ^ { k - 1 } ) - \nabla ^ { 2 } f _ { j } ( \widetilde { \mathbf { x } } ) ( \mathbf { x } ^ { k } - \mathbf { x } ^ { k - 1 } ) \| \arcsin ^ { k } } \\ & { \quad = \| \int _ { 0 } ^ { 1 } \left[ \nabla ^ { 2 } f _ { i } ( \mathbf { x } ^ { k - 1 } + t ( \mathbf { x } ^ { k } - \mathbf { x } ^ { k - 1 } ) ) - \nabla ^ { 2 } f _ { i } ( \widetilde { \mathbf { x } } ) \right] ( \mathbf { x } ^ { k } - \mathbf { x } ^ { k - 1 } ) d t \| } \\ & { \quad \leq \int _ { 0 } ^ { 1 } L _ { 2 } \bigg \| \mathbf { x } ^ { k } + ( 1 - t ) \mathbf { x } ^ { k - 1 } - \widetilde { \mathbf { x } } \bigg \| d t \cdot \| \mathbf { x } ^ { k } - \mathbf { x } ^ { k - 1 } \| } \\ & { \quad \leq \int _ { 0 } ^ { 1 } \big ( L \| \mathbf { x } ^ { k } - \widetilde { \mathbf { x } } \| + ( 1 - t ) \| \mathbf { x } ^ { k - 1 } - \widetilde { \mathbf { x } } \big \| d t \cdot \| \mathbf { x } ^ { k } - \mathbf { x } ^ { k - 1 } \| } \\ & \quad \leq \int _ { 0 } ^ { 1 } \big ( L \| \mathbf { x } ^ { k } - \widetilde { \mathbf { x } } \end{array} +$$ + +where the first inequality follows from Assumption 2.3 and the last inequality holds because $\| \mathbf { x } ^ { k } - $ $\tilde { \mathbf { x } } \| \leq k r$ and $\| \mathbf { x } ^ { k - 1 } - \bar { \tilde { \mathbf { x } } } \| \leq k r$ , where $r$ is the trust region radius. Similarly, we have $\| \nabla F ( \mathbf { x } ^ { k } ) -$ $\begin{array} { r } { \nabla F ( { \mathbf x } ^ { k - 1 } ) - \nabla ^ { 2 } F ( \tilde { { \mathbf x } } ) ( \dot { \mathbf x } ^ { k } - { \mathbf x } ^ { k - 1 } ) \| \leq L _ { 2 } k r ^ { 2 } } \end{array}$ . Thus, we bound + +$$ +\| \mathbf { b } _ { i } ^ { k } \| \leq 2 L _ { 2 } k r ^ { 2 } \leq 2 p _ { 1 } \epsilon +$$ + +From the construction of $\mathbf { g } ^ { k }$ , we have + +$$ +\mathbf { g } ^ { k } - \nabla F ( \mathbf { x } ^ { k } ) = \sum _ { j = 1 } ^ { k } \sum _ { i = 1 } ^ { s _ { 1 } } \frac { \mathbf { b } _ { i } ^ { j } } { s _ { 1 } } . +$$ + +We use $k \leq p _ { 1 }$ and the Azuma’s inequality to bound + +$$ +\begin{array} { r l } & { \quad P r \{ | | \mathbf { g } ^ { k } - \nabla F ( { \mathbf { x } } ^ { k } ) | | \geq t \} } \\ & { \leq \exp \{ - \frac { t ^ { 2 } / 8 } { \sum _ { j = 1 } ^ { k } \sum _ { i = 1 } ^ { s _ { 1 } } \frac { 4 p _ { 1 } ^ { 2 } \epsilon ^ { 2 } } { s _ { 1 } ^ { 2 } } } \} \leq \exp \{ - \frac { t ^ { 2 } / 8 } { 4 \epsilon ^ { 2 } p _ { 1 } ^ { 3 } / s _ { 1 } } \} . } \end{array} +$$ + +Thus, by taking $t = \epsilon / 6$ and $c = 1 1 5 2$ , we need $\begin{array} { r } { \frac { s _ { 1 } } { p _ { 1 } ^ { 3 } } \geq c \log \frac { K _ { 0 } } { \delta } } \end{array}$ . Further we want $s _ { 1 } p _ { 1 } \simeq \mathcal { O } ( n )$ and hence we take $p _ { 1 } = n ^ { 0 . 2 5 }$ and $\begin{array} { r } { s _ { 1 } = n ^ { 0 . 7 5 } c \log { \frac { \dot { K _ { 0 } } } { \delta } } } \end{array}$ . The amortized stochastic first-order oracle complexity is bounded by $2 s _ { 1 }$ . + +# C ANALYSIS OF METAALGORITHM 7 + +We first show that INEXACTTR $\mathbf { W E A K }$ finds an $\mathcal { O } ( \epsilon )$ -SOSP in $\mathcal { O } ( 1 / \epsilon ^ { 1 . 5 } )$ iterations with probability at least $2 / 3$ as stated in the following lemma. + +Lemma C.1. Consider problem $( ~ )$ under Assumptions 2.1-2.3. Suppose that the differential estimators $\mathbf { g } ^ { k }$ and $\mathbf { H } ^ { k }$ satisfy Eqn. (9) with probability at least $\begin{array} { r } { ( 1 - \frac { \zeta } { 4 K } ) } \end{array}$ . Besides, suppose that $\tilde { \mathbf { h } } ^ { k }$ is an approximate solution to (8) such that w.p. $\begin{array} { r } { ( 1 - \frac { \zeta } { 4 K } ) } \end{array}$ , + +$$ +\langle \mathbf { g } ^ { k } , \tilde { \mathbf { h } } ^ { k } \rangle + \frac { 1 } { 2 } \langle \mathbf { H } ^ { k } \tilde { \mathbf { h } } ^ { k } , \tilde { \mathbf { h } } ^ { k } \rangle \leq \langle \mathbf { g } ^ { k } , \mathbf { h } ^ { k } \rangle + \frac { 1 } { 2 } \langle \mathbf { H } ^ { k } \mathbf { h } ^ { k } , \mathbf { h } ^ { k } \rangle + \frac { \epsilon ^ { 1 . 5 } } { \sqrt { L _ { 2 } } } , +$$ + +where $\mathbf { h } ^ { k }$ is a global solution to (8). By setting $\zeta = 1 / 3$ , $r = \sqrt { \epsilon / L _ { 2 } }$ , and $K = 4 \sqrt { L _ { 2 } } \Delta / \epsilon ^ { 1 . 5 }$ , INEXACTTR $\mathbf { \partial } \cdot \mathbf { W } \mathbf { E } \mathbf { A } \mathbf { K }$ outputs a 500-SOSP w.p. at least $2 / 3$ . + +Proof. Combining (11) and (24), we have w.p. $\begin{array} { r } { ( 1 - \frac { \zeta } { 4 K } ) } \end{array}$ + +$$ +\begin{array} { r } { F ( { \mathbf x } ^ { k + 1 } ) \le F ( { \mathbf x } ^ { k } ) + \langle { \mathbf g } ^ { k } , { \mathbf h } ^ { k } \rangle + \displaystyle \frac { 1 } { 2 } \langle { \mathbf H } ^ { k } { \mathbf h } ^ { k } , { \mathbf h } ^ { k } \rangle + \displaystyle \frac { \epsilon ^ { 1 . 5 } } { \sqrt { L _ { 2 } } } } \\ { + \displaystyle \frac { L _ { 2 } } { 6 } \| \tilde { \mathbf h } ^ { k } \| ^ { 3 } + \| \nabla _ { k } \| \| \tilde { \mathbf h } ^ { k } \| + \displaystyle \frac { 1 } { 2 } \| \nabla _ { k } ^ { 2 } \| \| \tilde { \mathbf h } ^ { k } \| ^ { 2 } , } \end{array} +$$ + +where $\mathbf { h } ^ { k }$ is a global solution to the QCQP (8) and $\tilde { \mathbf { h } } ^ { k }$ is an approximate solution satisfying (24). We let $\lambda ^ { k }$ denote the dual variable corresponding to the global solution $\mathbf { h } ^ { k }$ as defined in Lemma 2.1. We note that $\mathbf { h } ^ { k }$ and $\lambda ^ { k }$ are used only in our analysis. The INEXACT $\mathrm { w e A K }$ algorithm only requires the approximate solution $\tilde { \mathbf { x } } ^ { k }$ without knowledge of $\mathbf { h } ^ { k }$ or $\lambda ^ { k }$ . + +By the assumption that (9) holds with probability $\textstyle { \left( { 1 - { \frac { \zeta } { 4 K } } } \right) }$ and the fact that $\| \tilde { \mathbf { h } } ^ { k } \| \le r = \sqrt { L _ { 2 } \epsilon }$ , we have w.p. $\textstyle { \left( { 1 - { \frac { \zeta } { 4 K } } } \right) }$ , + +$$ +\frac { L _ { 2 } } { 6 } \| \tilde { \mathbf { h } } ^ { k } \| ^ { 3 } + \| \nabla _ { k } \| \| \tilde { \mathbf { h } } ^ { k } \| + \frac { 1 } { 2 } \| \nabla _ { k } ^ { 2 } \| \| \tilde { \mathbf { h } } ^ { k } \| ^ { 2 } \leq \frac { \epsilon ^ { 1 . 5 } } { 2 \sqrt { L _ { 2 } } } . +$$ + +Plugging (16), (17), and (26) into (25) and applying the union bound, we have w.p. at least $\begin{array} { r } { ( 1 - \frac { \zeta } { 2 K } ) } \end{array}$ + +$$ +F ( { \mathbf x } ^ { k + 1 } ) \le F ( { \mathbf x } ^ { k } ) - \frac { L _ { 2 } \lambda ^ { k } \| { \mathbf h } ^ { k } \| ^ { 2 } } { 4 } + \frac { 3 \epsilon ^ { 1 . 5 } } { 2 \sqrt { L _ { 2 } } } = F ( { \mathbf x } ^ { k } ) - \frac { L _ { 2 } \lambda ^ { k } r ^ { 2 } } { 4 } + \frac { 3 \epsilon ^ { 1 . 5 } } { 2 \sqrt { L _ { 2 } } } , +$$ + +where the second inequality follows from (15): + +$$ +0 = \lambda ^ { k } ( \| \mathbf h ^ { k } \| - r ) = \lambda ^ { k } ( \| \mathbf h ^ { k } \| - r ) ( \| \mathbf h ^ { k } \| + r ) = \lambda ^ { k } ( \| \mathbf h ^ { k } \| ^ { 2 } - r ^ { 2 } ) . +$$ + +Summing inequality (27) from $k = 0$ to $K - 1$ and applying the union bound, we have w.p. at least $\left( 1 - \zeta / 2 \right)$ , + +$$ +\frac { 1 } { K } \sum _ { k = 0 } ^ { K - 1 } \lambda ^ { k } \le \frac { 4 ( F ( \mathbf { x } ^ { 0 } ) - F ( \mathbf { x } ^ { K + 1 } ) ) } { L _ { 2 } r ^ { 2 } K } + \frac { 6 \epsilon ^ { 1 . 5 } } { L _ { 2 } ^ { 1 . 5 } r ^ { 2 } } \le \frac { 4 \Delta } { \epsilon K } + \frac { 6 \sqrt { \epsilon } } { \sqrt { L _ { 2 } } } , +$$ + +where the second inequality follows from Assumption 2.1 and our choice of the trust region radius. By sampling $\bar { k }$ uniformly from $\{ 0 , \ldots , K - 1 \}$ , we obtain + +$$ +\mathbb { E } [ \lambda ^ { \bar { k } } ] = \frac { 1 } { K } \sum _ { k = 0 } ^ { K - 1 } \lambda ^ { k } , +$$ + +where the expectation is taken over the randomness of √ $\bar { k }$ . Combining (29) and (30) and taking $K = 4 \Delta \sqrt { L _ { 2 } } / \epsilon ^ { 1 . 5 }$ , we have w.p. at least $\left( 1 - \zeta / 2 \right)$ + +$$ +\mathbb { E } [ \lambda ^ { \bar { k } } ] \leq \frac { 7 \sqrt \epsilon } { \sqrt { L _ { 2 } } } . +$$ + +Since $\lambda ^ { k }$ is always no-negative, by Markov’s inequality and the union bound, with probability at least $1 - \zeta$ , we have + +$$ +\lambda ^ { \bar { k } } \leq \frac { 1 4 \sqrt { \epsilon } } { \zeta \sqrt { L _ { 2 } } } . +$$ + +By taking $\zeta = 1 / 3$ , we have w.p. at least $2 / 3 , \lambda ^ { \bar { k } } \leq 4 2 \sqrt { \epsilon / L _ { 2 } }$ . The rest of the proof is similar to Theorem 3.1 and we have the result. + +The following theorem shows that MetaAlgorithm 7 finds an $\mathcal { O } ( \epsilon )$ -SOSP w.p. $( 1 - \delta )$ after running INEXACTTR $\mathbf { \partial } \cdot \mathbf { W } \mathbf { E } \mathbf { A } \mathbf { K }$ for $\Theta ( \log ( 1 / \delta ) )$ times. + +Theorem C.1 (Iteration Complexity of MetaAlgorithm 7). In the same setting as Lemma C.1, let $\begin{array} { r } { T = \frac { 3 } { 2 } \mathrm { l o g } ( 2 / \delta ) } \end{array}$ , $c _ { 1 } = 6 0 0$ , $c _ { 2 } = 5 0 0$ , and 32L21L2 log(4d/δ). Then MetaAlgorithm 7 finds a 600-SOSP with probability at least $( 1 - \delta )$ . + +Proof. By Lemma C.1 and our choice of $T$ , with probability $\left( 1 - \delta / 2 \right)$ , at least one of $\mathbf { x } ^ { t }$ is a $5 0 0 \epsilon$ -SOSP. On the other hand, since √ $\psi _ { t } ( \tilde { \mathbf { v } } ^ { t } ) \leq \psi _ { t } ( \mathbf { v } ^ { t } ) + \sqrt { \epsilon L _ { 2 } }$ with probability $1 - \delta / 4$ , if $\psi _ { t } ( \tilde { \mathbf { v } } ^ { t } ) \geq - \sqrt { c _ { 2 } \epsilon L _ { 2 } }$ , then, w.p. $1 - \delta / 4$ , + +$$ +\psi _ { t } ( \mathbf { v } ^ { t } ) \geq \psi _ { t } ( \tilde { \mathbf { v } } ^ { t } ) - \sqrt { \epsilon L _ { 2 } } \geq - \sqrt { c _ { 2 } \epsilon L _ { 2 } } - \sqrt { \epsilon L _ { 2 } } \geq - \sqrt { 5 5 0 \epsilon L _ { 2 } } , +$$ + +where the last inequality follows from our choice of $c _ { 2 }$ . + +Option I: Since $\mathbf { H } ^ { t } = \nabla ^ { 2 } F ( \mathbf { x } ^ { t } )$ is the full Hessian, $\psi _ { t } ( \mathbf { v } ^ { t } )$ is the smallest eigenvalue of $\nabla ^ { 2 } F ( \mathbf { x } ^ { t } )$ . +Applying the union bound, we conclude that MetaAlgorithm 7 outputs a $6 0 0 \epsilon$ -SOSP w.p. $( 1 - \delta )$ . + +Option II: Let $\mathbf { B } _ { i } : = \nabla ^ { 2 } f _ { i } ( \mathbf { x } ^ { t } ) - \nabla ^ { 2 } F ( \mathbf { x } ^ { t } )$ for $i \in \mathcal { H }$ , then + +$$ +\mathbf { H } ^ { t } - \boldsymbol { \nabla } ^ { 2 } F ( \mathbf { x } ^ { t } ) = \frac { 1 } { s } \sum _ { i = 1 } ^ { s } \mathbf { B } _ { i } . +$$ + +By Assumption 2.2, we have + +$$ +\| \mathbf { B } _ { i } \| \leq \| \nabla ^ { 2 } f _ { i } ( \mathbf { x } ^ { t } ) \| + \| \nabla ^ { 2 } F ( \mathbf { x } ^ { t } ) \| \leq 2 L _ { 1 } . +$$ + +Applying the matrix Azuma’s Inequality in Theorem 7.1 of Tropp (2012) leads to + +$$ +P r \{ \| \mathbf { H } ^ { t } - \nabla ^ { 2 } F ( \mathbf { x } ^ { t } ) \| \ge \sqrt { \epsilon L _ { 2 } } \} \le d \cdot \exp ( \frac { - \epsilon L _ { 2 } s } { 3 2 L _ { 1 } ^ { 2 } } ) . +$$ + +By taking $\begin{array} { r } { s = \frac { 3 2 L _ { 1 } ^ { 2 } } { \epsilon L _ { 2 } } \mathrm { l o g } ( 4 d / \delta ) } \end{array}$ and applying the union bound, we have with probability $1 - \delta$ + +$$ +\nabla ^ { 2 } F ( \mathbf { x } ^ { t } ) \approx \mathbf { H } ^ { t } - \sqrt { \epsilon L _ { 2 } } \mathbf { I } \approx ( \psi _ { t } ( \mathbf { v } ^ { t } ) - \sqrt { \epsilon L _ { 2 } } ) \mathbf { I } \approx - \sqrt { 6 0 0 \epsilon L _ { 2 } } \mathbf { I } , +$$ + +where the last inequality follows from (33). This completes the proof. + +# D PROOF OF THEOREM 6.1 + +Proof. We first analyze the computational cost of Lanczos method. By Corollary 2 in (Carmon & Duchi, 2018), for any desired accuracy ˜, Lanczos method achieves this accuracy in $\begin{array} { r } { \mathcal { O } ( \frac { r } { \sqrt { \tilde { \epsilon } } } \mathrm { l o g } \frac { r \sqrt { d } } { \tilde { \epsilon } p } ) } \end{array}$ Lanczos iterations w.p. at least $( 1 - p )$ . Without loss of generality, we assume that the number of Lanczos iterations is strictly smaller than the dimension $d$ , otherwise the QCQP subproblem can be solved exactly. We note that each Lanczos iteration involves computation of one matrix-vector product. Therefore, to satisfy the condition (24) in Lemma C.1, one needs to evaluate $\tilde { \mathcal { O } } ( 1 / ( L _ { 2 } \epsilon ) ^ { 0 . 2 5 } )$ Hessian-vector products of the form $\mathbf { H } ^ { k } \mathbf { v }$ . Similarly, to solve (10) up to accuracy $\sqrt { \epsilon L _ { 2 } }$ w.h.p., one needs to evaluate $\tilde { \mathcal { O } } ( 1 / ( L _ { 2 } \epsilon ) ^ { 0 . 2 5 } )$ Hessian-vector products of the form $\mathbf { H } ^ { t } \mathbf { v }$ . + +In MetaAlgorithm 7, to verify whether the candidate solution $\mathbf { v } _ { } ^ { t }$ is indeed an $\mathcal { O } ( \epsilon )$ -SOSP, one needs at most ${ \mathcal { O } } ( n )$ stochastic gradient evaluations and $\mathcal { O } ( \operatorname* { m i n } \{ n , \log ( 4 d / \delta ) L _ { 1 } ^ { 2 } / ( L _ { 2 } \epsilon ) \} / ( L _ { 2 } \epsilon ) ^ { 0 . 2 5 } ) )$ stochastic Hessian-vector product evaluations, where the latter one follows from the proof of Theorem 7. We proceed to analyze the computational complexity of the INEXACTTRWEAK procedure. Recall that the iteration complexity of MetaAlgorithm 7 is $\mathcal { O } ( \log ( 1 / \delta ) / \epsilon ^ { 1 . 5 } )$ . Following Lemma 4.2 and Corollary 4.1, the stochastic first-order oracle complexity is $\tilde { \mathcal { O } } ( \operatorname* { m i n } \{ n / \epsilon ^ { 1 . 5 } , \sqrt { n } / \epsilon ^ { 2 } \} \mathrm { l o g } ( 1 / \delta ) )$ . Following the proof of Lemma 4.1 and Corollary 4.1, when $p _ { 2 } = 1$ , the overall stochastic Hessian sample complexity is $\tilde { \mathcal { O } } ( \operatorname* { m i n } \{ n / \epsilon ^ { 1 . 5 } , 1 / \epsilon ^ { 2 . 5 } \} \log ( 1 / \delta ) )$ . Since it takes $\tilde { \mathcal { O } } ( 1 / \epsilon ^ { 0 . 2 5 } )$ Lanczos iterations to meet the condition (24) as stated above, the overall stochastic Hessian-vector product oracle complexity is $\tilde { \mathcal { O } } ( \operatorname* { m i n } \{ n / \epsilon ^ { 1 . 7 5 } , 1 / \epsilon ^ { 2 . 7 5 } \} \log ( 1 / \delta ) )$ . Combining the stochastic first-order and Hessian-vector product complexities, the overall runtime is $\tilde { \mathcal { O } } ( d \operatorname* { m i n } \{ n / \epsilon ^ { 1 . 7 5 } , 1 / \epsilon ^ { 2 . 7 5 } + \sqrt { n } / \epsilon ^ { 2 } \} \log ( 1 / \delta ) )$ . □ + +# E A FASTER HESSIAN-VECTOR BASED QCQP SOLVER + +We recall from the previous section that, to approximately solve a quadratic subproblem in ${ \mathrm { { S T R } } } _ { \mathrm { { f r e e } } }$ , Lanczos method requires $\tilde { \mathcal { O } } ( \operatorname* { m i n } \{ n / \epsilon ^ { 0 . 2 5 } , 1 / \bar { \epsilon } ^ { 1 . 2 5 } \} )$ stochastic Hessian-vector product evaluations. In this section, we propose a faster QCQP solver with an $\tilde { \mathcal { O } } ( \operatorname* { m i n } \{ n + n ^ { 0 . 7 5 } / \epsilon ^ { 0 . 2 5 } , 1 / \epsilon \} )$ complexity. Replacing Lanczos method with this QCQP solver in $\operatorname { S T R } _ { \mathrm { f r e e } }$ results in a faster Hessian-free method, which we refer to as ${ \mathrm { S T R } } _ { \mathrm { f r e e } } +$ . + +# E.1 CONVEX REFORMULATION OF QCQP + +To begin with, we present a known result that is key to achieve faster algorithm than Lanczos method. We summarize this result in Lemma E.1 which shows that the trust region subproblem is equivalent to a convex QCQP. + +Lemma E.1. (Convex Reformulation of QCQP (Flippo & Jansen, 1996; Wang & Xia, 2017)) Denote $\lambda _ { \mathrm { m i n } }$ as the smallest eigenvalue of $\mathbf { H } ^ { k }$ . Let $\mathbf { u } _ { \mathrm { m i n } }$ be a corresponding eigenvector. W.l.o.g., we assume that $\langle \mathbf { g } ^ { k } , \mathbf { u } _ { \mathrm { m i n } } \rangle \leq 0$ . Let $\mu = \operatorname* { m i n } \{ \lambda _ { \operatorname* { m i n } } , 0 \}$ . Then the QCQP (8) is equivalent to the convex problem + +$$ +\operatorname* { m i n } _ { \substack { \mathbf { h } \in \mathbb { R } ^ { d } , \| \mathbf { h } \| \leq r } } q ^ { k } ( \mathbf { h } ) = \langle \mathbf { g } ^ { k } , \mathbf { h } \rangle + \frac { 1 } { 2 } \langle ( \mathbf { H } ^ { k } - \mu I ) \mathbf { h } , \mathbf { h } \rangle + \frac { 1 } { 2 } \mu r ^ { 2 } +$$ + +in the sense that (8) and (38) have the same minimum function value. Moreover, when $\lambda _ { \operatorname* { m i n } } < 0 ,$ , for any optimal solution of (38), denoted by $\mathbf { h } _ { c } ^ { k }$ , + +$$ +\mathbf { h } _ { c } ^ { k } + \frac { \sqrt { \langle \mathbf { h } _ { c } ^ { k } , \mathbf { u } _ { \mathrm { m i n } } \rangle ^ { 2 } - \| \mathbf { u } _ { \mathrm { m i n } } \| ^ { 2 } \big ( \| \mathbf { h } _ { c } ^ { k } \| ^ { 2 } - r ^ { 2 } \big ) } - \langle \mathbf { h } _ { c } ^ { k } , \mathbf { u } _ { \mathrm { m i n } } \rangle } { \| \mathbf { u } _ { \mathrm { m i n } } \| ^ { 2 } } \mathbf { u } _ { \mathrm { m i n } } +$$ + +is a global minimizer of the original QCQP (8). + +To perform the above reformulation, one needs to compute the exact eigenpair $\left( \lambda _ { \operatorname* { m i n } } , \mathbf { u } _ { \operatorname* { m i n } } \right)$ . Nevertheless, as we shall see, it is sufficient to compute an approximate eigenpair $( \tilde { \lambda } , \tilde { \mathbf { u } } )$ such that + +$$ +\lambda _ { \operatorname* { m i n } } \leq \tilde { \lambda } = \tilde { \mathbf { u } } ^ { T } \mathbf { H } ^ { k } \tilde { \mathbf { u } } \leq \lambda _ { \operatorname* { m i n } } + \tilde { \epsilon } , +$$ + +where ˜ is a target accuracy to be determined later. We note that $\tilde { \epsilon } \leq 2 L _ { 2 }$ w.l.o.g. since $\| \mathbf { H } ^ { k } \| \leq L _ { 2 }$ . With this approximate eigenpair, it remains to solve the following convex problem + +$$ +\operatorname* { m i n } _ { \mathbf { h } \in \mathbb { R } ^ { d } , \| \mathbf { h } \| \leq r } \tilde { q } ^ { k } ( \mathbf { h } ) = \langle \mathbf { g } ^ { k } , \mathbf { h } \rangle + \frac { 1 } { 2 } \langle ( \mathbf { H } ^ { k } - \tilde { \mu } I ) \mathbf { h } , \mathbf { h } \rangle + \frac { 1 } { 2 } \tilde { \mu } r ^ { 2 } +$$ + +where $\tilde { \mu } = \mathrm { m i n } \{ 0 , \tilde { \lambda } - \tilde { \epsilon } \}$ . One can check that the problem (41) well approximates (38). + +Corollary E.1. Let $q _ { * } ^ { k }$ and $\tilde { q } _ { * } ^ { k }$ be the minimum function value of (38) and (41), respectively. Assume $\lambda _ { \operatorname* { m i n } } \leq \tilde { \lambda } = \tilde { \mathbf { u } } ^ { T } \mathbf { H } ^ { k } \tilde { \mathbf { u } } \leq \lambda _ { \operatorname* { m i n } } + \tilde { \epsilon } .$ . Then + +$$ +| q _ { * } ^ { k } - \tilde { q } _ { * } ^ { k } | \leq \tilde { \epsilon } r ^ { 2 } . +$$ + +We note that the above convex reformulation approach divides an indefinite QCQP into two subproblems: (i) computation of an approximate eigenpair $( \tilde { \lambda } , \tilde { \mathbf { u } } )$ ; (ii) solving the convex problem (41). As we shall see, by exploiting the finite-sum structure of the Hessian $\mathbf { H } ^ { k }$ , these two subproblems can be efficiently solved. We treat these two subproblems in the following two subsections, respectively. + +# E.2 FINDING THE SMALLEST EIGENVECTOR + +To find a unit vector that satisfies requirement (40), we resort to the AppxPCA method (Allen-Zhu & Li, 2016), which first finds an approximate eigenvalue $\lambda = \lambda _ { \operatorname* { m i n } } - \tilde { \epsilon }$ via binary search and then applies Power method to the positive definite matrix $( \mathbf { H } ^ { k } - \lambda I ) ^ { - 1 }$ for a logarithmic number of iterations. Computing $( \mathbf { H } ^ { k } - \lambda I ) ^ { - 1 } \mathbf { v }$ for any vector $\mathbf { v }$ is equivalent to solving the $\tilde { \epsilon }$ -strongly convex problem (Allen-Zhu & Li, 2018) + +$$ +\operatorname* { m i n } _ { \mathbf { u } } \phi ^ { k } ( \mathbf { u } ) : = \frac { 1 } { 2 } \mathbf { u } ^ { T } ( \mathbf { H } ^ { k } - \lambda I ) \mathbf { u } - \langle \mathbf { v } , \mathbf { u } \rangle +$$ + +We note that $\begin{array} { r } { \mathbf { H } ^ { k } = \frac { 1 } { | \cal { S } | } \sum _ { i \in \cal { S } } \nabla ^ { 2 } f _ { i } ( \mathbf { x } ^ { k } ) } \end{array}$ . Specifically, in $\mathrm { { S T R } _ { \mathrm { { f r e e } } } }$ , either $| S | = n$ (i.e., $\mathbf { H } ^ { k }$ is the full Hessian) or $| S | = \tilde { \mathcal { O } } ( L _ { 1 } ^ { 2 } / ( L _ { 2 } \epsilon ) )$ by Lemma 4.1. Therefore, $\phi ^ { k } ( \cdot )$ can be expressed as sum of non-convex functions + +$$ +\phi ^ { k } ( \mathbf { u } ) = \frac { 1 } { | S | } \sum _ { i \in S } \phi _ { i } ^ { k } ( \mathbf { u } ) = \frac { 1 } { | S | } \sum _ { i \in S } \left( \frac { 1 } { 2 } \mathbf { u } ^ { T } ( \nabla ^ { 2 } f _ { i } ( \mathbf { x } ^ { k } ) - \lambda I ) \mathbf { u } - \langle \mathbf { v } , \mathbf { u } \rangle \right) . +$$ + +By observing that each $\phi _ { i } ^ { k }$ is non-convex and has $\left( 4 L _ { 2 } \right)$ -Lipschitz gradient, we can use KatyushaXS (Allen-Zhu, 2018a) to solve problem (43) in $\tilde { \mathcal { O } } ( | S | + | S | ^ { 3 / 4 } \sqrt { L _ { 2 } / \tilde { \epsilon } } )$ stochastic Hessianvector product (i.e., $\nabla ^ { 2 } f _ { i } ( \mathbf { x } ^ { k } ) \mathbf { u } )$ evaluations. The following result is taken from (Agarwal et al., 2017, Section G.3), which gives the overall computation complexity of AppxPCA. + +# Algorithm 9 Fast QCQP Solver + +Input: $\mathbf { H } ^ { k } , \mathbf { g } ^ { k } , r , \tilde { \epsilon } , \tilde { \epsilon } _ { 1 }$ + +1: Use AppxPCA to find $( \tilde { \lambda } , \tilde { \mathbf { u } } )$ satisfying (40), in which the matrix inverse is solved by KatyushaXS; +2: Use KatyushaXW to solve (41) up to accuracy $\tilde { \epsilon } _ { 1 }$ , i.e., find a vector $\tilde { \mathbf { h } }$ such that $\tilde { q } ^ { k } ( \mathbf { h } ) - \tilde { q } _ { * } ^ { k } \leq \tilde { \epsilon } _ { 1 }$ with high probability; +3: Return $\tilde { \mathbf { h } } + ( \sqrt { \langle \tilde { \mathbf { h } } , \tilde { \mathbf { u } } \rangle ^ { 2 } - \| \tilde { \mathbf { u } } \| ^ { 2 } ( \| \tilde { \mathbf { h } } \| ^ { 2 } - r ^ { 2 } ) } - \langle \tilde { \mathbf { h } } , \tilde { \mathbf { u } } \rangle ) \tilde { \mathbf { u } } / \| \tilde { \mathbf { u } } \| ^ { 2 } .$ + +# Algorithm 10 STRfree+ + +1: In the same setting as MetaAlgorithm 7, +2: construct gradient estimator $\mathbf { g } ^ { k }$ by Estimator 4; +3: construct Hessian estimator $\mathbf { H } ^ { k }$ by +4: Option I: $\mathbf { H } ^ { k } : = \nabla ^ { 2 } F ( \mathbf { x } ^ { k } )$ ; +5: Option II: Draw $s$ samples indexed by $\mathcal { H }$ and let $\mathbf { H } ^ { k } : = \nabla ^ { 2 } f ( \mathbf { x } ^ { k } ; \mathcal { H } )$ ; +6: use Algorithm 9 to solve QCQP subproblems. + +Lemma E.2. Let $\begin{array} { r } { \mathbf { H } ^ { k } = \frac { 1 } { | \mathcal { S } | } \sum _ { i \in \cal S } \nabla ^ { 2 } f _ { i } ( \mathbf { x } ^ { k } ) \in \mathbb { R } ^ { d \times d } } \end{array}$ , where $\| \nabla ^ { 2 } f _ { i } ( \mathbf { x } ^ { k } ) \| \leq L _ { 2 }$ . With probability at least $1 - p ,$ , AppxPCA produces a unit vector u satisfying $\mathbf { u } ^ { T } \mathbf { H } ^ { k } \mathbf { u } \leq \lambda _ { \operatorname* { m i n } } + \tilde { \epsilon } .$ . The total stochastic Hessian-vector product oracle complexity is $\tilde { \mathcal { O } } ( | S | + | S | ^ { 3 / 4 } \sqrt { L _ { 2 } / \tilde { \epsilon } } )$ . + +# E.3 SOLVING THE CONVEX QCQP + +In what follows, we show that the convex problem (41) can be solved efficiently. We first observe that problem (41) has a finite-sum structure and can be rewritten as an unconstrained problem of the form + +$$ +\operatorname* { m i n } _ { \mathbf { h \in \mathbb { R } } ^ { d } } \frac { 1 } { | \mathcal { S } | } \sum _ { i \in \mathcal { S } } \tilde { q } _ { i } ^ { k } ( \mathbf { h } ) + \Psi ( \mathbf { h } ) = \frac { 1 } { | \mathcal { S } | } \sum _ { i \in \mathcal { S } } \Big ( \langle \mathbf { g } ^ { k } , \mathbf { h } \rangle + \frac { 1 } { 2 } \langle ( \nabla ^ { 2 } f _ { i } ( \mathbf { x } ^ { k } ) - \tilde { \mu } I ) \mathbf { h } , \mathbf { h } \rangle + \frac { 1 } { 2 } \tilde { \mu } r ^ { 2 } \Big ) + \Psi ( \mathbf { h } ) , +$$ + +where $\Psi ( { \bf h } ) = 0$ if $\| \mathbf { h } \| \leq r$ , otherwise $\Psi ( \mathbf { h } ) \ = \ + \infty$ . We note that each $\tilde { q } _ { i } ^ { k } ( \mathbf { h } )$ in (45) has $\left( 4 L _ { 2 } \right)$ -Lipschitz continuous gradient since $\| \nabla ^ { 2 } f _ { i } ( \mathbf { x } ^ { k } ) \| \leq L _ { 2 }$ and $\tilde { \epsilon } \leq 2 L _ { 2 }$ . Therefore, we can use KatyushaX $\mathrm { w }$ (Allen-Zhu, 2018a) to solve (45). By (Allen-Zhu, 2018a, Theorem 4.6), KatyushaXW finds a point h such that $\mathbb { E } [ \tilde { q } ^ { k } ( \mathbf { h } ) - \tilde { q } _ { * } ^ { k } ] \leq \tilde { \epsilon } _ { 1 }$ using $\tilde { \mathcal { O } } ( | S | + | S | ^ { 3 / 4 } \sqrt { L _ { 2 } } \cdot r / \sqrt { \tilde { \epsilon } _ { 1 } } )$ stochastic Hessianvector products, where $\tilde { \epsilon } _ { 1 }$ is the target accuracy to be determined later. + +# E.4 PUTTING IT ALL TOGETHER + +The complete procedure of our fast QCQP solver is summarized in Algorithm 9. Combining all the above results and setting $r = \sqrt { \epsilon / L _ { 2 } }$ , $\tilde { \epsilon } = \sqrt { \epsilon L _ { 2 } } / 2$ , and $\tilde { \epsilon } _ { 1 } = \tilde { \epsilon } r ^ { 2 }$ , one can find an approximate solution to QCQP (8) satisfying requirement (24) in ${ \tilde { \mathcal { O } } } ( | S | + | S | ^ { 3 / 4 } L _ { 2 } ^ { 0 . 2 5 } / \epsilon ^ { 0 . 2 5 } )$ stochastic Hessianvector product evaluations. By replacing Lanczos method with this solver in $\operatorname { S T R } _ { \mathrm { f r e e } }$ , we derive a new Hessian-free method called ${ \mathrm { S T R } } _ { \mathrm { f r e e } } +$ , which is summarized in Algorithm 10. The following theorem establishes the overall runtime complexity of ${ \mathrm { S T R } } _ { \mathrm { f r e e } ^ { + } }$ for finding an $\epsilon$ -SOSP. + +Theorem E.1. Consider Algorithm 10 for solving problem (1). Let $\zeta = 1 / 3$ , $r = \sqrt { \epsilon / L _ { 2 } }$ , $K =$ $4 \sqrt { L _ { 2 } } \Delta / \epsilon ^ { 1 . 5 }$ , $\begin{array} { r } { T = \frac { 3 } { 2 } \mathrm { { l o g } } ( 2 / \delta ) } \end{array}$ , $c _ { 1 } = 6 0 0$ , $c _ { 2 } = 5 0 0 \mathrm { \Omega }$ , and 32L21L2 log(4d/δ). The hyper-parameters√ in Estimator 4 are set to the same values as those in Lemma 4.2. Besides, let $\tilde { \epsilon } = \sqrt { \epsilon L _ { 2 } } / 2$ and $\tilde { \epsilon } _ { 1 } = \tilde { \epsilon } r ^ { 2 }$ in Algorithm 9. To find an $O ( \epsilon ) { - } S O S P ~ w . p$ . at least $1 - \delta$ , the runtime complexity is $\tilde { \mathcal { O } } ( d \mathrm { m i n } \{ n / \epsilon ^ { 1 . 5 } + n ^ { 0 . 7 5 } / \epsilon ^ { 1 . 7 5 } , 1 / \epsilon ^ { 2 . 5 } + \sqrt { n } / \epsilon ^ { 2 } \} \mathrm { l o g } ( 1 / \delta ) )$ . + +Proof. The proof directly follows from that in Sec. D. + +We compare the runtime complexity of $\operatorname { S T R } _ { \mathrm { f r e e } }$ and ${ \mathrm { S T R } } _ { \mathrm { f r e e } } +$ with existing Hessian free methods in Table 2. One can see that $\mathrm { { S T R } _ { \mathrm { { f r e e } } } }$ strictly outperforms Hessian-free Cubic. Besides, $\mathrm { { S T R } _ { \mathrm { { f r e e } } } }$ outperforms Fast-Cubic if $n \ge \Omega ( 1 / \epsilon ^ { 4 / 3 } )$ , which is a mild condition for large-scale problems in the moderate accuracy case. ${ \mathrm { S T R } } _ { \mathrm { f r e e } ^ { + } }$ strictly outperforms both Hessian-free Cubic and Fast-Cubic. We note that the runtime analyses in (Tripuraneni et al., 2018; Zhou & Gu, 2019) rely on an additional assumption which states that for all $\mathbf { x }$ , with probability 1, + +![](images/95ff5517569e6471795a7c5ec47d8ab253bfdb1a9cc4c1823c6362cb6ef03d6f.jpg) +Figure 4: Comparison of gradient norm on both the non-convex logistic regression and nonlinear least square problems. + +$$ +\begin{array} { r } { \| \nabla f _ { i } ( \mathbf { x } ) - \nabla F ( \mathbf { x } ) \| \leq \sigma . } \end{array} +$$ + +Under this additional assumption, one can use the same argument as in Section B.2 and B.3 to prove that ${ \mathrm { { S T R } } } _ { \mathrm { { f r e e } } }$ achieves a runtime complexity of $\tilde { \mathcal { O } } ( d \operatorname* { m i n } \{ n / \epsilon ^ { 1 . 7 5 } , n ^ { 0 . 5 } / \epsilon ^ { 2 } + 1 / \epsilon ^ { 2 . 7 5 } , 1 / \epsilon ^ { 3 } \} ) ^ { 1 }$ . Similarly, the runtime complexity of ${ \mathrm { S T R } } _ { \mathrm { f r e e } ^ { + } }$ would be $\tilde { \mathcal { O } } ( d \operatorname* { m i n } \{ n / \epsilon ^ { 1 . 5 } + n ^ { 0 . 7 5 } / \epsilon ^ { 1 . 7 5 } , 1 / \epsilon ^ { 2 . 5 } +$ $\sqrt { n } / \epsilon ^ { 2 } , \dot { 1 } / \epsilon ^ { 3 } \} ,$ ). In this sense, both ${ \mathrm { { S T R } } } _ { \mathrm { { f r e e } } }$ and ${ \mathrm { S T R } } _ { \mathrm { f r e e } } +$ outperform Stochastic Cubic and $\mathrm { S R V R C _ { \mathrm { f r e e } } }$ . + +Table 3: Descriptions of the five testing datasets. + +
#sample#feature#sample#feature
a9a32.561123w8a49,749300
ijcnn49,99022phishing7,60468
codrna28.3058mnist60,000784
epsilon40,0002,000
+ +# F ADDITIONAL EXPERIMENTAL RESULTS + +# F.1 MORE EXPERIMENTAL DETAILS + +Descriptions of Testing Datasets. We briefly introduce the seven testing datasets in the manuscript. Among them, six datasets are provided by the LibSVM website2, including (a9a, ijcnn, codrna, phishing, w8a and epsilon). The detailed information is summarized in Table 3. We can observe that these datasets are different from each other in feature dimension, training samples, etc. + +Experimental Settings. In the manuscript, following SVRC (Zhou et al., 2018c) and LiteSVRC (Zhou et al., 2018b), we select hyper parameters from a set, namely $s _ { 1 }$ from $\{ 0 . 2 n , 0 . 6 n , n \}$ $s _ { 2 }$ from $\lbrace 0 . 0 1 n , 0 . 1 n , 0 . 2 n \rbrace$ , $p _ { 1 }$ and $p _ { 2 }$ from $\{ 0 . 0 1 n ^ { 0 . 5 } , 0 . 0 5 n ^ { 0 . 5 } , 0 . 1 n ^ { 0 . 5 } \}$ . For the Hessian estimation at the beginning of each $p _ { 2 }$ iterations, we use full Hessian. Similarly, for the gradient estimation at the beginning of each $p _ { 1 }$ iterations, we adopt the full gradient as the gradient estimation. + +Memory Analysis. SVRC, Lite-SVRC, and our method need to store the previous and current gradient and Hessian and thus their memory complexity is $2 ( d ^ { 2 } + d )$ . TR, CR (ARC) and SCR need to compute current and Hessian and thus has complexity $d ^ { 2 } + d$ . So these memory is of the same order but our method is much faster than TR, CR and SCR both validated by theory and experiments. + +# F.2 MORE EXPERIMENTS + +Here we give more experimental results on the gradient norm v.s. the algorithm running time and the Hessian sample complexity. Due to the space limit, in the manuscript we only provide the gradient-norm related results on the codrna dataset. Here we provide results of a9a and ijcnn datasets in Figure 4. One can observe that on both the logistic regression with non-convex regularizer and the nonlinear least square problems, the proposed algorithm always shows sharper convergence behavior in terms of both the running time and the Hessian sample complexity. These observations are consistent with the results in Figure 2 in the manuscript. All these results demonstrate the high efficiency of our proposed algorithm and also confirm our theoretical implication. \ No newline at end of file diff --git a/parse/train/HkxJpnVtPr/HkxJpnVtPr_content_list.json b/parse/train/HkxJpnVtPr/HkxJpnVtPr_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..e876000cbd99c2ab8453c8e9d037a04a8c5b1d44 --- /dev/null +++ b/parse/train/HkxJpnVtPr/HkxJpnVtPr_content_list.json @@ -0,0 +1,3744 @@ +[ + { + "type": "text", + "text": "A STOCHASTIC TRUST REGION METHOD FOR NONCONVEX MINIMIZATION ", + "text_level": 1, + "bbox": [ + 174, + 98, + 826, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 171, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 234, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We target the problem of finding a local minimum in non-convex finite-sum minimization. Towards this goal, we first prove that the trust region method with inexact gradient and Hessian estimation can achieve a convergence rate of order $\\mathcal { O } ( 1 / k ^ { 2 / 3 } )$ as long as those differential estimations are sufficiently accurate. Combining such result with a novel Hessian estimator, we propose a sample-efficient√ stochastic trust region (STR) algorithm which finds an $( \\epsilon , \\sqrt { \\epsilon } )$ -approximate local minimum within $\\tilde { \\mathcal { O } } ( \\sqrt { n } / \\epsilon ^ { 1 . 5 } )$ stochastic Hessian oracle queries. This improves the state-of-the-art result by a factor of $\\mathcal { O } ( n ^ { 1 / 6 } )$ . Finally, we also develop Hessian-free STR algorithms which achieve the lowest runtime complexity. Experiments verify theoretical conclusions and the efficiency of the proposed algorithms. ", + "bbox": [ + 233, + 266, + 766, + 411 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 436, + 336, + 453 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We consider the following finite-sum non-convex minimization problem ", + "bbox": [ + 173, + 468, + 647, + 483 + ], + "page_idx": 0 + }, + { + "type": "equation", + "img_path": "images/5bc941f3b76f7588e96135223fb099d13cb31a550c3fcc643fd514ba8ede29cb.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\mathbf { x } \\in \\mathbb { R } ^ { d } } F ( \\mathbf { x } ) = { \\frac { 1 } { n } } \\sum _ { i = 1 } ^ { n } f _ { i } ( \\mathbf { x } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 397, + 488, + 598, + 521 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "where each (non-convex) component function $f _ { i } : \\mathbb { R } ^ { d } \\mathbb { R }$ is assumed to have $L _ { 1 }$ -Lipschitz continuous gradient and $L _ { 2 }$ -Lipschitz continuous Hessian. Since first-order stationary points could be saddle points with inferior generalization performance (Dauphin et al., 2014), in this work we are√ particularly interested in computing $( \\epsilon , \\sqrt { \\epsilon } )$ -approximate second-order stationary points, $\\epsilon$ -SOSP: ", + "bbox": [ + 174, + 529, + 825, + 585 + ], + "page_idx": 0 + }, + { + "type": "equation", + "img_path": "images/e51134a96f8117ef906583db50c1c35f049a30e8b883be7e1e2aa53e9c2854bd.jpg", + "text": "$$\n\\begin{array} { r } { \\| \\nabla F ( { \\bf x } _ { \\epsilon } ) \\| \\le \\epsilon \\quad \\mathrm { a n d } \\quad \\nabla ^ { 2 } F ( { \\bf x } _ { \\epsilon } ) \\succcurlyeq - \\sqrt { L _ { 2 } \\epsilon } { \\bf I } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 343, + 592, + 653, + 612 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "To find a local minimum of problem (1), the cubic regularization approach (Nesterov & Polyak, 2006) and the trust region algorithm (Conn et al., 2000; Curtis et al., 2017) are two classical methods. Specifically, cubic regularization forms a cubic surrogate function for the objective $F ( \\mathbf { x } )$ by adding a third-order regularization term to the second-order Taylor expansion, and minimizes it iteratively. Such a method is proved to achieve an $\\mathcal { O } ( 1 / k ^ { 2 / 3 } )$ global convergence rate and thus needs $\\mathcal { O } ( n / \\epsilon ^ { 1 . 5 } )$ stochastic first- and second-order oracle queries, namely the evaluation number of stochastic gradient and Hessian, to achieve a point that satisfies (2). On the other hand, trust region algorithms estimate the objective with its second-order Taylor expansion but minimize it only within a local region. Recently, Curtis et al. (2017) proposes a trust region variant to achieve the same convergence rate as the cubic regularization approach. But both methods require computing full gradients and Hessians of $F ( \\mathbf { x } )$ and thus suffer from high computational cost in large-scale problems. ", + "bbox": [ + 173, + 618, + 826, + 773 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "To avoid costly exact differential evaluations, many works explore the finite-sum structure of problem (1) and develop stochastic cubic regularization approaches. Both Kohler & Lucchi (2017b) and Xu et al. (2017) propose to directly subsample the gradient and Hessian in the cubic surrogate function, and achieve $\\tilde { \\mathcal { O } } ( 1 / \\epsilon ^ { 3 . 5 } )$ and $\\tilde { \\mathcal { O } } ( 1 / \\epsilon ^ { 2 . 5 } )$ stochastic first- and second-order oracle complexities respectively. By plugging a stochastic variance reduced estimator (Johnson & Zhang, 2013) and the Hessian tracking technique (Gower et al., 2018) into the gradient and Hessian estimation, the approach in (Zhou et al., 2018a) improves both the stochastic first- and second-order oracle complexities to $\\tilde { \\mathcal { O } } ( n ^ { 0 . 8 } / \\epsilon ^ { 1 . 5 } )$ . Recently, Zhang et al. (2018) and Zhou et al. (2018b) develop more efficient stochastic cubic regularization variants, which further reduce the stochastic second-order oracle complexity to $\\tilde { \\mathcal { O } } ( n ^ { 2 / 3 } \\bar { / } \\epsilon ^ { 1 . 5 } )$ at the cost of increasing the stochastic first-order oracle complexity to ${ \\tilde { \\mathcal { O } } } ( n ^ { 2 \\bar { / } 3 } / \\epsilon ^ { 2 . \\bar { 5 } } )$ . ", + "bbox": [ + 173, + 779, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "table", + "img_path": "images/641aadcafe1f9fc794a90c86245d00127cccfe259998bf5b8664df3dd144c614.jpg", + "table_caption": [ + "Table 1: Stochastic first- and second-order oracle complexities, SFO and SSO for short respectively, of the proposed STR approaches and other state-of-the-art methods. When SSO is prioritized, our $\\mathrm { { S T R } _ { 1 } }$ has strictly better complexity than both SCR and Lite-SVRC. When SFO and SSO are treated equally, $\\mathrm { { S T R _ { 2 } } }$ improves the existing result in SVRC. " + ], + "table_footnote": [], + "table_body": "
AlgorithmSFOSSO
TR (Conn et al., 2000; Curtis et al., 2017)CR (Nesterov & Polyak, 2006)SCR (Kohler & Lucchi, 2017a)SVRC (Zhou et al., 2018c)Lite-SVRC (Zhou et al., 2018b)0(共n01.5013.52.5O n共0(1.50（1（.5）6国1.5
STRi (this paper)STR2 (this paper)O(min{n1.5，1.5O(min{1，1.5
", + "bbox": [ + 209, + 162, + 789, + 296 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Contributions: In this paper we propose and exploit a formulation in which we make explicit control of the step size in the trust region method. This idea is leveraged to develop two efficient stochastic trust region (STR) approaches. We tailor our methods to achieve state-of-the-art oracle complexities under the following two measurements: (i) the stochastic second-order oracle complexity is prioritized; (ii) the stochastic first- and second-order oracle complexities are treated equally. Specifically, in Setting (i), our method $\\mathrm { { S T R } _ { 1 } }$ employs a newly proposed estimator to approximate the Hessian and adopts the estimator in (Fang et al., 2018) for gradient approximation. Our novel Hessian estimator maintains an accurate second-order differential approximation with lower amortized oracle complexity. In this way, $\\operatorname { S T R } _ { 1 }$ achieves $\\tilde { \\mathcal { O } } ( \\operatorname* { m i n } \\{ 1 / \\epsilon ^ { 2 } , \\sqrt { n } / \\epsilon ^ { 1 . 5 } \\} )$ stochastic second-order oracle complexity. This is lower than existing results for solving problem (1). In Setting (ii), our method $\\mathrm { { S T R } _ { 2 } }$ substitutes the gradient estimator in $\\mathrm { { S T R } _ { 1 } }$ with one that integrates stochastic gradient and Hessian together to maintain an accurate gradient approximation. As a result, $\\mathrm { { S T R _ { 2 } } }$ achieves convergence in ${ \\tilde { \\mathcal { O } } } ( n ^ { 3 / 4 } / \\epsilon ^ { 1 . 5 } )$ overall stochastic first- and second-order oracle queries. Finally, based on STR, we further develop Hessian-free STR algorithms, namely $\\mathrm { { S T R } _ { \\mathrm { { f r e e } } } }$ and ${ \\mathrm { S T R } } _ { \\mathrm { f r e e } } +$ , which outperform existing Hessian-free algorithms theoretically. ", + "bbox": [ + 173, + 304, + 825, + 516 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1.1 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 529, + 331, + 542 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Computing local minimum to a non-convex optimization problem is gaining considerable amount of attentions in recent years. Both cubic regularization (CR) approaches (Nesterov & Polyak, 2006) and trust region (TR) algorithms (Conn et al., 2000; Curtis et al., 2017) can escape saddle points and find a local minimum by iterating the variable along the direction related to the eigenvector of the Hessian with the most negative eigenvalue. As the CR heavily depends on the regularization parameter for the cubic term, Cartis et al. (2011) propose an adaptive cubic regularization (ARC) approach to boost the efficiency by adaptively tuning the regularization parameter according to the current objective decrease. Noting the high cost of full gradient and Hessian computation in ARC, sub-sampled cubic regularization (SCR) (Kohler & Lucchi, 2017a) is developed for sampling partial data points to estimate the full gradient and Hessian. Recently, by exploring the finite-sum structure of the target problem, many works incorporate variance reduction technique (Johnson & Zhang, 2013) into CR and propose stochastic variance-reduced methods. For example, Zhou et al. (2018c) propose stochastic variance-reduced cubic (SVRC) in which they integrate the stochastic variance-reduced gradient estimator (Johnson & Zhang, 2013) and the Hessian tracking technique (Gower et al., 2018) with CR. Such a method is proved to be at least $\\mathcal { O } ( n ^ { 1 / 5 } )$ faster than CR and TR. Then Zhou et al. (2018b) use adaptive gradient batch size and constant Hessian batch size, and develop Lite-SVRC to further reduce the stochastic second-order oracle ${ \\tilde { \\mathcal { O } } } ( n ^ { 4 / 5 } / \\epsilon ^ { 1 . 5 } )$ of SVRC to $\\tilde { \\mathcal { O } } ( n ^ { 2 / 3 } / \\epsilon ^ { 1 . 5 } )$ at the cost of higher gradient computation cost. Similarly, except turning the gradient batch size, Zhang et al. (2018) further adaptively sample a certain number of data points to estimate the Hessian and prove the proposed method to have the same stochastic second-order oracle complexity as Lite-SVRC. ", + "bbox": [ + 173, + 545, + 826, + 825 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 PRELIMINARY ", + "text_level": 1, + "bbox": [ + 174, + 838, + 323, + 853 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Notation. We use $\\| \\mathbf { v } \\|$ to denote the Euclidean norm of vector $\\mathbf { v }$ and use $\\| \\mathbf { A } \\|$ to denote the spectral norm of matrix A. Let $s$ be the set of component indices. We define the minibatch average of component functions by $\\begin{array} { r } { f ( \\mathbf { x } ; \\mathcal { S } ) \\ \\stackrel { \\mathrm { d e f } } { = } \\ \\frac { 1 } { | \\mathcal { S } | } \\sum _ { i \\in \\mathcal { S } } f _ { i } ( \\mathbf { x } ) } \\end{array}$ . Then we specify the assumptions that are necessary to the analysis of our methods. ", + "bbox": [ + 174, + 859, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "MetaAlgorithm 1 Inexact Trust Region Method ", + "text_level": 1, + "bbox": [ + 174, + 103, + 493, + 118 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Input: initial point $\\mathbf { x } ^ { 0 }$ , step size $r$ , number of iterations $K$ , construction of differential estimators $\\mathbf { g } ^ { k }$ and $\\mathbf { H } ^ { k }$ \n1: for $k = 0$ to $K - 1$ do \n2: Compute $\\mathbf { h } ^ { k }$ and $\\lambda ^ { k }$ by solving (8); \n3: $\\mathbf { x } ^ { k + 1 } : = \\mathbf { x } ^ { k } + \\mathbf { h } ^ { k }$ ; \n4: if $\\lambda ^ { k } \\le 3 \\sqrt { \\epsilon / L _ { 2 } }$ then \n5: Output ${ \\bf x } _ { \\epsilon } = { \\bf x } ^ { k + 1 }$ ; \n6: end if \n7: end for ", + "bbox": [ + 174, + 125, + 823, + 252 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Assumption 2.1. $F$ is bounded from below and its global optimal is achieved at $\\mathbf { x } ^ { * }$ . We denote $\\Delta = \\bar { F ( \\mathbf { x } ^ { 0 } ) } - F ( \\mathbf { x } ^ { * } )$ . ", + "bbox": [ + 173, + 263, + 821, + 292 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Assumption 2.2. Each $f _ { i } : \\mathbb { R } ^ { d } \\mathbb { R }$ has $L _ { 1 }$ -Lipschitz continuous gradient: for any $\\mathbf { x } , \\mathbf { y } \\in \\mathbb { R } ^ { d }$ ", + "bbox": [ + 173, + 294, + 794, + 310 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/fd6bffcd4ac329e3d2d08df3cf883bd01f854e8785247937cef9f572b0103e89.jpg", + "text": "$$\n\\| \\nabla f _ { i } ( \\mathbf { x } ) - \\nabla f _ { i } ( \\mathbf { y } ) \\| \\leq L _ { 1 } \\| \\mathbf { x } - \\mathbf { y } \\| .\n$$", + "text_format": "latex", + "bbox": [ + 377, + 313, + 617, + 330 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Assumption 2.3. Each $f _ { i } : \\mathbb { R } ^ { d } \\mathbb { R }$ has $L _ { 2 }$ -Lipschitz continuous Hessian: for any $\\mathbf { x } , \\mathbf { y } \\in \\mathbb { R } ^ { d }$ ", + "bbox": [ + 176, + 333, + 790, + 348 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/f045f817af869389a938a90ea9e9cb9a5faa86aaab790f25f9fff951f42e8cbb.jpg", + "text": "$$\n\\| \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ) - \\nabla ^ { 2 } f _ { i } ( \\mathbf { y } ) \\| \\leq L _ { 2 } \\| \\mathbf { x } - \\mathbf { y } \\| .\n$$", + "text_format": "latex", + "bbox": [ + 370, + 349, + 625, + 368 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 TRUST REGION METHOD ", + "text_level": 1, + "bbox": [ + 174, + 382, + 390, + 397 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Here we briefly introduce the trust region method (Conn et al., 2000). In each step, it first solves the Quadratic Constraint Quadratic Program (QCQP) defined as ", + "bbox": [ + 173, + 409, + 823, + 438 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/0416e2f89ec5860a12a749b0d715f9cdd957cbbbf4133f5ba8eab08267fc6b24.jpg", + "text": "$$\n\\mathbf { h } ^ { k } : = \\operatorname * { a r g m i n } _ { \\mathbf { h } \\in \\mathbb { R } ^ { d } , \\| \\mathbf { h } \\| \\leq r } \\langle \\nabla F ( \\mathbf { x } ^ { k } ) , \\mathbf { h } \\rangle + \\frac { 1 } { 2 } \\langle \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) \\mathbf { h } , \\mathbf { h } \\rangle ,\n$$", + "text_format": "latex", + "bbox": [ + 321, + 440, + 673, + 476 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $r$ is the trust-region radius. Then it updates the new variable as ", + "bbox": [ + 176, + 478, + 630, + 493 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/c4ab9e77bd27ab82705bb5c6b81eae7b31e4a6605a63663107dc70ca7b75f4e0.jpg", + "text": "$$\n\\mathbf { x } ^ { k + 1 } : = \\mathbf { x } ^ { k } + \\mathbf { h } ^ { k } .\n$$", + "text_format": "latex", + "bbox": [ + 436, + 496, + 562, + 513 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Since $\\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } )$ is indefinite, the trust-region subproblem (5) is non-convex. But its global optimizer can be characterized by the following lemma (Corollary 7.2.2 in (Conn et al., 2000)). ", + "bbox": [ + 174, + 518, + 825, + 547 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Lemma 2.1. Any global minimizer of problem (5) satisfies the equation ", + "bbox": [ + 173, + 549, + 647, + 564 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/4a076c3a0ce11ed5344c9d8696446694bae00258e95f126ea3728ffe56ebd0f5.jpg", + "text": "$$\n\\left( \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) + \\lambda \\mathbf { I } \\right) \\mathbf { h } ^ { k } = - \\nabla F ( \\mathbf { x } ^ { k } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 380, + 568, + 616, + 587 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where the dual variable $\\lambda \\geq 0$ should satisfy $\\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) + \\lambda \\mathbf { I } \\succcurlyeq 0$ and $\\lambda ( \\| \\mathbf { h } ^ { k } \\| - r ) = 0 .$ . ", + "bbox": [ + 174, + 590, + 753, + 608 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In particular, the standard QCQP solver returns both the minimizer $\\mathbf { h } ^ { k }$ as well as the corresponding dual variable $\\lambda$ of subproblem (5). In the following section, we first prove that the deterministic trust-region update (5) and (6) converges at the rate of √ $\\mathcal { O } ( 1 / k ^ { 2 / 3 } )$ , much sharper than existing provable convergence rate $\\mathcal { O } ( 1 / \\sqrt { k } )$ (Conn et al., 2000), and then develop a more efficient stochastic trust-region approach. ", + "bbox": [ + 173, + 616, + 826, + 690 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 METHODOLOGY ", + "text_level": 1, + "bbox": [ + 176, + 709, + 341, + 726 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Here we first introduce a general inexact trust region method which is summarized in MetaAlgorithm 1. It accepts inexact gradient estimation $\\mathbf { g } ^ { k }$ and Hessian estimation $\\mathbf { H } ^ { k }$ as input to the QCQP subproblem ", + "bbox": [ + 171, + 739, + 823, + 770 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/383fa79b49e3a62fed484af8e46b537ce0c10629f2ea7bb37f5efe898d0174e2.jpg", + "text": "$$\n\\mathbf { h } ^ { k } : = \\operatorname * { a r g m i n } _ { \\mathbf { h } \\in \\mathbb { R } ^ { d } , \\| \\mathbf { h } \\| \\leq r } \\langle \\mathbf { g } ^ { k } , \\mathbf { h } \\rangle + \\frac { 1 } { 2 } \\langle \\mathbf { H } ^ { k } \\mathbf { h } , \\mathbf { h } \\rangle .\n$$", + "text_format": "latex", + "bbox": [ + 362, + 777, + 633, + 813 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Similar to (5), Lemma 2.1 characterizes the global optimum to problem (8) which can be efficiently solved by Lanczos method (Gould et al., 1999). Assume the dual variable of the minimizer $\\mathbf { h } ^ { k }$ is $\\lambda ^ { \\dot { k } }$ ", + "bbox": [ + 173, + 814, + 823, + 843 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We prove that such inexact trust region method achieves the optimal $\\mathcal { O } ( 1 / k ^ { 2 / 3 } )$ convergence rate when the estimation $\\mathbf { g } ^ { k }$ and $\\mathbf { H } ^ { k }$ at each iteration are sufficiently close to their full (exact) counterparts $\\nabla F ( \\mathbf { x } ^ { k } )$ and $\\nabla ^ { 2 } F ( \\bar { \\mathbf { x } } ^ { k } )$ respectively: ", + "bbox": [ + 173, + 849, + 826, + 893 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/f963400f2268043dadd66165d09d23c7c2c691ca8d928a843ba29b60a2d5e794.jpg", + "text": "$$\n\\| \\mathbf { g } ^ { k } - \\nabla F ( \\mathbf { x } ^ { k } ) \\| \\leq \\frac { \\epsilon } { 6 } , \\| \\mathbf { H } ^ { k } - \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) \\| \\leq \\frac { \\sqrt { \\epsilon L _ { 2 } } } { 3 } .\n$$", + "text_format": "latex", + "bbox": [ + 323, + 896, + 674, + 929 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Algorithm 2 STR1 ", + "text_level": 1, + "bbox": [ + 174, + 103, + 299, + 117 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/ef2c9b6f7a41269eb6b756ffff844c0d4332196f5ba7452e1276d9c1a2b52c00.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
1:for k= 1 to K do 2: 3: Compute hk and Xk by solving (8);Input:initial point x°,step size r,number of iterations K Construct gradient estimator gk by Estimator 4; Construct Hessian estimator Hk by Estimator 3;
", + "bbox": [ + 176, + 123, + 558, + 267 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Such result allows us to derive stochastic trust-region variants with novel differential estimators that are tailored to ensure the optimal convergence rate. We state our formal results in Theorem 3.1, whose proof is deferred to Appendix B.1 due to the space limit. ", + "bbox": [ + 176, + 277, + 823, + 319 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem 3.1 (Main Result). Consider problem (1) under Assumption 2.1-2.3. If the differential estimators $\\mathbf { g } ^ { k }$ and $\\mathbf { H } ^ { k }$ satisfy Eqn. (9) for all $k$ , MetaAlgorithm 1 finds an $\\mathcal { O } ( \\epsilon )$ -SOSP in less than $K = \\mathcal { O } ( \\sqrt { L _ { 2 } } \\Delta / \\epsilon ^ { 1 . 5 } )$ iterations by setting the trust-region radius as $r = \\sqrt { \\epsilon / L _ { 2 } }$ . ", + "bbox": [ + 173, + 323, + 826, + 369 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Remark 3.1. We emphasize that MetaAlgorithm 1 degenerates to the exact trust region method by taking $\\mathbf { g } ^ { k } = \\nabla F ( \\mathbf { x } ^ { k } )$ and $\\mathbf { H } ^ { k } = \\nabla ^ { 2 } F ( \\mathbf { \\bar { x } } ^ { k } )$ . Such result is of its own interest because this is the first proof to show that the vanilla trust region method has the optimal $\\mathcal { O } ( 1 / k ^ { 2 / 3 } )$ convergence rate. Similar rate is achieved by Curtis et al. (2017) but with a much more complicated trust region variant. ", + "bbox": [ + 173, + 372, + 825, + 431 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem 3.1 shows the explicit step size control of the trust region method: Since the dual variable√ satisfies $\\lambda ^ { k } > 3 \\epsilon ^ { 0 . 5 } / \\sqrt { L _ { 2 } } ^ { \\bullet } > 0$ for all but the last iteration, we always find the solution to the trustregion subproblem (8) in the boundary, i.e. $\\| \\mathbf { h } ^ { k } \\| = r$ , according to the complementary condition (15) in Appendix B.1. Such exact step size control property is missing in the cubic-regularization method where the step size is implicitly decided by the cubic regularization parameter. ", + "bbox": [ + 173, + 443, + 825, + 512 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "More importantly, we emphasize that such explicit step size control is crucial to the sample efficiency of our variance reduced differential estimators. The essence of variance reduction is to exploit the correlations between the differentials in consecutive iterations. Intuitively, when two neighboring iterates are close, so are their differentials due to the Lipschitz continuity, and hence a smaller number of samples suffice to maintain the accuracy of the estimators. On the other hand, smaller step size reduces the per-iteration objective decrease which harms the convergence rate of the algorithm (see proof of Theorem 3.1). Therefore, the explicit step size control in trust region method allows us to well trade-off the per-iteration sample complexity and convergence rate, from which we can derive stochastic trust region approaches with the state-of-the-art sample efficiency. In contrast, existing trust region methods change the step size at every iteration according to progress made, which requires loss evaluations that can be as expensive as gradient computations (e.g. the non-convex linear model in Section 7) and is thus prohibitive for large-scale problems. ", + "bbox": [ + 173, + 520, + 825, + 686 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 STOCHASTIC TRUST REGION METHOD: TYPE I ", + "text_level": 1, + "bbox": [ + 174, + 707, + 599, + 723 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Having the inexact trust region method as prototype, we now present our first sample-efficient stochastic trust region method, namely $\\mathrm { { S T R } _ { 1 } }$ , in Algorithm 2 which emphasizes cheaper stochastic second-order oracle complexity. As Theorem 3.1 already guarantees the optimal convergence rate of MetaAlgorithm 1 when the gradient estimator $\\mathbf { g } ^ { k }$ and the Hessian estimator $\\mathbf { H } ^ { k }$ meet the requirement (9), here we focus on constructing such novel differential estimators. Specifically, we first present our Hessian estimator in Estimator 3 and our first gradient estimator in Estimator 4, both of which exploit the trust region radius $r = \\sqrt { \\epsilon / L _ { 2 } }$ to reduce their variances. ", + "bbox": [ + 174, + 738, + 825, + 839 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1 HESSIAN ESTIMATOR ", + "text_level": 1, + "bbox": [ + 176, + 856, + 364, + 869 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Our epoch-wise Hessian estimator $\\mathbf { H } ^ { k }$ is given in Estimator 3, where $p _ { 2 }$ controls the epoch length and $s _ { 2 }$ (and optionally $s _ { 2 } ^ { \\prime }$ ) controls the minibatch size. At the beginning of each epoch, Estimator 3 has two options, designed for different target accuracy: Option I is preferable for the high accuracy ", + "bbox": [ + 176, + 881, + 823, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Estimator 3 Hessian Estimator ", + "text_level": 1, + "bbox": [ + 174, + 103, + 382, + 117 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Input: Epoch length $p _ { 2 }$ , sample size $s _ { 2 }$ , $s _ { 2 } ^ { \\prime }$ (optional) \n1: if $\\mod ( k , p _ { 2 } ) = 0$ then \n2: Option I: $\\diamond$ high accuracy case (small $\\epsilon$ ) \n3: $\\mathbf { H } ^ { k } : = \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } )$ ; \n4: Option II: $\\diamond$ low accuracy case (moderate $\\epsilon$ ) \n5: Draw $s _ { 2 } ^ { \\prime }$ samples indexed by $\\mathcal { H } ^ { \\prime }$ and let $\\mathbf { H } ^ { k } : = \\nabla ^ { 2 } f ( \\mathbf { x } ^ { k } ; \\mathcal { H } ^ { \\prime } )$ ; \n6: else \n7: Draw $s _ { 2 }$ samples indexed by $\\mathcal { H }$ and let $\\mathbf { H } ^ { k } : = \\nabla ^ { 2 } f ( \\mathbf { x } ^ { k } ; \\mathcal { H } ) - \\nabla ^ { 2 } f ( \\mathbf { x } ^ { k - 1 } ; \\mathcal { H } ) + \\mathbf { H } ^ { k - 1 }$ ; \n8: end if ", + "bbox": [ + 176, + 121, + 803, + 247 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Estimator 4 Gradient Estimator: Case (1) ", + "text_level": 1, + "bbox": [ + 178, + 257, + 450, + 272 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "1: if $\\mod ( k , p _ { 1 } ) = 0$ then \n2: $\\mathbf { g } ^ { k } : = \\nabla F ( \\mathbf { x } ^ { k } )$ ; \n3: else \n4: Draw $s _ { 1 }$ samples indexed by $\\mathcal { G }$ and $\\mathbf { g } ^ { k } = \\nabla f ( \\mathbf { x } ^ { k } ; \\mathcal { G } ) - \\nabla f ( \\mathbf { x } ^ { k - 1 } ; \\mathcal { G } ) + \\mathbf { g } ^ { k - 1 } ;$ ; \n5: end if ", + "bbox": [ + 179, + 276, + 741, + 347 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "case $( \\epsilon < \\mathcal { O } ( 1 / n ) )$ where we compute the full Hessian to avoid approximation error, and Option II is designed for the moderate accuracy case $( \\epsilon > \\mathcal { O } ( 1 / n ) )$ where we only need an approximate Hessian estimator. Then, $p _ { 2 }$ iterations follow with $\\mathbf { H } ^ { k }$ defined in a recurrent manner. These recurrent estimators exist for the first-order case (Nguyen et al., 2017; Fang et al., 2018), but their bound only holds under the vector $\\ell _ { 2 }$ norm. Here we generalize them into Hessian estimation with matrix spectral norm bound. ", + "bbox": [ + 173, + 358, + 825, + 443 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The following lemma analyzes the amortized stochastic second-order oracle (Hessian) complexity for Algorithm 3 to meet the requirement in Theorem 3.1. As we need an error bound under the spectral norm, we will appeal to the matrix Azuma’s inequality (Tropp, 2012). The proof is deferred to Appendix B.2. ", + "bbox": [ + 173, + 449, + 825, + 505 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Lemma 4.1. Assume Algorithm 2 takes the trust region radius $r = \\sqrt { \\epsilon / L _ { 2 } }$ as in Theorem 3.1. For any $k \\geq 0$ , Estimator 3 produces estimator $\\mathbf { H } ^ { k }$ for the second order differential $\\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } )$ such that $\\| \\mathbf { H } ^ { k } - \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) \\| \\leq \\sqrt { \\epsilon L _ { 2 } } / 3$ with probability at least $1 - \\delta / K _ { 0 }$ if we set (1) $\\boldsymbol { p } _ { 2 } = \\sqrt { n }$ and $s _ { 2 } = 3 2 \\sqrt { n } \\log ( d K _ { 0 } / \\delta )$ in option $I ,$ or (2) $p _ { 2 } = L _ { 1 } / ( 2 \\sqrt { \\epsilon L _ { 2 } } )$ , $s _ { 2 } ^ { \\prime } = 1 6 L _ { 1 } ^ { 2 } / ( \\epsilon L _ { 2 } ) \\log ( d K _ { 0 } / \\delta )$ , and $s _ { 2 } = 3 2 L _ { 1 } / ( \\sqrt { \\epsilon L _ { 2 } } ) \\log ( d K _ { 0 } / \\delta )$ in option $\\boldsymbol { { I I } }$ . Here $K _ { 0 }$ is a constant to be determined later. Consequently the amortized per-iteration stochastic second-order oracle complexity to construct is no more than $\\begin{array} { r } { 2 s _ { 2 } = \\operatorname* { m i n } \\{ 6 4 \\sqrt { n } \\log \\frac { d K _ { 0 } } { \\delta } } \\end{array}$ , $\\begin{array} { r } { \\frac { 6 4 L _ { 1 } } { \\sqrt { \\epsilon L _ { 2 } } } \\log \\frac { d K _ { 0 } } { \\delta } \\biggr \\} } \\end{array}$ $\\mathbf { H } ^ { k }$ ", + "bbox": [ + 173, + 508, + 825, + 619 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.2 GRADIENT ESTIMATOR: CASE (1) ", + "text_level": 1, + "bbox": [ + 176, + 633, + 450, + 648 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "When the stochastic second-order oracle complexity is prioritized, we directly employ the SPIDER gradient estimator to construct $\\mathbf { g } ^ { k }$ (Fang et al., 2018). Similar to the construction for $\\bar { \\mathbf { H } } ^ { k }$ , the estimator $\\bar { \\mathbf { g } } ^ { k }$ is also constructed in an epoch-wise manner as presented in Estimator 4, where $p _ { 1 }$ controls the epoch length and $s _ { 1 }$ controls the minibatch size. ", + "bbox": [ + 174, + 660, + 825, + 715 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We now analyze the stochastic first-order oracle complexity to meet the requirement in Theorem 3.1. Lemma 4.2. Assume Algorithm 2 takes the trust region radius $r = \\sqrt { \\epsilon / L _ { 2 } }$ . Estimator $\\cdot$ produces estimator $\\mathbf { g } ^ { k }$ of the first order differential $\\nabla F ( \\mathbf { x } ^ { k } )$ such that $\\| \\mathbf { g } ^ { k } - \\nabla F ( \\mathbf { x } ^ { k } ) \\| \\leq \\epsilon / 6$ with probability at least $1 - \\delta / K _ { 0 }$ for any $k \\geq 0 ,$ , if we set $\\begin{array} { r } { p _ { 1 } = \\operatorname* { m a x } \\lbrace 1 , \\sqrt { n \\epsilon L _ { 2 } / ( c L _ { 1 } ^ { 2 } \\log \\frac { K _ { 0 } } { \\delta } ) } \\rbrace } \\end{array}$ and $s _ { 1 } = \\operatorname* { m i n } \\{ n , \\sqrt { c n L _ { 1 } ^ { 2 } \\log ( K _ { 0 } / \\delta ) / ( \\epsilon L _ { 2 } ) } \\}$ , where the constant $c = 1 1 5 2$ and $K _ { 0 }$ is a constant to be determined later. Consequently, the amortized per-iteration stochastic first-order oracle complexity to construct $\\mathbf { g } ^ { k } i s \\operatorname* { m i n } \\{ n , \\sqrt { 4 c n L _ { 1 } ^ { 2 } \\log { ( K _ { 0 } / \\delta ) } / ( \\epsilon L _ { 2 } ) } \\}$ . ", + "bbox": [ + 171, + 723, + 826, + 845 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The proof of Lemma 4.2 is similar to the one of Lemma 4.1 and is deferred to Appendix B.3. These two lemmas only guarantee that the differential estimators satisfy the requirement (9) in a single iteration and can be extended to hold for all $k$ by using the union bound with $K _ { 0 } = 2 K$ , where $K$ denotes the number of iterations. Combining such lifted result with Theorem 3.1, we can establish the computational complexity bound as follows. ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Algorithm 5 STR2 ", + "text_level": 1, + "bbox": [ + 174, + 103, + 300, + 117 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Input: initial point $\\mathbf { x } ^ { 0 }$ , step size $r$ , number of iterations $K$ ", + "bbox": [ + 173, + 123, + 562, + 138 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "1: for $k = 1$ to $K$ do \n2: Construct gradient estimator $\\mathbf { g } ^ { k }$ by Estimator 6; \n3: Construct Hessian estimator $\\bar { \\mathbf { H } } ^ { k }$ by Estimator 3; \n4: Compute $\\mathbf { h } ^ { k }$ and $\\lambda ^ { k }$ by solving (8); \n5: $\\mathbf { x } ^ { k + 1 } : = \\mathbf { x } ^ { k } + \\mathbf { h } ^ { k }$ ; \n6: if $\\lambda ^ { k } \\le 3 \\sqrt { \\epsilon / L _ { 2 } }$ then \n7: Output ${ \\bf x } _ { \\epsilon } = { \\bf x } ^ { k + 1 }$ ; \n8: end if \n9: end for ", + "bbox": [ + 178, + 137, + 555, + 265 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Estimator 6 Gradient Estimator: Case (2) ", + "text_level": 1, + "bbox": [ + 176, + 276, + 450, + 291 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "1: if $\\mod ( k , p _ { 1 } ) = 0$ then \n2: Let $\\widetilde \\mathbf { x } : = \\mathbf { x } ^ { k }$ , $\\mathbf { g } ^ { k } : = \\nabla F ( \\tilde { \\mathbf { x } } )$ \n3: else \n4: Draw $s _ { 1 }$ samples indexed by $\\mathcal { G }$ ; \n5: $\\mathbf { g } ^ { k } = \\bar { \\nabla } f ( \\mathbf { x } ^ { k ^ { \\prime } } ; \\mathcal { G } ) - \\nabla f ( \\mathbf { x } ^ { k ^ { \\prime } - 1 } ; \\mathcal { G } ) + \\mathbf { g } ^ { k - 1 } + [ \\nabla ^ { 2 } F ( \\tilde { \\mathbf { x } } ) - \\nabla ^ { 2 } f ( \\tilde { \\mathbf { x } } ; \\mathcal { G } ) ] ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) ;$ \n6: end if ", + "bbox": [ + 179, + 296, + 769, + 381 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Corollary 4.1 (Computational Complexity of $\\mathrm { S T R _ { 1 } }$ ). Assume Algorithm 2 will use Estimator 4 to construct the first-order differential estimator $\\mathbf { g } ^ { k }$ and use Estimator $^ 3$ to construct the second-order differential estimator $\\mathbf { H } ^ { k }$ . To find a $1 2 \\epsilon$ -SOSP with probability at least $1 - \\delta$ , the overall stochastic first-order oracle complexity is O(min{ n L2∆\u000f1.5 , $\\begin{array} { r l } { \\mathcal { O } ( \\operatorname* { m i n } \\{ \\frac { n \\sqrt { L _ { 2 } } \\Delta } { \\epsilon ^ { 1 . 5 } } , \\frac { \\sqrt { n } L _ { 1 } } { \\epsilon ^ { 2 } } \\log ( \\frac { L _ { 2 } \\Delta } { \\delta \\epsilon ^ { 1 . 5 } } ) \\} ) } \\end{array}$ and the overall stochastic secondorder oracle complexity is $\\begin{array} { r l } { \\mathcal { O } ( \\operatorname* { m i n } \\{ \\frac { \\sqrt { n L _ { 2 } } \\Delta } { \\epsilon ^ { 1 . 5 } } , \\frac { L _ { 1 } \\Delta } { \\epsilon ^ { 2 } } \\} \\log ( \\frac { d \\sqrt { L _ { 2 } } \\Delta } { \\delta \\epsilon ^ { 1 . 5 } } ) ) } \\end{array}$ ", + "bbox": [ + 173, + 390, + 826, + 472 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "From Corollary 4.1 we see that $\\tilde { \\mathcal { O } } ( \\operatorname* { m i n } \\{ \\sqrt { n } / \\epsilon ^ { 1 . 5 } , 1 / \\epsilon ^ { 2 } \\} )$ stochastic second-order oracle queries are sufficient for $\\operatorname { S T R } _ { 1 }$ to find an $\\epsilon$ -SOSP which is significantly better than both the subsampled cubic regularization method $\\tilde { \\mathcal { O } } ( 1 / \\epsilon ^ { 2 . 5 } )$ (Kohler & Lucchi, 2017a) and the variance reduction based ones $\\tilde { \\mathcal { O } } \\bar { ( } n ^ { 2 / 3 } / \\epsilon ^ { 1 . 5 } )$ (Zhou et al., 2018b; Zhang et al., 2018). Recently, Zhou & Gu (2019) developed a√ stochastic recursive variance-reduced cubic regularization (SRVRC) method which finds an $( \\epsilon , \\sqrt { \\epsilon } )$ - approximate local minimum with $\\tilde { \\mathcal { O } } ( n / \\epsilon ^ { 1 . 5 } , 1 / \\epsilon ^ { 3 } )$ SFO and $\\tilde { \\mathcal { O } } ( \\sqrt { n } / \\epsilon ^ { 1 . 5 } , 1 / \\epsilon ^ { 2 } )$ SSO. But the result of SRVRC needs to assume stochastic gradient to be bounded, i.e., $\\| \\nabla f _ { i } ( { \\mathbf { x } } ) - \\nabla F ( { \\mathbf { x } } ) \\| \\leq \\sigma$ . With this extra assumption, STR1 enjoys $\\tilde { \\mathcal { O } } ( n / \\epsilon ^ { 1 . 5 } , n / \\epsilon ^ { 2 } , 1 / \\epsilon ^ { 3 } )$ SFO and $\\tilde { \\mathcal { O } } ( \\sqrt { n } / \\epsilon ^ { 1 . 5 } , 1 / \\epsilon ^ { 2 } )$ SSO. Thus, if $1 / { \\epsilon } \\le n \\le 1 / \\bar { \\epsilon } ^ { 2 }$ , STR1 outperforms SRVRC; otherwise they have the same complexities. ", + "bbox": [ + 173, + 483, + 826, + 619 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 STOCHASTIC TRUST REGION METHOD: TYPE II ", + "text_level": 1, + "bbox": [ + 174, + 640, + 607, + 659 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In the above section, we focus on the setting where the stochastic second-order oracle complexity is prioritized over the stochastic first-order oracle complexity. In this setting, $\\mathrm { { S T R } _ { 1 } }$ achieves the stateof-the-art efficiency. In this section, we consider a different complexity measure where first-order and second-order oracle complexities are treated equally and our goal is to minimize the maximum of them. We note that, currently the best result is $\\tilde { \\mathcal { O } } ( n ^ { 4 / 5 } / \\epsilon ^ { 1 . 5 } )$ of the SVRC method (Zhou et al., 2018c). ", + "bbox": [ + 173, + 674, + 826, + 761 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Since the Hessian estimator $\\mathbf { H } ^ { k }$ of $\\operatorname { S T R } _ { 1 }$ already delivers the superior $\\tilde { \\mathcal { O } } ( \\sqrt { n } / \\epsilon ^ { 1 . 5 } )$ stochastic Hessian complexity, in $\\mathrm { { S T R _ { 2 } } }$ (see Algorithm 5), we retain Estimator 3 for second-order differential estimation and use Estimator 6 to further reduce the stochastic gradient complexity. ", + "bbox": [ + 174, + 765, + 825, + 809 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1 GRADIENT ESTIMATOR: CASE (2) ", + "text_level": 1, + "bbox": [ + 176, + 828, + 450, + 843 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "When stochastic gradient and Hessian complexities are equally important, we use Hessian to improve the gradient estimation. Denote $\\mathbf { x } ( a ) = a \\bar { \\mathbf { x } } ^ { t } + ( 1 - a ) \\tilde { \\mathbf { x } }$ . From Assumption 2.3, we have ", + "bbox": [ + 173, + 854, + 825, + 885 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/ebcb777e8d807878d2f376a9a1ef58e3733ab936c83c3451ca41ba983bd3a78b.jpg", + "text": "$$\n\\nabla f _ { i } ( \\mathbf { x } ^ { t } ) - \\nabla f _ { i } ( \\widetilde { \\mathbf { x } } ) - \\nabla ^ { 2 } f _ { i } ( \\widetilde { \\mathbf { x } } ) ( \\mathbf { x } ^ { t } - \\widetilde { \\mathbf { x } } ) \\| = \\| \\int _ { 0 } ^ { 1 } [ \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ( a ) ) - \\nabla ^ { 2 } f _ { i } ( \\widetilde { \\mathbf { x } } ) ] ( \\mathbf { x } ^ { t } - \\widetilde { \\mathbf { x } } ) \\mathbf { d } a \\| \\leq \\frac { L _ { 2 } } { 2 } \\| \\mathbf { x } ^ { t } - \\widetilde { \\mathbf { x } } \\| ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 181, + 892, + 839, + 929 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Such property can be used to improve Lemma 4.2 of Estimator 4. Specifically, define the correction ", + "bbox": [ + 173, + 103, + 823, + 118 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/374f1a04ace9edfc2eff14647f0696751e7030cb0adc31d0cd8e315d6054623a.jpg", + "text": "$$\n\\mathbf { c } ^ { k } = [ \\nabla ^ { 2 } F ( \\tilde { \\mathbf { x } } ) - \\nabla ^ { 2 } f ( \\tilde { \\mathbf { x } } ; \\mathcal { G } ) ] ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 354, + 125, + 642, + 143 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $\\tilde { \\bf x }$ is some reference point updated in an epoch-wise manner. Estimator 6 adds $\\mathbf { c } ^ { k }$ to the estimator in Estimator 4. Note that in Estimator 6, the first- and second-order oracle complexities are the same. We now analyze the first-order (and second-order) oracle complexity to meet requirement (9). ", + "bbox": [ + 173, + 150, + 825, + 205 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Lemma 5.1. Assume Algorithm $\\cdot$ takes the trust region radius $r = \\sqrt { \\epsilon / L _ { 2 } }$ as in Theorem 3.1. For any $k \\geq 0$ , Estimator 6 produces estimator $\\mathbf { g } ^ { k }$ for the first order differential $\\nabla F ( \\mathbf { x } ^ { k } )$ such that $\\| \\mathbf { g } ^ { k } - \\mathbf { \\mu }$ $\\nabla F ( \\mathbf { x } ^ { k } ) \\| \\leq \\epsilon / 6$ with probability at least $1 - \\delta / K _ { 0 }$ , if we set $p _ { 1 } = n ^ { 0 . 2 5 }$ and $s _ { 1 } \\doteq \\dot { n } ^ { 0 . 7 5 } c \\log ( \\ddot { K _ { 0 } } / \\delta )$ where $c = 1 1 5 2$ and $K _ { 0 }$ is a constant to be determined. Consequently, the amortized per-iteration stochastic first-order oracle complexity to construct $\\mathbf { g } ^ { k }$ is $2 s _ { 1 } = 2 n ^ { 0 . \\dot { 7 } 5 } c \\log ( K _ { 0 } / \\delta )$ . ", + "bbox": [ + 173, + 210, + 825, + 285 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The proof of Lemma 5.1 is similar to the one of Lemma 4.1 and is deferred to Appendix B.4. Similar to the previous section, Lemma 5.1 only guarantees that the gradient estimator satisfies the requirement (9) in a single iteration. Such result can be extended to hold for all $k$ by using the union bound with $K _ { 0 } = 2 K$ , which together with Theorem 3.1 gives the following corollary. ", + "bbox": [ + 174, + 295, + 825, + 352 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Corollary 5.1 (Computational Complexity of $\\mathrm { { S T R } _ { 2 } }$ ). Algorithm 5 finds a $1 2 \\epsilon$ -SOSP with probability at least $1 - \\delta$ , within $\\begin{array} { r } { \\mathcal { O } ( \\frac { n ^ { 0 . 7 5 } \\sqrt { L _ { 2 } } \\Delta } { \\epsilon ^ { 1 . 5 } } \\log ( \\frac { \\sqrt { L _ { 2 } } \\Delta } { \\delta \\epsilon ^ { 1 . 5 } } ) ) } \\end{array}$ ) overall stochastic first-order oracle queries and $\\begin{array} { r } { \\mathcal { O } ( \\frac { n ^ { 0 . 7 5 } \\sqrt { L _ { 2 } } \\Delta } { \\epsilon ^ { 1 . 5 } } \\log ( \\frac { d \\sqrt { L _ { 2 } } \\Delta } { \\delta \\epsilon ^ { 1 . 5 } } ) ) } \\end{array}$ \u000f1.5 δ\u000f1.5 overall stochastic second-order oracle queries. ", + "bbox": [ + 173, + 354, + 825, + 409 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Corollary 5.1 shows that to find an $\\epsilon$ -SOSP, both SFO and SSO of $\\mathrm { { S T R _ { 2 } } }$ are $\\tilde { \\mathcal { O } } ( n ^ { 3 / 4 } / \\epsilon ^ { 1 . 5 } )$ which surpasses the best existing one ${ \\tilde { \\mathcal { O } } } ( n ^ { 4 / 5 } / \\epsilon ^ { 1 . 5 } )$ in (Zhou et al., 2018c). ", + "bbox": [ + 174, + 420, + 823, + 453 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6 PRACTICAL STOCHASTIC TRUST REGION VARIANTS ", + "text_level": 1, + "bbox": [ + 176, + 472, + 642, + 489 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6.1 HANDLING INEXACT QCQP SOLUTIONS ", + "text_level": 1, + "bbox": [ + 176, + 502, + 496, + 518 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "One drawback of MetaAlgorithm 1 is that it requires the exact solution to the QCQP subproblem (8) and uses the dual variable as stopping criterion. We address this problem by developing a practical variant, MetaAlgorithm 7, which admits inexact QCQP solutions without access to the dual variable. This algorithm repeatedly invokes a procedure called INEXACTTR $\\mathbf { W E A K }$ , which, as we shall see, outputs an $\\mathcal { O } ( \\epsilon )$ -SOSP with a constant probability of $2 / 3$ in $\\mathcal { O } ( 1 / \\epsilon ^ { 1 . 5 } )$ iterations. By repeatedly invoking INEXACTTR $\\mathbf { \\partial } \\cdot \\mathbf { W E A K }$ for $\\Theta ( \\log ( 1 / \\bar { \\delta } ) )$ times, MetaAlgorithm 7 boosts the probability to $( 1 - \\delta )$ for any desired $\\delta$ . This repeating technique has been studied by, e.g., (Allen-Zhu & Li, 2018; AllenZhu, 2018b). To test whether the $t$ -th run outputs an $\\mathcal { O } ( \\epsilon )$ -SOSP, we need to compute $\\| \\nabla F ( \\mathbf { x } ^ { t } ) \\|$ and the smallest eigenvalue of $\\nabla ^ { 2 } F ( \\mathbf { x } ^ { t } )$ . The latter one can be approximated by solving the QCQP ", + "bbox": [ + 173, + 529, + 826, + 656 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/fecab025599a069ccd8344e4f1adcf4e617058a069250ece8202a60bfb7555b8.jpg", + "text": "$$\n\\mathbf { v } ^ { t } : = \\operatorname * { a r g m i n } _ { \\| \\mathbf { v } \\| \\leq 1 } \\psi _ { t } ( \\mathbf { v } ) = \\langle \\mathbf { H } ^ { t } \\mathbf { v } , \\mathbf { v } \\rangle ,\n$$", + "text_format": "latex", + "bbox": [ + 385, + 660, + 611, + 689 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $\\mathbf { H } ^ { t }$ is the full Hessian $\\nabla ^ { 2 } F ( \\mathbf { x } ^ { t } )$ or its estimation. One can show that MetaAlgorithm 7 finds an $\\mathcal { O } ( \\epsilon )$ -SOSP w.p. at least $( 1 - \\delta )$ in $\\tilde { \\mathcal { O } } ( 1 / \\epsilon ^ { 1 . 5 } )$ iterations. We defer the detailed analysis to Appendix C. ", + "bbox": [ + 174, + 696, + 825, + 729 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6.2 HESSIAN-FREE IMPLEMENTATION ", + "text_level": 1, + "bbox": [ + 176, + 744, + 452, + 758 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Based on MetaAlgorithm 7, we propose a Hessian-free method named ${ \\mathrm { { S T R } } } _ { \\mathrm { { f r e e } } }$ , which is summarized in Algorithm 8. $\\mathrm { S T R } _ { \\mathrm { f r e e } }$ leverages the full/stochastic Hessian and Estimator 4 to construct $\\mathbf { H } ^ { k }$ and $\\mathbf { g } ^ { k }$ , respectively. Besides, it uses Lanczos method (Gould et al., 1999; Carmon & Duchi, 2018) as the QCQP solver, which can be implemented in a Hessian-free manner (i.e., using only Hessian-vector products without explicit Hessian matrix evaluations). Thus, $\\mathbf { H } ^ { k }$ is only accessed through Hessianvector products and is never explicitly constructed. Since Hessian-vector products can be computed in linear time (in terms of the dimension $d$ ) for many machine learning problems (Allen-Zhu, 2018b; Agarwal et al., 2017), Hessian-free methods are usually more practical than Hessian based ones for high dimensional problems. The following theorem, whose proof can be found in Appendix D, establishes the runtime complexity (i.e., the total complexity of stochastic gradient and Hessian-vector product evaluations (Zhou & Gu, 2019)) of ${ \\bf S T R } _ { \\mathrm { f r e e } }$ . ", + "bbox": [ + 173, + 770, + 826, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "MetaAlgorithm 7 Inexact Trust Region Method II ", + "text_level": 1, + "bbox": [ + 176, + 103, + 508, + 118 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Input: initial point $\\mathbf { x } ^ { 0 }$ , step size $r$ , number of inner iterations $K$ , constants $\\delta , \\zeta \\in ( 0 , 1 )$ , c1, c2, \nnumber of outer iterations $T = \\Theta ( \\log ( 1 / \\delta ) )$ , sample size $s$ (optional) \n1: for $t = 1$ to $T$ do \n2: $\\mathbf { x } ^ { t } \\gets \\mathrm { I N E X A C T T R } _ { \\mathrm { W E A K } } ( \\mathbf { x } ^ { 0 } , r , K , \\zeta )$ ; \n3: Option I: $\\diamond$ high accuracy case (small $\\epsilon$ ) \n4: $\\mathbf { H } ^ { t } : = \\nabla ^ { 2 } F ( \\mathbf { x } ^ { t } )$ ; \n5: Option II: $\\diamond$ low accuracy case (moderate $\\epsilon$ ) \n6: Draw $s$ samples indexed by $\\mathcal { H }$ and let $\\mathbf { H } ^ { t } : = \\nabla ^ { 2 } f ( \\mathbf { x } ^ { t } ; \\mathcal { H } )$ ; \n7: Compute $\\tilde { \\mathbf { v } } ^ { t }$ by solving (10) up to accuracy $\\sqrt { \\epsilon L _ { 2 } }$ with probability $1 - \\delta / 4$ ; \n8: if $\\| \\nabla F ( \\mathbf { x } ^ { t } ) \\| \\leq c _ { 1 } \\epsilon _ { . }$ and $\\psi _ { t } ( \\tilde { \\mathbf { v } } ^ { t } ) \\geq - \\sqrt { c _ { 2 } \\epsilon L _ { 2 } }$ then \n9: return $\\mathbf { x } _ { \\epsilon } : = \\mathbf { x } ^ { t }$ ; \n10: end if \n11: end for \n12: procedure INEXACTTR $\\mathbf { w } _ { \\mathrm { E A K } } ( \\mathbf { x } ^ { 0 } , r , K , \\zeta )$ \n13: for $k = 0$ to $K - 1$ do \n14: Compute $\\mathbf { g } ^ { k }$ and $\\mathbf { H } ^ { k }$ such that (9) holds with probability $\\textstyle { 1 - \\frac { \\zeta } { 4 K } }$ \n15: Compute $\\tilde { \\mathbf { h } } ^ { k }$ by solving (8) up to accuracy $\\epsilon ^ { 1 . 5 } / \\sqrt { L _ { 2 } }$ 4Kwith probability $\\textstyle 1 - { \\frac { \\zeta } { 4 K } }$ ; \n16: $\\mathbf { x } ^ { k + 1 } : = \\mathbf { x } ^ { k } + \\tilde { \\mathbf { h } } ^ { k }$ ; \n17: end for \n18: Randomly select $\\bar { k }$ from $\\{ 0 , \\ldots , K - 1 \\}$ ; \n19: return xk¯+1; \n20: end procedure ", + "bbox": [ + 173, + 123, + 823, + 457 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Algorithm 8 STRfree ", + "text_level": 1, + "bbox": [ + 174, + 469, + 312, + 484 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "1: In the same setting as MetaAlgorithm 7, \n2: construct gradient estimator $\\mathbf { g } ^ { k }$ by Estimator 4; \n3: construct Hessian estimator $\\bar { \\mathbf { H } } ^ { k }$ by \n4: Option I: $\\mathbf { H } ^ { k } : = \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } )$ ; \n5: Option II: Draw $s$ samples indexed by $\\mathcal { H }$ and let $\\mathbf { H } ^ { k } : = \\nabla ^ { 2 } f ( \\mathbf { x } ^ { k } ; \\mathcal { H } )$ ; \n6: use Lanczos method to solve QCQP subproblems. ", + "bbox": [ + 178, + 488, + 692, + 574 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Theorem 6.1. Consider Algorithm 8 for solving problem (1). Let $\\zeta = 1 / 3$ , $r = \\sqrt { \\epsilon / L _ { 2 } }$ , $K =$ $4 \\sqrt { L _ { 2 } } \\Delta / \\epsilon ^ { 1 . 5 }$ , $\\begin{array} { r } { T = \\frac { 3 } { 2 } \\mathrm { { l o g } ( 2 / \\delta ) } } \\end{array}$ , $c _ { 1 } = 6 0 0$ , $c _ { 2 } = 5 0 0 \\mathrm { \\Omega }$ , and 32L21\u000fL2 log(4d/δ). The hyper-parameters in Estimator $^ { 4 }$ are set to the same values as those in Lemma 4.2. The number of iterations of Lanczos method is set to $\\tilde { \\mathcal { O } } ( 1 / ( L _ { 2 } \\epsilon ) ^ { 0 . 2 5 } )$ . To find an $\\mathcal { O } ( \\epsilon )$ -SOSP w.p. at least $1 - \\delta$ , the runtime complexity is $\\tilde { \\mathcal { O } } ( d \\operatorname* { m i n } \\{ n / \\epsilon ^ { 1 . 7 5 } , 1 / \\epsilon ^ { 2 . 7 5 } + \\sqrt { n } / \\epsilon ^ { 2 } \\} \\log ( 1 / \\delta ) )$ . ", + "bbox": [ + 173, + 585, + 826, + 669 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To solve the QCQP more efficiently, here we develop a faster solver which is based on the AppxPCA method (Allen-Zhu & Li, 2016) and KatyushaXW (Allen-Zhu, 2018a). See details in Appendix E. By replacing Lanczos method with this solver in ${ \\bf S T R } _ { \\mathrm { f r e e } }$ , we further improve the runtime complexity to $\\tilde { \\mathcal { O } } ( d \\operatorname* { m i n } \\{ n / \\epsilon ^ { 1 . 5 } + n ^ { 0 . 7 5 } / \\epsilon ^ { 1 . 7 5 } , 1 / \\epsilon ^ { 2 . 5 } + \\sqrt { n } / \\epsilon ^ { 2 } \\}$ ). We call this new algorithm ${ \\mathrm { S T R } } _ { \\mathrm { f r e e } } +$ whose details can be found in Appendix E. Table 2 shows that for the runtime complexities, both $\\mathrm { { S T R } _ { \\mathrm { { f r e e } } } }$ and ${ \\mathrm { S T R } } _ { \\mathrm { f r e e } } +$ outperform existing methods. See more comparison and discussion in Appendix E.4. ", + "bbox": [ + 174, + 676, + 825, + 763 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "7 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 782, + 326, + 797 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Here we compare the proposed STR with several state-of-the-art (stochastic) cubic regularized algorithms and trust region approaches, including trust region (TR) algorithm (Conn et al., 2000), adaptive cubic regularization (ARC) (Cartis et al., 2011), sub-sampled cubic regularization (SCR) (Kohler & Lucchi, 2017a), stochastic variance-reduced cubic (SVRC) (Zhou et al., 2018c), Lite-SVRC (Zhou et al., 2018b), and SRVRC (Zhou & Gu, 2019). For STR, we estimate the gradient as the way in case (1). This is because such a method enjoys lower Hessian computational complexity over the way in case (2) and for most problems, computing their Hessian matrices is much more time-consuming than computing their gradients. For the subproblems in these compared methods, we use Lanczos method (Gould et al., 1999; Kohler & Lucchi, 2017a) to solve the subproblem approximately in a Hessian-related Krylov subspace. We run simulations on seven datasets from LibSVM (a9a, ijcnn, codrna, phishing, w8a, epsilon and mnist). We run our algorithm for 40 epochs and use the output as the optimal value $f ^ { * }$ for sub-optimality estimation. Note the output has very small gradient already verified by Figure 2 and 4 in appendix. For all the considered algorithms, we set their initializations as zeros and tune their hyper-parameters optimally. For more experimental settings, e.g. details of testing datasets and algorithm parameter settings, please refer to Appendix F. ", + "bbox": [ + 173, + 811, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/5fb60b49e8c69b788f6bf743f2b22c706151d6ad4829c3a871b14cfa1cc11853.jpg", + "table_caption": [ + "Table 2: Runtime complexities of $\\operatorname { S T R } _ { \\mathrm { f r e e } }$ , ${ \\mathrm { S T R } } _ { \\mathrm { f r e e } } +$ , and other state-of-the-art methods. " + ], + "table_footnote": [ + "These entries rely on an additional assumption: $\\| \\nabla f _ { i } ( { \\mathbf { x } } ) - \\nabla F ( { \\mathbf { x } } ) \\| \\leq \\sigma$ a.s. " + ], + "table_body": "
AlgorithmRuntime
Hessian-free Cubic (Carmon & Duchi, 2016)Fast-Cubic (Agarwal et al., 2017)Stochastic Cubic (Tripuraneni et al., 2018)SRVRCfree (Zhou & Gu, 2019)（）0dn dn0.75+151.756d *35O(min{ d}）*
STRfree (this paper)STRfree+ (this paper)O(min{dn d dn0.5e1.75，2.752O(min{ dn0.75dn d dn0.51.5+e1.75，2.5+20)
", + "bbox": [ + 209, + 121, + 789, + 256 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/8013d004ab6213afb97a494ee9b3163ce86d5b5c7e6f77d3cb61238a0eee4a13.jpg", + "image_caption": [ + "Figure 1: Comparison on the logistic regression with non-convex regularizer. " + ], + "image_footnote": [], + "bbox": [ + 178, + 284, + 821, + 541 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 570, + 826, + 667 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Two evaluation non-convex problems. Following (Kohler & Lucchi, 2017a; Zhou et al., 2018c), we evaluate all considered algorithms on two learning tasks: the logistic regression with nonconvex regularizer and the nonlinear least square. Given $n$ data points $( { \\pmb x } _ { i } , y _ { i } )$ where $\\pmb { x } _ { i } \\in \\mathbb { R } ^ { d }$ is the sample vector and $y _ { i } ~ \\in ~ \\{ - 1 , 1 \\}$ is the label, logistic regression with non-convex regularizer aims at distinguishing these two kinds of samples by solving the following problem $\\begin{array} { r } { \\operatorname* { m i n } _ { \\pmb { w } } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\log ( 1 + \\mathrm { { e x p } } ( - y _ { i } \\pmb { w } ^ { T } \\pmb { x } _ { i } ) ) + \\lambda R ( \\pmb { w } ; \\alpha ) } \\end{array}$ , where the non-convex regularizer $R ( { \\boldsymbol { w } } ; \\alpha )$ is defined as $\\begin{array} { r } { R ( \\pmb { w } ; \\alpha ) = \\sum _ { i = 1 } ^ { d } \\alpha \\pmb { w } _ { i } ^ { 2 } / ( 1 + \\alpha \\pmb { w } _ { i } ^ { 2 } ) } \\end{array}$ . The nonlinear least square problem fits the nonlinear data by minimizing $\\begin{array} { r } { \\operatorname* { m i n } _ { \\pmb { w } } \\frac { 1 } { 2 n } \\sum _ { i = 1 } ^ { n } \\left[ y _ { i } - \\phi ( \\pmb { w } ^ { T } \\pmb { x } _ { i } ) \\right] ^ { 2 } + \\lambda R ( \\pmb { w } , \\alpha ) } \\end{array}$ . For these two kinds of problems, we set the parameters $\\lambda = 1 0 ^ { - 3 }$ and $\\alpha = 1 0$ for all testing datasets. ", + "bbox": [ + 173, + 674, + 826, + 810 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Comparison of Hessian based algorithms. Figure 1 summarizes testing results on the non-convex logistic regression problem. For each dataset, we report the function value gap v.s. the overall algorithm running time which can reflect the overall computational complexity of an algorithm, and also show the function value gap v.s. Hessian sample complexity which reveals the complexity of Hessian computation. From Figure 1, one can observe that our proposed STR algorithm runs faster than the compared algorithms in terms of the algorithm running time, showing the overall superiority of STR. Furthermore, STR also reveals much sharper convergence curves in terms of the Hessian sample complexity which is consistent with our theory. This is because to achieve an $\\epsilon$ -accuracy local minimum, the Hessian sample complexity of the proposed STR is $\\tilde { \\mathcal { O } } ( n ^ { 0 . 5 } / \\epsilon ^ { 1 . 5 } )$ and is superior over the complexity of the compared methods (see the comparison in Sec. 4.2). Indeed, this also explains why our algorithm is also faster in terms of algorithm running time, since for most optimization problems, Hessian matrix is much more computationally expensive than the gradient and thus more efficient Hessian sample complexity means faster overall convergence speed. Note, as all compared methods need to compute the Hessian and gradient, their memory complexity are all $\\mathcal { O } ( d ^ { 2 } + \\dot { d } )$ . ", + "bbox": [ + 173, + 816, + 825, + 915 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/1f795eb6ac39b80b39a1210d8ceb789239034df7ee57a3eb2a229984d19c4413.jpg", + "image_caption": [ + "Figure 2: Comparison on the nonlinear least square problem. " + ], + "image_footnote": [], + "bbox": [ + 178, + 104, + 821, + 359 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 390, + 825, + 489 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Figure 2 displays results of the compared algorithms on the nonlinear least square problem. STR shows very similar behaviors as those in Figure 1. Specifically, STR achieves fastest convergence rate in terms of both algorithm running time and Hessian sample complexity. On the codrna dataset we further plot the gradient norm versus running time and Hessian sample complexity. One can obverse that the gradient in STR vanishes significantly faster than other algorithms which means that STR can find the stationary point with high efficiency. See Figure 4 in Appendix F.2 for more experimental results on gradient norm comparison. All these results confirm the superiority of the proposed STR. ", + "bbox": [ + 173, + 496, + 826, + 593 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Comparison of Hessian-free algorithms. Here we compare our proposed Hessian-free STR, namely $\\mathrm { { S T R } _ { \\mathrm { { f r e e } } } }$ , with other state-ofthe-art Hessian-free algorithms on the two high-dimensional datasets, including epsilon and mnist (see details in Appendix F). Here we do not compare ${ \\mathrm { S T R } } _ { \\mathrm { f r e e } } +$ , as it is based on AppxPCA method (Allen-Zhu & Li, 2016) and KatyushaXW (Allen-Zhu, 2018a) which require tuning a lot of hyper-parameters. From the results in Figure 3, one can observe that compared with other algorithms, our ${ \\mathrm { { S T R } } } _ { \\mathrm { { f r e e } } }$ ", + "bbox": [ + 174, + 601, + 485, + 767 + ], + "page_idx": 9 + }, + { + "type": "image", + "img_path": "images/6ae2d6fe5712438b857d2a33bd755443d66d2c7c1e1911c6e91971831b384a67.jpg", + "image_caption": [ + "Figure 3: Comparison of Hessian-free algorithms on non-convex logistic and soft-max regressions. " + ], + "image_footnote": [], + "bbox": [ + 503, + 603, + 826, + 727 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "achieves the best convergence speed which demonstrates its high efficiency in realistic applications. Besides, one also can find that $\\operatorname { S T R } _ { \\mathrm { f r e e } }$ is much faster than Hessian based STR since computing full Hessian is actually much computationally expensive than the computation of the Hessian vector. ", + "bbox": [ + 176, + 767, + 825, + 809 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "8 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 835, + 318, + 852 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We proposed two stochastic trust region variants. Under two settings (whether stochastic first- and second-order oracle complexities are treated equally), the proposed methods achieve state-of-theart oracle complexities. We also propose Hessian-free variants with lowest runtime complexity. Experimental results testify our theoretical implications and the efficiency of the proposed algorithms. 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Then, we analyze MetaAlgorithm 7 and $\\mathrm { { S T R } _ { \\mathrm { { f r e e } } } }$ in Sec. C and Sec. D, respectively. Next, in Sec. E, we develop a fast QCQP solver to further improve the computational complexity of $\\operatorname { S T R } _ { \\mathrm { f r e e } }$ . Finally, more experimental details and results are presented in Sec. F. ", + "bbox": [ + 174, + 132, + 825, + 189 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B DEFERRED PROOFS ", + "text_level": 1, + "bbox": [ + 174, + 209, + 372, + 226 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B.1 PROOF OF THEOREM 3.1 ", + "text_level": 1, + "bbox": [ + 174, + 241, + 388, + 255 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Proof. For simplicity of notation, we denote ", + "bbox": [ + 176, + 266, + 467, + 281 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/1616005f39564fe9d71d3519e3e67722604349daf5642951c666821e901afad5.jpg", + "text": "$$\n\\nabla _ { k } \\overset { \\mathrm { d e f } } { = } \\nabla F ( { \\mathbf x } ^ { k } ) - { \\mathbf g } ^ { k } \\mathrm { a n d } \\nabla _ { k } ^ { 2 } \\overset { \\mathrm { d e f } } { = } \\nabla ^ { 2 } F ( { \\mathbf x } ^ { k } ) - { \\mathbf H } ^ { k } .\n$$", + "text_format": "latex", + "bbox": [ + 328, + 286, + 663, + 309 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "From Assumption 2.3 we have ", + "bbox": [ + 173, + 314, + 377, + 329 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/60776325382d53ffc379302cc2bd007615bafade13a29997a5454b14ac1cb486.jpg", + "text": "$$\n\\begin{array} { r } { F ( { \\mathbf x } ^ { k + 1 } ) \\leq F ( { \\mathbf x } ^ { k } ) + \\langle \\nabla F ( { \\mathbf x } ^ { k } ) , \\mathbf { h } ^ { k } \\rangle + \\displaystyle \\frac { 1 } { 2 } \\langle \\nabla ^ { 2 } F ( { \\mathbf x } ^ { k } ) { \\mathbf h } ^ { k } , \\mathbf { h } ^ { k } \\rangle + \\displaystyle \\frac { L _ { 2 } } { 6 } \\| { \\mathbf h } ^ { k } \\| ^ { 3 } } \\\\ { = F ( { \\mathbf x } ^ { k } ) + \\langle \\nabla _ { k } + \\mathbf { g } ^ { k } , \\mathbf { h } ^ { k } \\rangle + \\displaystyle \\frac { 1 } { 2 } \\langle [ \\nabla _ { k } ^ { 2 } + \\mathbf { H } ^ { k } ] { \\mathbf h } ^ { k } , \\mathbf { h } ^ { k } \\rangle + \\displaystyle \\frac { L _ { 2 } } { 6 } \\| { \\mathbf h } ^ { k } \\| ^ { 3 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 259, + 334, + 736, + 397 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Use the CauchySchwarz inequality to obtain ", + "bbox": [ + 174, + 398, + 467, + 415 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/724a682108a329e0e34f13c56f9ed7bb22bc39ff9e358f0842cac42f169c8c9e.jpg", + "text": "$$\nF ( { \\mathbf x } ^ { k + 1 } ) \\leq F ( { \\mathbf x } ^ { k } ) + \\langle { \\mathbf g } ^ { k } , { \\mathbf h } ^ { k } \\rangle + \\frac { 1 } { 2 } \\langle \\mathbf H ^ { k } { \\mathbf h } ^ { k } , { \\mathbf h } ^ { k } \\rangle + \\frac { L _ { 2 } } { 6 } \\| { \\mathbf h } ^ { k } \\| ^ { 3 } + \\| \\nabla _ { k } \\| \\| { \\mathbf h } ^ { k } \\| + \\frac { 1 } { 2 } \\| \\nabla _ { k } ^ { 2 } \\| \\| { \\mathbf h } ^ { k } \\| ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 192, + 420, + 777, + 450 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The requirement (9) together with the trust region radius $\\| \\mathbf { h } \\| \\le r = \\sqrt { \\epsilon / L _ { 2 } }$ allows us to bound ", + "bbox": [ + 174, + 455, + 800, + 473 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/20f59b80b78f28c39cababf76e097dc8503b4bdd19e4adab58f76d072d24133d.jpg", + "text": "$$\n\\| \\nabla _ { k } \\| \\| \\mathbf h ^ { k } \\| + \\frac { 1 } { 2 } \\| \\nabla _ { k } ^ { 2 } \\| \\| \\mathbf h ^ { k } \\| ^ { 2 } \\leq \\frac { 1 } { 3 } \\cdot \\frac { \\epsilon ^ { 1 . 5 } } { \\sqrt { L _ { 2 } } } .\n$$", + "text_format": "latex", + "bbox": [ + 364, + 478, + 633, + 513 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The optimality of (5) indicates that there exists a dual variable $\\lambda ^ { k } \\geq 0$ so that (Corollary 7.2.2 in (Conn et al., 2000)) ", + "bbox": [ + 173, + 520, + 825, + 549 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/76c1bfa5079d2ca5ab495230ca27f14a0fb325aa2f4fbe19b88aeb68dd1ab56f.jpg", + "text": "$$\n\\begin{array} { c } { { \\displaystyle { \\mathrm { F i r s t ~ O r d e r : ~ } } { \\bf { g } } ^ { k } + { \\bf H } ^ { k } { \\bf h } ^ { k } + \\frac { \\lambda ^ { k } L _ { 2 } } { 2 } { \\bf h } ^ { k } = 0 , } } \\\\ { { \\displaystyle \\mathrm { S e c o n d ~ O r d e r : ~ } { \\bf H } ^ { k } + \\frac { \\lambda ^ { k } L _ { 2 } } { 2 } \\cdot { \\bf I } \\approx 0 , } } \\\\ { { \\displaystyle \\mathrm { C o m p l e m e n t a r y : ~ } \\lambda ^ { k } \\cdot ( \\| { \\bf h } ^ { k } \\| - r ) = 0 . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 343, + 553, + 655, + 640 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Multiplying (13) by $\\mathbf { h } ^ { k }$ , we have ", + "bbox": [ + 173, + 645, + 392, + 661 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/3ca4b496905c8f950c02fddce5bfead8062c1e2d6890f6ca671ead521c898670.jpg", + "text": "$$\n\\langle \\mathbf { g } ^ { k } + \\mathbf { H } ^ { k } \\mathbf { h } ^ { k } + \\frac { \\lambda ^ { k } L _ { 2 } } { 2 } \\mathbf { h } ^ { k } , \\mathbf { h } ^ { k } \\rangle = 0 .\n$$", + "text_format": "latex", + "bbox": [ + 382, + 666, + 616, + 699 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Additionally, using (14) we have ", + "bbox": [ + 174, + 703, + 390, + 718 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/9fe5336125356da135765759de8b320f4b0b0bb83a80b24651b692e574126573.jpg", + "text": "$$\n\\langle ( \\mathbf { H } ^ { k } + \\frac { \\lambda ^ { k } L _ { 2 } } { 2 } \\mathbf { I } ) \\mathbf { h } ^ { k } , \\mathbf { h } ^ { k } \\rangle \\geq 0 ,\n$$", + "text_format": "latex", + "bbox": [ + 400, + 722, + 596, + 756 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "which together with (16) gives ", + "bbox": [ + 174, + 760, + 377, + 775 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/946aa56b1d75ac209766be15d8244e904b486e7db7d2e1551e1c66e04611ef36.jpg", + "text": "$$\n\\langle \\mathbf { g } ^ { k } , \\mathbf { h } ^ { k } \\rangle \\leq 0 .\n$$", + "text_format": "latex", + "bbox": [ + 452, + 772, + 545, + 791 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Moreover, the complementary property (15) indicates $\\| \\mathbf { h } ^ { k } \\| = \\sqrt { \\epsilon / L _ { 2 } }$ as we have $\\lambda ^ { k } > 3 \\sqrt { \\epsilon / L _ { 2 } } > 0$ before MetaAlgorithm 1 terminates. Plug (12), (16), and (17) into (11) and use $\\| \\mathbf { h } ^ { k } \\| = \\sqrt { \\epsilon / L _ { 2 } }$ : ", + "bbox": [ + 174, + 795, + 828, + 828 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/5d9f43775d126fd33c19908719dde4c8f9bdab9615d428224acaa9852c3a134b.jpg", + "text": "$$\nF ( \\mathbf { x } ^ { k + 1 } ) \\leq F ( \\mathbf { x } ^ { k } ) - \\frac { L _ { 2 } \\lambda ^ { k } } { 4 } \\cdot \\frac { \\epsilon } { L _ { 2 } } + \\frac { 1 } { 2 } \\cdot \\frac { \\epsilon ^ { 1 . 5 } } { \\sqrt { L _ { 2 } } } .\n$$", + "text_format": "latex", + "bbox": [ + 344, + 833, + 653, + 868 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Therefore, if we have $\\lambda ^ { k } > 3 \\epsilon ^ { 0 . 5 } / \\sqrt { L _ { 2 } }$ , then ", + "bbox": [ + 174, + 875, + 468, + 890 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/d2678914ca47a96601f43b408625494bc06fbe5a3e6fe963e2b298c694e0db95.jpg", + "text": "$$\nF ( \\mathbf { x } ^ { k + 1 } ) \\leq F ( \\mathbf { x } ^ { k } ) - { \\frac { 1 } { 4 { \\sqrt { L _ { 2 } } } } } \\cdot \\epsilon ^ { 1 . 5 } .\n$$", + "text_format": "latex", + "bbox": [ + 382, + 895, + 616, + 929 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Using Assumption 2.1, we find $\\lambda ^ { k } \\le 3 \\epsilon ^ { 0 . 5 } / \\sqrt { L _ { 2 } }$ in no more than $4 \\sqrt { L _ { 2 } } \\cdot ( F ( \\mathbf { x } ^ { 0 } ) - F ( \\mathbf { x } ^ { * } ) ) / \\epsilon ^ { 1 . 5 }$ iterations. ", + "bbox": [ + 171, + 102, + 823, + 133 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We now show that once $\\lambda ^ { k } \\le 3 \\epsilon ^ { 0 . 5 } / \\sqrt { L _ { 2 } }$ , then $\\mathbf { x } ^ { k + 1 }$ is already an $\\mathcal { O } ( \\epsilon )$ -SOSP: From (13), we have ", + "bbox": [ + 176, + 137, + 818, + 155 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/fe5176f9e4fc73913bf17cea59355d763b3609c4969c2274a129d80ca54486ee.jpg", + "text": "$$\n\\| \\mathbf { g } ^ { k } + \\mathbf { H } ^ { k } \\mathbf { h } ^ { k } \\| = \\frac { L _ { 2 } \\lambda ^ { k } } { 2 } \\cdot \\| \\mathbf { h } ^ { k } \\| \\leq 2 \\epsilon .\n$$", + "text_format": "latex", + "bbox": [ + 375, + 159, + 622, + 191 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The assumptions $\\lVert \\nabla _ { k } \\rVert \\leq \\epsilon / 6$ and $\\| \\nabla _ { k } ^ { 2 } \\| \\le \\sqrt { \\epsilon L _ { 2 } } / 3$ together with the trust region radius $\\| \\mathbf { h } \\| \\leq$ $\\sqrt { \\epsilon / L _ { 2 } }$ imply ", + "bbox": [ + 173, + 196, + 826, + 229 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/361b3602948ad49869427833b2f7dc2ffec583dca112dd5db25b66e21207d4d3.jpg", + "text": "$$\n\\| \\nabla F ( { \\mathbf x } ^ { k } ) + \\nabla ^ { 2 } F ( { \\mathbf x } ^ { k } ) { \\mathbf h } ^ { k } \\| \\leq \\| { \\mathbf g } ^ { k } + { \\mathbf H } ^ { k } { \\mathbf h } ^ { k } \\| + \\| \\nabla _ { k } \\| + \\| \\nabla _ { k } ^ { 2 } \\cdot { \\mathbf h } ^ { k } \\| \\leq 2 . 5 \\epsilon .\n$$", + "text_format": "latex", + "bbox": [ + 250, + 236, + 746, + 256 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "On the other hand use Assumption 2.3 to bound ", + "bbox": [ + 173, + 261, + 488, + 275 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/48e7f410cff77d58233676d4808238ec882e3f54bf80340cf3f8c99867ddc477.jpg", + "text": "$$\n\\| \\nabla F ( { \\mathbf { x } } ^ { k + 1 } ) - \\nabla F ( { \\mathbf { x } } ^ { k } ) - \\nabla ^ { 2 } F ( { \\mathbf { x } } ^ { k } ) { \\mathbf { h } } ^ { k } \\| \\leq \\frac { L _ { 2 } } { 2 } \\| { \\mathbf { h } } ^ { k } \\| ^ { 2 } \\leq \\frac { \\epsilon } { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 299, + 280, + 696, + 310 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Combining these two results gives $\\| \\nabla F ( \\mathbf { x } ^ { k + 1 } ) \\| \\leq 3 \\epsilon$ ", + "bbox": [ + 174, + 316, + 532, + 332 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Besides, using Assumption 2.3, $\\| \\nabla _ { k } ^ { 2 } \\| \\le \\sqrt { \\epsilon L _ { 2 } } / 3$ , and (14), we derive the Hessian lower bound ", + "bbox": [ + 176, + 332, + 799, + 348 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/e36cc1c7ae4617bfd606174d203031bbdf0f504a87031e63a7bc9653d9b34b71.jpg", + "text": "$$\n\\nabla ^ { 2 } F ( \\mathbf { x } ^ { k + 1 } ) \\approx \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) - L _ { 2 } \\cdot \\Vert \\mathbf { h } ^ { k } \\Vert \\mathbf { I } \\approx \\mathbf { H } ^ { k } - \\sqrt { \\epsilon L _ { 2 } } / 3 \\mathbf { I } - L _ { 2 } \\Vert \\mathbf { h } ^ { k } \\Vert \\mathbf { I } \\approx - \\sqrt { 1 2 \\epsilon L _ { 2 } } \\mathbf { I } .\n$$", + "text_format": "latex", + "bbox": [ + 210, + 353, + 769, + 375 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Hence $\\mathbf { x } ^ { k + 1 }$ is a $1 2 \\epsilon$ -stationary point. Additionally, we have $\\| \\mathbf { h } ^ { k } \\| = r$ according to the complementary condition (15) for all but the last iteration. □ ", + "bbox": [ + 173, + 381, + 825, + 410 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B.2 PROOF OF LEMMA 4.1 ", + "text_level": 1, + "bbox": [ + 174, + 426, + 372, + 440 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Proof. Without loss of generality, we analyze the case $0 \\leq k < q _ { 2 }$ for ease of notation. We first focus on Option II. The proof for Option I follows the similar argument. ", + "bbox": [ + 171, + 452, + 825, + 481 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Option II: Define for $k = 0$ and $i \\in [ s _ { 2 } ^ { \\prime } ]$ ", + "bbox": [ + 176, + 481, + 442, + 494 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/5b59805a4af053ee4fb95c5b42b27008a7e356ff92846c332e497cfc3ff31f58.jpg", + "text": "$$\n\\mathbf { B } _ { i } ^ { 0 } \\overset { \\mathrm { d e f } } { = } \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { 0 } ) - \\nabla ^ { 2 } F ( \\mathbf { x } ^ { 0 } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 395, + 501, + 599, + 523 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "and define for $k \\geq 1$ and $i \\in [ s _ { 2 } ]$ ", + "bbox": [ + 173, + 527, + 392, + 544 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/1289057aa766bac4d1c804c34e177cedc7345fe754faf9636136f226ac77e204.jpg", + "text": "$$\n\\mathbf { B } _ { i } ^ { k } \\stackrel { \\mathrm { d e f } } { = } \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) - \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k - 1 } ) - \\big ( \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) - \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k - 1 } ) \\big ) .\n$$", + "text_format": "latex", + "bbox": [ + 287, + 550, + 709, + 571 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "$\\{ \\mathbf { B } _ { i } ^ { k } \\}$ is a martingale difference sequence. We have for all $k$ and $i$ ", + "bbox": [ + 174, + 575, + 611, + 592 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/1ab1248d3ac8b56df88bfe2bdb5750638f45686b37cd3daf9ffb75238c215762.jpg", + "text": "$$\n\\mathbb { E } [ \\mathbf { B } _ { i } ^ { k } | \\mathbf { x } ^ { k } ] = 0 .\n$$", + "text_format": "latex", + "bbox": [ + 447, + 597, + 549, + 616 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Besides, we use Assumption 2.2 for $k = 0$ to bound ", + "bbox": [ + 173, + 621, + 516, + 636 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/a96ab3c9c32ec3a722ba047b878da8bddd46e3b9d866c8b308c6d60bf12ac7a8.jpg", + "text": "$$\n\\| \\mathbf { B } _ { i } ^ { 0 } \\| \\leq \\| \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { 0 } ) \\| + \\| \\nabla ^ { 2 } F ( \\mathbf { x } ^ { 0 } ) \\| = 2 L _ { 1 } ,\n$$", + "text_format": "latex", + "bbox": [ + 351, + 640, + 645, + 660 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "and use Assumption 2.3 for $k \\geq 1$ to bound ", + "bbox": [ + 174, + 664, + 462, + 679 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/b76551264722383e915a5aec6eda7153275e6f93c5b9292ca46b4842c46f24cc.jpg", + "text": "$$\n\\| \\mathbf { B } _ { i } ^ { k } \\| \\leq \\| \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) - \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k - 1 } ) \\| + \\| \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) - \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k - 1 } ) \\| \\leq 2 \\sqrt { \\epsilon L _ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 232, + 685, + 748, + 705 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "From the construction of $\\mathbf { H } ^ { k }$ , we have ", + "bbox": [ + 174, + 712, + 426, + 727 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/31737fe57bead9ab4a5a9f1ab377ff3604127e5e0167ea5d6d85fd8f50827453.jpg", + "text": "$$\n\\mathbf { H } ^ { k } - \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) = \\sum _ { i = 1 } ^ { s _ { 2 } ^ { \\prime } } { \\frac { \\mathbf { B } _ { i } ^ { 0 } } { s _ { 2 } ^ { \\prime } } } + \\sum _ { j = 1 } ^ { k } \\sum _ { i = 1 } ^ { s _ { 2 } } { \\frac { \\mathbf { B } _ { i } ^ { j } } { s _ { 2 } } } .\n$$", + "text_format": "latex", + "bbox": [ + 356, + 733, + 642, + 780 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Thus using the matrix Azuma’s Inequality in Theorem 7.1 of (Tropp, 2012) and $k \\leq p _ { 2 }$ , we have ", + "bbox": [ + 173, + 784, + 805, + 800 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/f24e6e87a452eaf0ec8450b5212fb193ef6015d4900c4dabd7c2aca6fe52901e.jpg", + "text": "$$\n\\begin{array} { r l } & { P r \\{ \\| \\mathbf H ^ { k } - \\nabla ^ { 2 } F ( { \\mathbf x } ^ { k } ) \\| \\ge t \\} \\le d \\cdot \\exp \\{ - \\frac { t ^ { 2 } / 8 } { \\sum _ { i = 1 } ^ { s _ { 2 } ^ { \\prime } } 4 L _ { 1 } ^ { 2 } / s _ { 2 } ^ { \\prime 2 } + \\sum _ { j = 1 } ^ { k } \\sum _ { i = 1 } ^ { s _ { 2 } } 4 \\epsilon L _ { 2 } / s _ { 2 } ^ { 2 } } \\} } \\\\ & { \\qquad \\le d \\cdot \\exp \\{ - \\frac { t ^ { 2 } / 8 } { 4 L _ { 1 } ^ { 2 } / s _ { 2 } ^ { \\prime } + 4 p _ { 2 } \\epsilon L _ { 2 } / s _ { 2 } } \\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 225, + 804, + 771, + 883 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Consequently, we have ", + "bbox": [ + 173, + 886, + 328, + 900 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/0aadfd3a3d67afea7174dde94c4e59b87792f85f47f57d09eb4924bbfb771bad.jpg", + "text": "$$\nP r \\{ \\| \\mathbf { H } ^ { k } - \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) \\| \\leq \\sqrt { \\epsilon L _ { 2 } } \\} \\geq 1 - \\delta / K _ { 0 } .\n$$", + "text_format": "latex", + "bbox": [ + 331, + 906, + 648, + 926 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "by taking √ $t = \\sqrt { \\epsilon L _ { 2 } }$ , $s _ { 2 } ^ { \\prime } = 1 6 L _ { 1 } ^ { 2 } / ( \\epsilon L _ { 2 } ) \\log ( d K _ { 0 } / \\delta )$ , $s _ { 2 } = 3 2 L _ { 1 } / ( \\sqrt { \\epsilon L _ { 2 } } ) \\log ( d K _ { 0 } / \\delta )$ , and $p _ { 2 } =$ $L _ { 1 } / ( 2 \\sqrt { \\epsilon L _ { 2 } } )$ . ", + "bbox": [ + 168, + 102, + 825, + 133 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Option I: The proof is similar to the one of Option $\\mathrm { I I }$ except that we replace $\\mathbf { B } _ { i } ^ { 0 }$ with zero matrix. In such case, the matrix Azuma’s Inequality implies ", + "bbox": [ + 171, + 138, + 825, + 170 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/5105acae8fe1efbc1894a40154a10678befebbbcf1d7c774d44a663b7a40f1b8.jpg", + "text": "$$\nP r \\{ \\| \\mathbf H ^ { k } - \\nabla ^ { 2 } F ( \\mathbf x ^ { k } ) \\| \\geq t \\} \\leq d \\cdot \\exp \\{ - \\frac { t ^ { 2 } / 8 } { \\sum _ { j = 1 } ^ { k } \\sum _ { i = 1 } ^ { s _ { 2 } } 4 \\epsilon L _ { 2 } / s _ { 2 } ^ { 2 } } \\} \\leq d \\cdot \\exp \\{ - \\frac { t ^ { 2 } / 8 } { 4 p _ { 2 } \\epsilon L _ { 2 } / s _ { 2 } } \\} .\n$$", + "text_format": "latex", + "bbox": [ + 192, + 174, + 803, + 215 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Thus by taking $t = \\sqrt { \\epsilon L _ { 2 } }$ , $s _ { 2 } = 3 2 \\sqrt { n } \\log ( d K _ { 0 } / \\delta )$ , and $p _ { 2 } = { \\sqrt { n } }$ , we have the result. ", + "bbox": [ + 174, + 222, + 736, + 239 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Amortized Complexity: In option I, the choice of parameters ensures that: $s _ { 2 } ^ { \\prime } \\leq p _ { 2 } \\times s _ { 2 }$ and in option II: $n \\leq p _ { 2 } \\times s _ { 2 }$ . Consequently, the amortized stochastic second-order oracle complexity is bounded from above by $2 s _ { 2 }$ . □ ", + "bbox": [ + 173, + 244, + 826, + 286 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "B.3 PROOF OF LEMMA 4.2 ", + "text_level": 1, + "bbox": [ + 174, + 303, + 374, + 318 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Without loss of generality, we analyze the case $0 \\leq k < q _ { 1 }$ for ease of notation. Define for $k \\geq 1$ and $i \\in [ s _ { 1 } ]$ ", + "bbox": [ + 171, + 329, + 825, + 358 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/9e3fb8e7847a8fe361829154850844ae45dd30a0d6fe4819f9ed0712e27fd699.jpg", + "text": "$$\n\\mathbf { a } _ { i } ^ { k } \\overset { \\mathrm { d e f } } { = } \\nabla f _ { i } ( \\mathbf { x } ^ { k } ) - \\nabla f _ { i } ( \\mathbf { x } ^ { k - 1 } ) - ( \\nabla F ( \\mathbf { x } ^ { k } ) - \\nabla F ( \\mathbf { x } ^ { k - 1 } ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 303, + 357, + 691, + 378 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "$\\{ \\mathbf { a } _ { i } ^ { k } \\}$ is a martingale difference sequence: for all $k$ and $i$ ", + "bbox": [ + 174, + 381, + 544, + 396 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/f93eff03b18062bc0b677ebdaf30d4cae89642076fbd0c72a353abd16cb8e347.jpg", + "text": "$$\n\\mathbb { E } [ \\mathbf { a } _ { i } ^ { k } | \\mathbf { x } ^ { k } ] = 0 .\n$$", + "text_format": "latex", + "bbox": [ + 449, + 402, + 547, + 421 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Besides, $\\mathbf { a } _ { i } ^ { k }$ has bounded norm: ", + "bbox": [ + 173, + 428, + 382, + 443 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/f715a3de97eec7becc8c9300e7ae24b987d9d93081cc43e8995f128f1b3ee07a.jpg", + "text": "$$\n\\begin{array} { r l } & { \\| \\mathbf { a } _ { i } ^ { k } \\| \\leq \\| \\nabla f _ { i } ( \\mathbf { x } ^ { k } ) - \\nabla f _ { i } ( \\mathbf { x } ^ { k - 1 } ) \\| + \\| \\nabla F ( \\mathbf { x } ^ { k } ) - \\nabla F ( \\mathbf { x } ^ { k - 1 } ) \\| } \\\\ & { \\qquad \\leq L _ { 1 } \\| \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } \\| + L _ { 1 } \\| \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } \\| } \\\\ & { \\qquad \\leq 2 L _ { 1 } \\sqrt { \\epsilon / L _ { 2 } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 303, + 449, + 694, + 513 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "From the construction of $\\mathbf { g } ^ { k }$ , we have ", + "bbox": [ + 173, + 520, + 421, + 535 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/48297e473473875c7e965a5d0d8eb7481ba2dd03849ae39f01c631ccaa95e91f.jpg", + "text": "$$\n\\mathbf { g } ^ { k } - \\nabla F ( \\mathbf { x } ^ { k } ) = \\sum _ { j = 1 } ^ { k } \\sum _ { i = 1 } ^ { s _ { 1 } } \\frac { \\mathbf { a } _ { i } ^ { j } } { s _ { 1 } } .\n$$", + "text_format": "latex", + "bbox": [ + 398, + 541, + 598, + 587 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Recall the Azuma’s Inequality. Using $k \\leq p _ { 1 }$ , we have ", + "bbox": [ + 174, + 592, + 534, + 608 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/6610708d9395a9ff6faf13517d35ce1d336cce3c78fa09f4c5fae8c8b605b99a.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad P r \\{ | | \\mathbf { g } ^ { k } - \\nabla F ( \\mathbf { x } ^ { k } ) | | \\geq t \\} } \\\\ & { \\leq \\exp \\{ - \\frac { t ^ { 2 } / 8 } { \\sum _ { j = 1 } ^ { k } \\sum _ { i = 1 } ^ { s _ { 1 } } \\frac { 4 \\epsilon L _ { 1 } ^ { 2 } } { L _ { 2 } s _ { 1 } ^ { 2 } } } \\} \\leq \\exp \\{ - \\frac { t ^ { 2 } / 8 } { 4 \\epsilon L _ { 1 } ^ { 2 } p _ { 1 } / \\left( s _ { 1 } L _ { 2 } \\right) } \\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 307, + 613, + 689, + 678 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Take $t = \\epsilon / 6$ and denote $c = 1 1 5 2$ . To ensure that ", + "bbox": [ + 173, + 684, + 504, + 699 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/219e48ab6211306275bfa208f98d7c0bf6a9540fcfc0b9811d58976c5755b7a1.jpg", + "text": "$$\n\\begin{array} { r } { P r \\{ \\| \\mathbf { g } ^ { k } - \\nabla F ( \\mathbf { x } ^ { k } ) \\| \\ge \\epsilon / 6 \\} \\le \\delta / K _ { 0 } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 369, + 705, + 627, + 726 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "we need $\\begin{array} { r } { \\frac { c L _ { 1 } ^ { 2 } } { L _ { 2 } } \\log \\frac { K _ { 0 } } { \\delta } \\ \\leq \\ \\frac { \\epsilon s _ { 1 } } { p _ { 1 } } } \\end{array}$ . The best amortized stochastic first-order oracle complexity can be obtained by solving the following two-dimensional programming: ", + "bbox": [ + 173, + 733, + 823, + 767 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/cee4f3f149c9fea3c2a8a24b315552645ff0586f655c92f8c97cc53095e0de72.jpg", + "text": "$$\n\\begin{array} { r l } { \\underset { p _ { 1 } \\geq 1 , s _ { 1 } \\geq 1 } { \\operatorname* { m i n } } } & { ( n + s _ { 1 } ( p _ { 1 } - 1 ) ) / p _ { 1 } } \\\\ { s . t . } & { \\frac { c L _ { 1 } ^ { 2 } } { L _ { 2 } } \\log \\frac { K _ { 0 } } { \\delta } \\leq \\frac { \\epsilon s _ { 1 } } { p _ { 1 } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 387, + 772, + 609, + 835 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "which has the soluthat when we take $\\begin{array} { r } { s _ { 1 } = \\operatorname* { m i n } \\{ n , \\sqrt { \\frac { n } { \\epsilon } \\cdot \\frac { c L _ { 1 } ^ { 2 } \\log \\frac { K _ { 0 } } { \\delta } } { L _ { 2 } } } \\} } \\end{array}$ $\\begin{array} { r } { p _ { 1 } = \\operatorname* { m a x } \\lbrace 1 , \\sqrt { n \\epsilon \\cdot \\frac { L _ { 2 } } { c L _ { 1 } ^ { 2 } \\log \\frac { K _ { 0 } } { \\delta } } } \\rbrace } \\end{array}$ . Note $s _ { 1 } = n$ $\\mathbf { g } ^ { k } = \\nabla F ( \\mathbf { x } ^ { k } )$ ", + "bbox": [ + 173, + 842, + 826, + 890 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The amortized stochastic first-order oracle complexity is obtained by plugging in the choice of $s _ { 1 }$ and $p _ { 1 }$ , which completes the proof. ", + "bbox": [ + 173, + 895, + 825, + 925 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "B.4 PROOF OF LEMMA 5.1 ", + "text_level": 1, + "bbox": [ + 174, + 103, + 372, + 117 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Without loss of generality, we analyze the case $0 \\leq k < q _ { 1 }$ for ease of notation. Define for $k \\geq 1$ and $i \\in [ s _ { 1 } ]$ ", + "bbox": [ + 174, + 128, + 825, + 159 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/e52ebd66685c9729a0c3e478a34d1f0d5d56f5fedb103736c94d30b06dbe61d0.jpg", + "text": "$$\n\\begin{array} { r l } { \\mathbf { b } _ { i } ^ { k } \\stackrel { \\mathrm { d e f } } { = } \\nabla f _ { i } ( \\mathbf { x } ^ { k } ) - \\nabla f _ { i } ( \\mathbf { x } ^ { k - 1 } ) - \\nabla ^ { 2 } f _ { i } ( \\tilde { \\mathbf { x } } ) ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) } & { } \\\\ { - \\left[ \\nabla F ( \\mathbf { x } ^ { k } ) - \\nabla F ( \\mathbf { x } ^ { k - 1 } ) - \\nabla ^ { 2 } F ( \\tilde { \\mathbf { x } } ) ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) \\right] . } & { } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 302, + 156, + 694, + 200 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "$\\{ \\mathbf { b } _ { i } ^ { k } \\}$ is a martingale difference sequence: for all $k$ and $i$ ", + "bbox": [ + 176, + 200, + 544, + 215 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/f660f0e6f977b918184ec2c064d86ac6b34604eab53e949ab1a4d4fe716f9881.jpg", + "text": "$$\n\\mathbb { E } [ \\mathbf { b } _ { i } ^ { k } | \\mathbf { x } ^ { k } ] = 0 .\n$$", + "text_format": "latex", + "bbox": [ + 449, + 218, + 547, + 236 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Besides, $\\mathbf { b } _ { i } ^ { k }$ has bounded norm: ", + "bbox": [ + 173, + 237, + 382, + 253 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/07dcf33b984b0e393b017fb1bdbd957b6bc029cdd60d7705e31f39932eefb42d.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\| \\mathbf { b } _ { \\star } ^ { k } \\| \\leq \\| \\nabla f _ { i } ( \\mathbf { x } ^ { k } ) - \\nabla f _ { 1 } ( \\mathbf { x } ^ { k - 1 } ) - \\nabla ^ { 2 } f _ { i } ( \\widetilde { \\mathbf { x } } ) ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) \\| } \\\\ & { \\qquad + \\| \\nabla F ( \\mathbf { x } ^ { k } ) - \\nabla F ( \\mathbf { x } ^ { k - 1 } ) - \\nabla ^ { 2 } F ( \\widetilde { \\mathbf { x } } ) ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) \\| . } \\\\ & { \\quad + \\| \\nabla f _ { i } ( \\mathbf { x } ^ { k } ) - \\nabla f _ { \\widetilde { \\mathbf { x } } } ( \\mathbf { x } ^ { k - 1 } ) - \\nabla ^ { 2 } f _ { i } ( \\widetilde { \\mathbf { x } } ) ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) \\| \\arcsin ^ { k } , } \\\\ & { \\| \\nabla f _ { i } ( \\mathbf { x } ^ { k } ) - \\nabla f _ { i } ( \\mathbf { x } ^ { k - 1 } ) - \\nabla ^ { 2 } f _ { j } ( \\widetilde { \\mathbf { x } } ) ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) \\| \\arcsin ^ { k } } \\\\ & { \\quad = \\| \\int _ { 0 } ^ { 1 } \\left[ \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k - 1 } + t ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) ) - \\nabla ^ { 2 } f _ { i } ( \\widetilde { \\mathbf { x } } ) \\right] ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) d t \\| } \\\\ & { \\quad \\leq \\int _ { 0 } ^ { 1 } L _ { 2 } \\bigg \\| \\mathbf { x } ^ { k } + ( 1 - t ) \\mathbf { x } ^ { k - 1 } - \\widetilde { \\mathbf { x } } \\bigg \\| d t \\cdot \\| \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } \\| } \\\\ & { \\quad \\leq \\int _ { 0 } ^ { 1 } \\big ( L \\| \\mathbf { x } ^ { k } - \\widetilde { \\mathbf { x } } \\| + ( 1 - t ) \\| \\mathbf { x } ^ { k - 1 } - \\widetilde { \\mathbf { x } } \\big \\| d t \\cdot \\| \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } \\| } \\\\ & \\quad \\leq \\int _ { 0 } ^ { 1 } \\big ( L \\| \\mathbf { x } ^ { k } - \\widetilde { \\mathbf { x } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 174, + 255, + 732, + 465 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where the first inequality follows from Assumption 2.3 and the last inequality holds because $\\| \\mathbf { x } ^ { k } - $ $\\tilde { \\mathbf { x } } \\| \\leq k r$ and $\\| \\mathbf { x } ^ { k - 1 } - \\bar { \\tilde { \\mathbf { x } } } \\| \\leq k r$ , where $r$ is the trust region radius. Similarly, we have $\\| \\nabla F ( \\mathbf { x } ^ { k } ) -$ $\\begin{array} { r } { \\nabla F ( { \\mathbf x } ^ { k - 1 } ) - \\nabla ^ { 2 } F ( \\tilde { { \\mathbf x } } ) ( \\dot { \\mathbf x } ^ { k } - { \\mathbf x } ^ { k - 1 } ) \\| \\leq L _ { 2 } k r ^ { 2 } } \\end{array}$ . Thus, we bound ", + "bbox": [ + 173, + 465, + 825, + 508 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/181c05d64df01cda8dc29b06a074116dec1fc8866c54aaa09648aedfa82ae906.jpg", + "text": "$$\n\\| \\mathbf { b } _ { i } ^ { k } \\| \\leq 2 L _ { 2 } k r ^ { 2 } \\leq 2 p _ { 1 } \\epsilon\n$$", + "text_format": "latex", + "bbox": [ + 416, + 511, + 581, + 530 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "From the construction of $\\mathbf { g } ^ { k }$ , we have ", + "bbox": [ + 173, + 531, + 421, + 546 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/0d859d459e644d295ac6fee6bd28d3a95a5f459bca8bfbe286fb71b07ab7df7f.jpg", + "text": "$$\n\\mathbf { g } ^ { k } - \\nabla F ( \\mathbf { x } ^ { k } ) = \\sum _ { j = 1 } ^ { k } \\sum _ { i = 1 } ^ { s _ { 1 } } \\frac { \\mathbf { b } _ { i } ^ { j } } { s _ { 1 } } .\n$$", + "text_format": "latex", + "bbox": [ + 398, + 547, + 598, + 592 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We use $k \\leq p _ { 1 }$ and the Azuma’s inequality to bound ", + "bbox": [ + 174, + 593, + 519, + 608 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/b5064eab451bb9795910961730cc6cfd5f285422035967e6fc64ebe9dce20a50.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad P r \\{ | | \\mathbf { g } ^ { k } - \\nabla F ( { \\mathbf { x } } ^ { k } ) | | \\geq t \\} } \\\\ & { \\leq \\exp \\{ - \\frac { t ^ { 2 } / 8 } { \\sum _ { j = 1 } ^ { k } \\sum _ { i = 1 } ^ { s _ { 1 } } \\frac { 4 p _ { 1 } ^ { 2 } \\epsilon ^ { 2 } } { s _ { 1 } ^ { 2 } } } \\} \\leq \\exp \\{ - \\frac { t ^ { 2 } / 8 } { 4 \\epsilon ^ { 2 } p _ { 1 } ^ { 3 } / s _ { 1 } } \\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 326, + 609, + 669, + 672 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Thus, by taking $t = \\epsilon / 6$ and $c = 1 1 5 2$ , we need $\\begin{array} { r } { \\frac { s _ { 1 } } { p _ { 1 } ^ { 3 } } \\geq c \\log \\frac { K _ { 0 } } { \\delta } } \\end{array}$ . Further we want $s _ { 1 } p _ { 1 } \\simeq \\mathcal { O } ( n )$ and hence we take $p _ { 1 } = n ^ { 0 . 2 5 }$ and $\\begin{array} { r } { s _ { 1 } = n ^ { 0 . 7 5 } c \\log { \\frac { \\dot { K _ { 0 } } } { \\delta } } } \\end{array}$ . The amortized stochastic first-order oracle complexity is bounded by $2 s _ { 1 }$ . ", + "bbox": [ + 173, + 676, + 828, + 727 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "C ANALYSIS OF METAALGORITHM 7 ", + "text_level": 1, + "bbox": [ + 173, + 744, + 498, + 761 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We first show that INEXACTTR $\\mathbf { W E A K }$ finds an $\\mathcal { O } ( \\epsilon )$ -SOSP in $\\mathcal { O } ( 1 / \\epsilon ^ { 1 . 5 } )$ iterations with probability at least $2 / 3$ as stated in the following lemma. ", + "bbox": [ + 174, + 773, + 825, + 804 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Lemma C.1. Consider problem $( ~ )$ under Assumptions 2.1-2.3. Suppose that the differential estimators $\\mathbf { g } ^ { k }$ and $\\mathbf { H } ^ { k }$ satisfy Eqn. (9) with probability at least $\\begin{array} { r } { ( 1 - \\frac { \\zeta } { 4 K } ) } \\end{array}$ . Besides, suppose that $\\tilde { \\mathbf { h } } ^ { k }$ is an approximate solution to (8) such that w.p. $\\begin{array} { r } { ( 1 - \\frac { \\zeta } { 4 K } ) } \\end{array}$ , ", + "bbox": [ + 173, + 805, + 826, + 856 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/2c16ec5c53bf827dae529153d298ea592a3a6e2d9f69bce18185bac35265930a.jpg", + "text": "$$\n\\langle \\mathbf { g } ^ { k } , \\tilde { \\mathbf { h } } ^ { k } \\rangle + \\frac { 1 } { 2 } \\langle \\mathbf { H } ^ { k } \\tilde { \\mathbf { h } } ^ { k } , \\tilde { \\mathbf { h } } ^ { k } \\rangle \\leq \\langle \\mathbf { g } ^ { k } , \\mathbf { h } ^ { k } \\rangle + \\frac { 1 } { 2 } \\langle \\mathbf { H } ^ { k } \\mathbf { h } ^ { k } , \\mathbf { h } ^ { k } \\rangle + \\frac { \\epsilon ^ { 1 . 5 } } { \\sqrt { L _ { 2 } } } ,\n$$", + "text_format": "latex", + "bbox": [ + 287, + 857, + 709, + 890 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where $\\mathbf { h } ^ { k }$ is a global solution to (8). By setting $\\zeta = 1 / 3$ , $r = \\sqrt { \\epsilon / L _ { 2 } }$ , and $K = 4 \\sqrt { L _ { 2 } } \\Delta / \\epsilon ^ { 1 . 5 }$ , INEXACTTR $\\mathbf { \\partial } \\cdot \\mathbf { W } \\mathbf { E } \\mathbf { A } \\mathbf { K }$ outputs a 500\u000f-SOSP w.p. at least $2 / 3$ . ", + "bbox": [ + 171, + 893, + 825, + 925 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Proof. Combining (11) and (24), we have w.p. $\\begin{array} { r } { ( 1 - \\frac { \\zeta } { 4 K } ) } \\end{array}$ ", + "bbox": [ + 173, + 102, + 552, + 119 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/2f16a65cfa817451421902b5bee412c3789951b3c8f68cdb9bbf2e16f4347e7f.jpg", + "text": "$$\n\\begin{array} { r } { F ( { \\mathbf x } ^ { k + 1 } ) \\le F ( { \\mathbf x } ^ { k } ) + \\langle { \\mathbf g } ^ { k } , { \\mathbf h } ^ { k } \\rangle + \\displaystyle \\frac { 1 } { 2 } \\langle { \\mathbf H } ^ { k } { \\mathbf h } ^ { k } , { \\mathbf h } ^ { k } \\rangle + \\displaystyle \\frac { \\epsilon ^ { 1 . 5 } } { \\sqrt { L _ { 2 } } } } \\\\ { + \\displaystyle \\frac { L _ { 2 } } { 6 } \\| \\tilde { \\mathbf h } ^ { k } \\| ^ { 3 } + \\| \\nabla _ { k } \\| \\| \\tilde { \\mathbf h } ^ { k } \\| + \\displaystyle \\frac { 1 } { 2 } \\| \\nabla _ { k } ^ { 2 } \\| \\| \\tilde { \\mathbf h } ^ { k } \\| ^ { 2 } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 310, + 127, + 686, + 195 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where $\\mathbf { h } ^ { k }$ is a global solution to the QCQP (8) and $\\tilde { \\mathbf { h } } ^ { k }$ is an approximate solution satisfying (24). We let $\\lambda ^ { k }$ denote the dual variable corresponding to the global solution $\\mathbf { h } ^ { k }$ as defined in Lemma 2.1. We note that $\\mathbf { h } ^ { k }$ and $\\lambda ^ { k }$ are used only in our analysis. The INEXACT $\\mathrm { w e A K }$ algorithm only requires the approximate solution $\\tilde { \\mathbf { x } } ^ { k }$ without knowledge of $\\mathbf { h } ^ { k }$ or $\\lambda ^ { k }$ . ", + "bbox": [ + 173, + 200, + 826, + 260 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "By the assumption that (9) holds with probability $\\textstyle { \\left( { 1 - { \\frac { \\zeta } { 4 K } } } \\right) }$ and the fact that $\\| \\tilde { \\mathbf { h } } ^ { k } \\| \\le r = \\sqrt { L _ { 2 } \\epsilon }$ , we have w.p. $\\textstyle { \\left( { 1 - { \\frac { \\zeta } { 4 K } } } \\right) }$ , ", + "bbox": [ + 174, + 265, + 825, + 303 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/f4fa9225b6f536e955cf7b583aa298251879902ffc6088129ea61db36e3512bd.jpg", + "text": "$$\n\\frac { L _ { 2 } } { 6 } \\| \\tilde { \\mathbf { h } } ^ { k } \\| ^ { 3 } + \\| \\nabla _ { k } \\| \\| \\tilde { \\mathbf { h } } ^ { k } \\| + \\frac { 1 } { 2 } \\| \\nabla _ { k } ^ { 2 } \\| \\| \\tilde { \\mathbf { h } } ^ { k } \\| ^ { 2 } \\leq \\frac { \\epsilon ^ { 1 . 5 } } { 2 \\sqrt { L _ { 2 } } } .\n$$", + "text_format": "latex", + "bbox": [ + 326, + 308, + 671, + 344 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Plugging (16), (17), and (26) into (25) and applying the union bound, we have w.p. at least $\\begin{array} { r } { ( 1 - \\frac { \\zeta } { 2 K } ) } \\end{array}$ ", + "bbox": [ + 173, + 359, + 825, + 377 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/8a02aa228f0aec3132cde4a2fc73e578b05458ea77195a0b69b1c21caf94d387.jpg", + "text": "$$\nF ( { \\mathbf x } ^ { k + 1 } ) \\le F ( { \\mathbf x } ^ { k } ) - \\frac { L _ { 2 } \\lambda ^ { k } \\| { \\mathbf h } ^ { k } \\| ^ { 2 } } { 4 } + \\frac { 3 \\epsilon ^ { 1 . 5 } } { 2 \\sqrt { L _ { 2 } } } = F ( { \\mathbf x } ^ { k } ) - \\frac { L _ { 2 } \\lambda ^ { k } r ^ { 2 } } { 4 } + \\frac { 3 \\epsilon ^ { 1 . 5 } } { 2 \\sqrt { L _ { 2 } } } ,\n$$", + "text_format": "latex", + "bbox": [ + 246, + 386, + 750, + 421 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where the second inequality follows from (15): ", + "bbox": [ + 174, + 424, + 482, + 440 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/8cfbf8890382660d3b47c333a2048ce9c50cfd1b23059f2c3065e521e2bb78ad.jpg", + "text": "$$\n0 = \\lambda ^ { k } ( \\| \\mathbf h ^ { k } \\| - r ) = \\lambda ^ { k } ( \\| \\mathbf h ^ { k } \\| - r ) ( \\| \\mathbf h ^ { k } \\| + r ) = \\lambda ^ { k } ( \\| \\mathbf h ^ { k } \\| ^ { 2 } - r ^ { 2 } ) .\n$$", + "text_format": "latex", + "bbox": [ + 274, + 446, + 723, + 467 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Summing inequality (27) from $k = 0$ to $K - 1$ and applying the union bound, we have w.p. at least $\\left( 1 - \\zeta / 2 \\right)$ , ", + "bbox": [ + 173, + 473, + 826, + 502 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/bf123e3aef16dd9879bec6b5bb1b84059c93b84a438af694c98cb738aac4d7d7.jpg", + "text": "$$\n\\frac { 1 } { K } \\sum _ { k = 0 } ^ { K - 1 } \\lambda ^ { k } \\le \\frac { 4 ( F ( \\mathbf { x } ^ { 0 } ) - F ( \\mathbf { x } ^ { K + 1 } ) ) } { L _ { 2 } r ^ { 2 } K } + \\frac { 6 \\epsilon ^ { 1 . 5 } } { L _ { 2 } ^ { 1 . 5 } r ^ { 2 } } \\le \\frac { 4 \\Delta } { \\epsilon K } + \\frac { 6 \\sqrt { \\epsilon } } { \\sqrt { L _ { 2 } } } ,\n$$", + "text_format": "latex", + "bbox": [ + 287, + 502, + 710, + 545 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where the second inequality follows from Assumption 2.1 and our choice of the trust region radius. By sampling $\\bar { k }$ uniformly from $\\{ 0 , \\ldots , K - 1 \\}$ , we obtain ", + "bbox": [ + 173, + 549, + 823, + 585 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/107fcb40c963dcd1c5c33403eea00221a5f4ae50e2ed106e29f62798bef2b787.jpg", + "text": "$$\n\\mathbb { E } [ \\lambda ^ { \\bar { k } } ] = \\frac { 1 } { K } \\sum _ { k = 0 } ^ { K - 1 } \\lambda ^ { k } ,\n$$", + "text_format": "latex", + "bbox": [ + 429, + 593, + 566, + 637 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where the expectation is taken over the randomness of √ $\\bar { k }$ . Combining (29) and (30) and taking $K = 4 \\Delta \\sqrt { L _ { 2 } } / \\epsilon ^ { 1 . 5 }$ , we have w.p. at least $\\left( 1 - \\zeta / 2 \\right)$ ", + "bbox": [ + 174, + 645, + 825, + 674 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/2ef0ee6dcaed518fd4bee5fe578fc7e61dd7447a85f9346fbe07a3431e91c938.jpg", + "text": "$$\n\\mathbb { E } [ \\lambda ^ { \\bar { k } } ] \\leq \\frac { 7 \\sqrt \\epsilon } { \\sqrt { L _ { 2 } } } .\n$$", + "text_format": "latex", + "bbox": [ + 446, + 681, + 550, + 717 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Since $\\lambda ^ { k }$ is always no-negative, by Markov’s inequality and the union bound, with probability at least $1 - \\zeta$ , we have ", + "bbox": [ + 174, + 724, + 826, + 752 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/08ed327e45b2a8ba5c293d38c0b17968eabac4292c4c3c7f0fed7a71e005cc76.jpg", + "text": "$$\n\\lambda ^ { \\bar { k } } \\leq \\frac { 1 4 \\sqrt { \\epsilon } } { \\zeta \\sqrt { L _ { 2 } } } .\n$$", + "text_format": "latex", + "bbox": [ + 452, + 752, + 545, + 787 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "By taking $\\zeta = 1 / 3$ , we have w.p. at least $2 / 3 , \\lambda ^ { \\bar { k } } \\leq 4 2 \\sqrt { \\epsilon / L _ { 2 } }$ . The rest of the proof is similar to Theorem 3.1 and we have the result. ", + "bbox": [ + 173, + 792, + 826, + 823 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "The following theorem shows that MetaAlgorithm 7 finds an $\\mathcal { O } ( \\epsilon )$ -SOSP w.p. $( 1 - \\delta )$ after running INEXACTTR $\\mathbf { \\partial } \\cdot \\mathbf { W } \\mathbf { E } \\mathbf { A } \\mathbf { K }$ for $\\Theta ( \\log ( 1 / \\delta ) )$ times. ", + "bbox": [ + 173, + 840, + 825, + 871 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Theorem C.1 (Iteration Complexity of MetaAlgorithm 7). In the same setting as Lemma C.1, let $\\begin{array} { r } { T = \\frac { 3 } { 2 } \\mathrm { l o g } ( 2 / \\delta ) } \\end{array}$ , $c _ { 1 } = 6 0 0$ , $c _ { 2 } = 5 0 0$ , and 32L21\u000fL2 log(4d/δ). Then MetaAlgorithm 7 finds a 600\u000f-SOSP with probability at least $( 1 - \\delta )$ . ", + "bbox": [ + 173, + 875, + 825, + 925 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Proof. By Lemma C.1 and our choice of $T$ , with probability $\\left( 1 - \\delta / 2 \\right)$ , at least one of $\\mathbf { x } ^ { t }$ is a $5 0 0 \\epsilon$ -SOSP. On the other hand, since √ $\\psi _ { t } ( \\tilde { \\mathbf { v } } ^ { t } ) \\leq \\psi _ { t } ( \\mathbf { v } ^ { t } ) + \\sqrt { \\epsilon L _ { 2 } }$ with probability $1 - \\delta / 4$ , if $\\psi _ { t } ( \\tilde { \\mathbf { v } } ^ { t } ) \\geq - \\sqrt { c _ { 2 } \\epsilon L _ { 2 } }$ , then, w.p. $1 - \\delta / 4$ , ", + "bbox": [ + 173, + 102, + 826, + 148 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/5431faf34f9f7e6007dd3f9fa5c9b878277fae0c6e37d2c7365e0e69be97bd8f.jpg", + "text": "$$\n\\psi _ { t } ( \\mathbf { v } ^ { t } ) \\geq \\psi _ { t } ( \\tilde { \\mathbf { v } } ^ { t } ) - \\sqrt { \\epsilon L _ { 2 } } \\geq - \\sqrt { c _ { 2 } \\epsilon L _ { 2 } } - \\sqrt { \\epsilon L _ { 2 } } \\geq - \\sqrt { 5 5 0 \\epsilon L _ { 2 } } ,\n$$", + "text_format": "latex", + "bbox": [ + 282, + 154, + 714, + 172 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where the last inequality follows from our choice of $c _ { 2 }$ . ", + "bbox": [ + 174, + 176, + 537, + 191 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Option I: Since $\\mathbf { H } ^ { t } = \\nabla ^ { 2 } F ( \\mathbf { x } ^ { t } )$ is the full Hessian, $\\psi _ { t } ( \\mathbf { v } ^ { t } )$ is the smallest eigenvalue of $\\nabla ^ { 2 } F ( \\mathbf { x } ^ { t } )$ . \nApplying the union bound, we conclude that MetaAlgorithm 7 outputs a $6 0 0 \\epsilon$ -SOSP w.p. $( 1 - \\delta )$ . ", + "bbox": [ + 173, + 196, + 825, + 227 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Option II: Let $\\mathbf { B } _ { i } : = \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { t } ) - \\nabla ^ { 2 } F ( \\mathbf { x } ^ { t } )$ for $i \\in \\mathcal { H }$ , then ", + "bbox": [ + 173, + 232, + 571, + 250 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/82330d1f335568caf45cf8e4281942893a73d002f570e62e7e857cb02656db0c.jpg", + "text": "$$\n\\mathbf { H } ^ { t } - \\boldsymbol { \\nabla } ^ { 2 } F ( \\mathbf { x } ^ { t } ) = \\frac { 1 } { s } \\sum _ { i = 1 } ^ { s } \\mathbf { B } _ { i } .\n$$", + "text_format": "latex", + "bbox": [ + 401, + 252, + 596, + 294 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "By Assumption 2.2, we have ", + "bbox": [ + 174, + 296, + 364, + 311 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/a66b657f6d5e008d7e819d605f1b1407257caed0799a469954f3e07b2a273a77.jpg", + "text": "$$\n\\| \\mathbf { B } _ { i } \\| \\leq \\| \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { t } ) \\| + \\| \\nabla ^ { 2 } F ( \\mathbf { x } ^ { t } ) \\| \\leq 2 L _ { 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 352, + 314, + 643, + 333 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Applying the matrix Azuma’s Inequality in Theorem 7.1 of Tropp (2012) leads to ", + "bbox": [ + 173, + 337, + 709, + 352 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/93b547fd9a178797b861f3f6b660d4c1e3c7334f23d4c7a6543ffda48ce10664.jpg", + "text": "$$\nP r \\{ \\| \\mathbf { H } ^ { t } - \\nabla ^ { 2 } F ( \\mathbf { x } ^ { t } ) \\| \\ge \\sqrt { \\epsilon L _ { 2 } } \\} \\le d \\cdot \\exp ( \\frac { - \\epsilon L _ { 2 } s } { 3 2 L _ { 1 } ^ { 2 } } ) .\n$$", + "text_format": "latex", + "bbox": [ + 320, + 354, + 676, + 388 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "By taking $\\begin{array} { r } { s = \\frac { 3 2 L _ { 1 } ^ { 2 } } { \\epsilon L _ { 2 } } \\mathrm { l o g } ( 4 d / \\delta ) } \\end{array}$ and applying the union bound, we have with probability $1 - \\delta$ ", + "bbox": [ + 173, + 393, + 784, + 414 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/82fc59a55d1163015ccb3dcb872124de25d906f486abf3f8a4e8261b20f94afb.jpg", + "text": "$$\n\\nabla ^ { 2 } F ( \\mathbf { x } ^ { t } ) \\approx \\mathbf { H } ^ { t } - \\sqrt { \\epsilon L _ { 2 } } \\mathbf { I } \\approx ( \\psi _ { t } ( \\mathbf { v } ^ { t } ) - \\sqrt { \\epsilon L _ { 2 } } ) \\mathbf { I } \\approx - \\sqrt { 6 0 0 \\epsilon L _ { 2 } } \\mathbf { I } ,\n$$", + "text_format": "latex", + "bbox": [ + 281, + 419, + 715, + 439 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where the last inequality follows from (33). This completes the proof. ", + "bbox": [ + 174, + 443, + 630, + 458 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "D PROOF OF THEOREM 6.1 ", + "text_level": 1, + "bbox": [ + 176, + 477, + 413, + 493 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Proof. We first analyze the computational cost of Lanczos method. By Corollary 2 in (Carmon & Duchi, 2018), for any desired accuracy \u000f˜, Lanczos method achieves this accuracy in $\\begin{array} { r } { \\mathcal { O } ( \\frac { r } { \\sqrt { \\tilde { \\epsilon } } } \\mathrm { l o g } \\frac { r \\sqrt { d } } { \\tilde { \\epsilon } p } ) } \\end{array}$ Lanczos iterations w.p. at least $( 1 - p )$ . Without loss of generality, we assume that the number of Lanczos iterations is strictly smaller than the dimension $d$ , otherwise the QCQP subproblem can be solved exactly. We note that each Lanczos iteration involves computation of one matrix-vector product. Therefore, to satisfy the condition (24) in Lemma C.1, one needs to evaluate $\\tilde { \\mathcal { O } } ( 1 / ( L _ { 2 } \\epsilon ) ^ { 0 . 2 5 } )$ Hessian-vector products of the form $\\mathbf { H } ^ { k } \\mathbf { v }$ . Similarly, to solve (10) up to accuracy $\\sqrt { \\epsilon L _ { 2 } }$ w.h.p., one needs to evaluate $\\tilde { \\mathcal { O } } ( 1 / ( L _ { 2 } \\epsilon ) ^ { 0 . 2 5 } )$ Hessian-vector products of the form $\\mathbf { H } ^ { t } \\mathbf { v }$ . ", + "bbox": [ + 173, + 507, + 826, + 632 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "In MetaAlgorithm 7, to verify whether the candidate solution $\\mathbf { v } _ { } ^ { t }$ is indeed an $\\mathcal { O } ( \\epsilon )$ -SOSP, one needs at most ${ \\mathcal { O } } ( n )$ stochastic gradient evaluations and $\\mathcal { O } ( \\operatorname* { m i n } \\{ n , \\log ( 4 d / \\delta ) L _ { 1 } ^ { 2 } / ( L _ { 2 } \\epsilon ) \\} / ( L _ { 2 } \\epsilon ) ^ { 0 . 2 5 } ) )$ stochastic Hessian-vector product evaluations, where the latter one follows from the proof of Theorem 7. We proceed to analyze the computational complexity of the INEXACTTRWEAK procedure. Recall that the iteration complexity of MetaAlgorithm 7 is $\\mathcal { O } ( \\log ( 1 / \\delta ) / \\epsilon ^ { 1 . 5 } )$ . Following Lemma 4.2 and Corollary 4.1, the stochastic first-order oracle complexity is $\\tilde { \\mathcal { O } } ( \\operatorname* { m i n } \\{ n / \\epsilon ^ { 1 . 5 } , \\sqrt { n } / \\epsilon ^ { 2 } \\} \\mathrm { l o g } ( 1 / \\delta ) )$ . Following the proof of Lemma 4.1 and Corollary 4.1, when $p _ { 2 } = 1$ , the overall stochastic Hessian sample complexity is $\\tilde { \\mathcal { O } } ( \\operatorname* { m i n } \\{ n / \\epsilon ^ { 1 . 5 } , 1 / \\epsilon ^ { 2 . 5 } \\} \\log ( 1 / \\delta ) )$ . Since it takes $\\tilde { \\mathcal { O } } ( 1 / \\epsilon ^ { 0 . 2 5 } )$ Lanczos iterations to meet the condition (24) as stated above, the overall stochastic Hessian-vector product oracle complexity is $\\tilde { \\mathcal { O } } ( \\operatorname* { m i n } \\{ n / \\epsilon ^ { 1 . 7 5 } , 1 / \\epsilon ^ { 2 . 7 5 } \\} \\log ( 1 / \\delta ) )$ . Combining the stochastic first-order and Hessian-vector product complexities, the overall runtime is $\\tilde { \\mathcal { O } } ( d \\operatorname* { m i n } \\{ n / \\epsilon ^ { 1 . 7 5 } , 1 / \\epsilon ^ { 2 . 7 5 } + \\sqrt { n } / \\epsilon ^ { 2 } \\} \\log ( 1 / \\delta ) )$ . □ ", + "bbox": [ + 173, + 637, + 826, + 800 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "E A FASTER HESSIAN-VECTOR BASED QCQP SOLVER ", + "text_level": 1, + "bbox": [ + 174, + 818, + 648, + 835 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We recall from the previous section that, to approximately solve a quadratic subproblem in ${ \\mathrm { { S T R } } } _ { \\mathrm { { f r e e } } }$ , Lanczos method requires $\\tilde { \\mathcal { O } } ( \\operatorname* { m i n } \\{ n / \\epsilon ^ { 0 . 2 5 } , 1 / \\bar { \\epsilon } ^ { 1 . 2 5 } \\} )$ stochastic Hessian-vector product evaluations. In this section, we propose a faster QCQP solver with an $\\tilde { \\mathcal { O } } ( \\operatorname* { m i n } \\{ n + n ^ { 0 . 7 5 } / \\epsilon ^ { 0 . 2 5 } , 1 / \\epsilon \\} )$ complexity. Replacing Lanczos method with this QCQP solver in $\\operatorname { S T R } _ { \\mathrm { f r e e } }$ results in a faster Hessian-free method, which we refer to as ${ \\mathrm { S T R } } _ { \\mathrm { f r e e } } +$ . ", + "bbox": [ + 173, + 849, + 826, + 924 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "E.1 CONVEX REFORMULATION OF QCQP ", + "text_level": 1, + "bbox": [ + 176, + 103, + 477, + 118 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "To begin with, we present a known result that is key to achieve faster algorithm than Lanczos method. We summarize this result in Lemma E.1 which shows that the trust region subproblem is equivalent to a convex QCQP. ", + "bbox": [ + 174, + 128, + 823, + 171 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Lemma E.1. (Convex Reformulation of QCQP (Flippo & Jansen, 1996; Wang & Xia, 2017)) Denote $\\lambda _ { \\mathrm { m i n } }$ as the smallest eigenvalue of $\\mathbf { H } ^ { k }$ . Let $\\mathbf { u } _ { \\mathrm { m i n } }$ be a corresponding eigenvector. W.l.o.g., we assume that $\\langle \\mathbf { g } ^ { k } , \\mathbf { u } _ { \\mathrm { m i n } } \\rangle \\leq 0$ . Let $\\mu = \\operatorname* { m i n } \\{ \\lambda _ { \\operatorname* { m i n } } , 0 \\}$ . Then the QCQP (8) is equivalent to the convex problem ", + "bbox": [ + 173, + 174, + 826, + 218 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/3c259b5f80ea3f9312154d89b17ab811f991e2da8e57c54e7a7d957cbd040c5d.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\substack { \\mathbf { h } \\in \\mathbb { R } ^ { d } , \\| \\mathbf { h } \\| \\leq r } } q ^ { k } ( \\mathbf { h } ) = \\langle \\mathbf { g } ^ { k } , \\mathbf { h } \\rangle + \\frac { 1 } { 2 } \\langle ( \\mathbf { H } ^ { k } - \\mu I ) \\mathbf { h } , \\mathbf { h } \\rangle + \\frac { 1 } { 2 } \\mu r ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 300, + 226, + 694, + 258 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "in the sense that (8) and (38) have the same minimum function value. Moreover, when $\\lambda _ { \\operatorname* { m i n } } < 0 ,$ , for any optimal solution of (38), denoted by $\\mathbf { h } _ { c } ^ { k }$ , ", + "bbox": [ + 173, + 260, + 826, + 289 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/4a84f0a66a70d1fd6e8264a455b9841b44c7da5a8599d395dd6b5468a6e797b6.jpg", + "text": "$$\n\\mathbf { h } _ { c } ^ { k } + \\frac { \\sqrt { \\langle \\mathbf { h } _ { c } ^ { k } , \\mathbf { u } _ { \\mathrm { m i n } } \\rangle ^ { 2 } - \\| \\mathbf { u } _ { \\mathrm { m i n } } \\| ^ { 2 } \\big ( \\| \\mathbf { h } _ { c } ^ { k } \\| ^ { 2 } - r ^ { 2 } \\big ) } - \\langle \\mathbf { h } _ { c } ^ { k } , \\mathbf { u } _ { \\mathrm { m i n } } \\rangle } { \\| \\mathbf { u } _ { \\mathrm { m i n } } \\| ^ { 2 } } \\mathbf { u } _ { \\mathrm { m i n } }\n$$", + "text_format": "latex", + "bbox": [ + 282, + 291, + 715, + 328 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "is a global minimizer of the original QCQP (8). ", + "bbox": [ + 173, + 330, + 485, + 345 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "To perform the above reformulation, one needs to compute the exact eigenpair $\\left( \\lambda _ { \\operatorname* { m i n } } , \\mathbf { u } _ { \\operatorname* { m i n } } \\right)$ . Nevertheless, as we shall see, it is sufficient to compute an approximate eigenpair $( \\tilde { \\lambda } , \\tilde { \\mathbf { u } } )$ such that ", + "bbox": [ + 174, + 354, + 823, + 386 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/a85cc13309ad658a543ec13c6a3f9f3866bf41c82b4eb5baea8eb426643121dc.jpg", + "text": "$$\n\\lambda _ { \\operatorname* { m i n } } \\leq \\tilde { \\lambda } = \\tilde { \\mathbf { u } } ^ { T } \\mathbf { H } ^ { k } \\tilde { \\mathbf { u } } \\leq \\lambda _ { \\operatorname* { m i n } } + \\tilde { \\epsilon } ,\n$$", + "text_format": "latex", + "bbox": [ + 385, + 395, + 611, + 414 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "where \u000f˜ is a target accuracy to be determined later. We note that $\\tilde { \\epsilon } \\leq 2 L _ { 2 }$ w.l.o.g. since $\\| \\mathbf { H } ^ { k } \\| \\leq L _ { 2 }$ . With this approximate eigenpair, it remains to solve the following convex problem ", + "bbox": [ + 173, + 416, + 825, + 444 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/92370e579ccde63752f9debd6649e54b60e3b2dda0048a056aaecb071c1a9b48.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\mathbf { h } \\in \\mathbb { R } ^ { d } , \\| \\mathbf { h } \\| \\leq r } \\tilde { q } ^ { k } ( \\mathbf { h } ) = \\langle \\mathbf { g } ^ { k } , \\mathbf { h } \\rangle + \\frac { 1 } { 2 } \\langle ( \\mathbf { H } ^ { k } - \\tilde { \\mu } I ) \\mathbf { h } , \\mathbf { h } \\rangle + \\frac { 1 } { 2 } \\tilde { \\mu } r ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 300, + 446, + 694, + 479 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "where $\\tilde { \\mu } = \\mathrm { m i n } \\{ 0 , \\tilde { \\lambda } - \\tilde { \\epsilon } \\}$ . One can check that the problem (41) well approximates (38). ", + "bbox": [ + 174, + 486, + 754, + 501 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Corollary E.1. Let $q _ { * } ^ { k }$ and $\\tilde { q } _ { * } ^ { k }$ be the minimum function value of (38) and (41), respectively. Assume $\\lambda _ { \\operatorname* { m i n } } \\leq \\tilde { \\lambda } = \\tilde { \\mathbf { u } } ^ { T } \\mathbf { H } ^ { k } \\tilde { \\mathbf { u } } \\leq \\lambda _ { \\operatorname* { m i n } } + \\tilde { \\epsilon } .$ . Then ", + "bbox": [ + 173, + 503, + 821, + 534 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/144b1097a8e6cfa77799b3ee5d9151f9f89fa95e3828e733548c43633f5362a3.jpg", + "text": "$$\n| q _ { * } ^ { k } - \\tilde { q } _ { * } ^ { k } | \\leq \\tilde { \\epsilon } r ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 442, + 536, + 555, + 555 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We note that the above convex reformulation approach divides an indefinite QCQP into two subproblems: (i) computation of an approximate eigenpair $( \\tilde { \\lambda } , \\tilde { \\mathbf { u } } )$ ; (ii) solving the convex problem (41). As we shall see, by exploiting the finite-sum structure of the Hessian $\\mathbf { H } ^ { k }$ , these two subproblems can be efficiently solved. We treat these two subproblems in the following two subsections, respectively. ", + "bbox": [ + 173, + 564, + 826, + 623 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "E.2 FINDING THE SMALLEST EIGENVECTOR", + "text_level": 1, + "bbox": [ + 176, + 638, + 495, + 654 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "To find a unit vector that satisfies requirement (40), we resort to the AppxPCA method (Allen-Zhu & Li, 2016), which first finds an approximate eigenvalue $\\lambda = \\lambda _ { \\operatorname* { m i n } } - \\tilde { \\epsilon }$ via binary search and then applies Power method to the positive definite matrix $( \\mathbf { H } ^ { k } - \\lambda I ) ^ { - 1 }$ for a logarithmic number of iterations. Computing $( \\mathbf { H } ^ { k } - \\lambda I ) ^ { - 1 } \\mathbf { v }$ for any vector $\\mathbf { v }$ is equivalent to solving the $\\tilde { \\epsilon }$ -strongly convex problem (Allen-Zhu & Li, 2018) ", + "bbox": [ + 173, + 665, + 825, + 736 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/dac887be88ba8de15c2391acc4b84c9a52c388b533f2345e8611cffc16c8fef5.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\mathbf { u } } \\phi ^ { k } ( \\mathbf { u } ) : = \\frac { 1 } { 2 } \\mathbf { u } ^ { T } ( \\mathbf { H } ^ { k } - \\lambda I ) \\mathbf { u } - \\langle \\mathbf { v } , \\mathbf { u } \\rangle\n$$", + "text_format": "latex", + "bbox": [ + 357, + 738, + 637, + 768 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We note that $\\begin{array} { r } { \\mathbf { H } ^ { k } = \\frac { 1 } { | \\cal { S } | } \\sum _ { i \\in \\cal { S } } \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) } \\end{array}$ . Specifically, in $\\mathrm { { S T R } _ { \\mathrm { { f r e e } } } }$ , either $| S | = n$ (i.e., $\\mathbf { H } ^ { k }$ is the full Hessian) or $| S | = \\tilde { \\mathcal { O } } ( L _ { 1 } ^ { 2 } / ( L _ { 2 } \\epsilon ) )$ by Lemma 4.1. Therefore, $\\phi ^ { k } ( \\cdot )$ can be expressed as sum of non-convex functions ", + "bbox": [ + 174, + 771, + 825, + 820 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/eec1192412c143e85db823e72aa5225f155ff2f2d760377afc4b2cc2ceb28aab.jpg", + "text": "$$\n\\phi ^ { k } ( \\mathbf { u } ) = \\frac { 1 } { | S | } \\sum _ { i \\in S } \\phi _ { i } ^ { k } ( \\mathbf { u } ) = \\frac { 1 } { | S | } \\sum _ { i \\in S } \\left( \\frac { 1 } { 2 } \\mathbf { u } ^ { T } ( \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) - \\lambda I ) \\mathbf { u } - \\langle \\mathbf { v } , \\mathbf { u } \\rangle \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 259, + 820, + 738, + 858 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "By observing that each $\\phi _ { i } ^ { k }$ is non-convex and has $\\left( 4 L _ { 2 } \\right)$ -Lipschitz gradient, we can use KatyushaXS (Allen-Zhu, 2018a) to solve problem (43) in $\\tilde { \\mathcal { O } } ( | S | + | S | ^ { 3 / 4 } \\sqrt { L _ { 2 } / \\tilde { \\epsilon } } )$ stochastic Hessianvector product (i.e., $\\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) \\mathbf { u } )$ evaluations. The following result is taken from (Agarwal et al., 2017, Section G.3), which gives the overall computation complexity of AppxPCA. ", + "bbox": [ + 174, + 862, + 826, + 924 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Algorithm 9 Fast QCQP Solver ", + "text_level": 1, + "bbox": [ + 174, + 103, + 387, + 118 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Input: $\\mathbf { H } ^ { k } , \\mathbf { g } ^ { k } , r , \\tilde { \\epsilon } , \\tilde { \\epsilon } _ { 1 }$ ", + "bbox": [ + 174, + 123, + 328, + 138 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "1: Use AppxPCA to find $( \\tilde { \\lambda } , \\tilde { \\mathbf { u } } )$ satisfying (40), in which the matrix inverse is solved by KatyushaXS; \n2: Use KatyushaXW to solve (41) up to accuracy $\\tilde { \\epsilon } _ { 1 }$ , i.e., find a vector $\\tilde { \\mathbf { h } }$ such that $\\tilde { q } ^ { k } ( \\mathbf { h } ) - \\tilde { q } _ { * } ^ { k } \\leq \\tilde { \\epsilon } _ { 1 }$ with high probability; \n3: Return $\\tilde { \\mathbf { h } } + ( \\sqrt { \\langle \\tilde { \\mathbf { h } } , \\tilde { \\mathbf { u } } \\rangle ^ { 2 } - \\| \\tilde { \\mathbf { u } } \\| ^ { 2 } ( \\| \\tilde { \\mathbf { h } } \\| ^ { 2 } - r ^ { 2 } ) } - \\langle \\tilde { \\mathbf { h } } , \\tilde { \\mathbf { u } } \\rangle ) \\tilde { \\mathbf { u } } / \\| \\tilde { \\mathbf { u } } \\| ^ { 2 } .$ ", + "bbox": [ + 178, + 133, + 826, + 209 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Algorithm 10 STRfree+ ", + "text_level": 1, + "bbox": [ + 174, + 223, + 330, + 238 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "1: In the same setting as MetaAlgorithm 7, \n2: construct gradient estimator $\\mathbf { g } ^ { k }$ by Estimator 4; \n3: construct Hessian estimator $\\mathbf { H } ^ { k }$ by \n4: Option I: $\\mathbf { H } ^ { k } : = \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } )$ ; \n5: Option II: Draw $s$ samples indexed by $\\mathcal { H }$ and let $\\mathbf { H } ^ { k } : = \\nabla ^ { 2 } f ( \\mathbf { x } ^ { k } ; \\mathcal { H } )$ ; \n6: use Algorithm 9 to solve QCQP subproblems. ", + "bbox": [ + 178, + 242, + 692, + 327 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Lemma E.2. Let $\\begin{array} { r } { \\mathbf { H } ^ { k } = \\frac { 1 } { | \\mathcal { S } | } \\sum _ { i \\in \\cal S } \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) \\in \\mathbb { R } ^ { d \\times d } } \\end{array}$ , where $\\| \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) \\| \\leq L _ { 2 }$ . With probability at least $1 - p ,$ , AppxPCA produces a unit vector u satisfying $\\mathbf { u } ^ { T } \\mathbf { H } ^ { k } \\mathbf { u } \\leq \\lambda _ { \\operatorname* { m i n } } + \\tilde { \\epsilon } .$ . The total stochastic Hessian-vector product oracle complexity is $\\tilde { \\mathcal { O } } ( | S | + | S | ^ { 3 / 4 } \\sqrt { L _ { 2 } / \\tilde { \\epsilon } } )$ . ", + "bbox": [ + 173, + 338, + 820, + 390 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "E.3 SOLVING THE CONVEX QCQP ", + "text_level": 1, + "bbox": [ + 176, + 406, + 428, + 422 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In what follows, we show that the convex problem (41) can be solved efficiently. We first observe that problem (41) has a finite-sum structure and can be rewritten as an unconstrained problem of the form ", + "bbox": [ + 173, + 434, + 823, + 463 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/3801b173622be69c84e2ae4917f4dd8581c66820bf6fb61fa7f318f2e17b01f4.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\mathbf { h \\in \\mathbb { R } } ^ { d } } \\frac { 1 } { | \\mathcal { S } | } \\sum _ { i \\in \\mathcal { S } } \\tilde { q } _ { i } ^ { k } ( \\mathbf { h } ) + \\Psi ( \\mathbf { h } ) = \\frac { 1 } { | \\mathcal { S } | } \\sum _ { i \\in \\mathcal { S } } \\Big ( \\langle \\mathbf { g } ^ { k } , \\mathbf { h } \\rangle + \\frac { 1 } { 2 } \\langle ( \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) - \\tilde { \\mu } I ) \\mathbf { h } , \\mathbf { h } \\rangle + \\frac { 1 } { 2 } \\tilde { \\mu } r ^ { 2 } \\Big ) + \\Psi ( \\mathbf { h } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 179, + 469, + 787, + 508 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "where $\\Psi ( { \\bf h } ) = 0$ if $\\| \\mathbf { h } \\| \\leq r$ , otherwise $\\Psi ( \\mathbf { h } ) \\ = \\ + \\infty$ . We note that each $\\tilde { q } _ { i } ^ { k } ( \\mathbf { h } )$ in (45) has $\\left( 4 L _ { 2 } \\right)$ -Lipschitz continuous gradient since $\\| \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) \\| \\leq L _ { 2 }$ and $\\tilde { \\epsilon } \\leq 2 L _ { 2 }$ . Therefore, we can use KatyushaX $\\mathrm { w }$ (Allen-Zhu, 2018a) to solve (45). By (Allen-Zhu, 2018a, Theorem 4.6), KatyushaXW finds a point h such that $\\mathbb { E } [ \\tilde { q } ^ { k } ( \\mathbf { h } ) - \\tilde { q } _ { * } ^ { k } ] \\leq \\tilde { \\epsilon } _ { 1 }$ using $\\tilde { \\mathcal { O } } ( | S | + | S | ^ { 3 / 4 } \\sqrt { L _ { 2 } } \\cdot r / \\sqrt { \\tilde { \\epsilon } _ { 1 } } )$ stochastic Hessianvector products, where $\\tilde { \\epsilon } _ { 1 }$ is the target accuracy to be determined later. ", + "bbox": [ + 173, + 516, + 826, + 590 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "E.4 PUTTING IT ALL TOGETHER ", + "text_level": 1, + "bbox": [ + 176, + 608, + 413, + 623 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "The complete procedure of our fast QCQP solver is summarized in Algorithm 9. Combining all the above results and setting $r = \\sqrt { \\epsilon / L _ { 2 } }$ , $\\tilde { \\epsilon } = \\sqrt { \\epsilon L _ { 2 } } / 2$ , and $\\tilde { \\epsilon } _ { 1 } = \\tilde { \\epsilon } r ^ { 2 }$ , one can find an approximate solution to QCQP (8) satisfying requirement (24) in ${ \\tilde { \\mathcal { O } } } ( | S | + | S | ^ { 3 / 4 } L _ { 2 } ^ { 0 . 2 5 } / \\epsilon ^ { 0 . 2 5 } )$ stochastic Hessianvector product evaluations. By replacing Lanczos method with this solver in $\\operatorname { S T R } _ { \\mathrm { f r e e } }$ , we derive a new Hessian-free method called ${ \\mathrm { S T R } } _ { \\mathrm { f r e e } } +$ , which is summarized in Algorithm 10. The following theorem establishes the overall runtime complexity of ${ \\mathrm { S T R } } _ { \\mathrm { f r e e } ^ { + } }$ for finding an $\\epsilon$ -SOSP. ", + "bbox": [ + 173, + 635, + 826, + 724 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Theorem E.1. Consider Algorithm 10 for solving problem (1). Let $\\zeta = 1 / 3$ , $r = \\sqrt { \\epsilon / L _ { 2 } }$ , $K =$ $4 \\sqrt { L _ { 2 } } \\Delta / \\epsilon ^ { 1 . 5 }$ , $\\begin{array} { r } { T = \\frac { 3 } { 2 } \\mathrm { { l o g } } ( 2 / \\delta ) } \\end{array}$ , $c _ { 1 } = 6 0 0$ , $c _ { 2 } = 5 0 0 \\mathrm { \\Omega }$ , and 32L21\u000fL2 log(4d/δ). The hyper-parameters√ in Estimator 4 are set to the same values as those in Lemma 4.2. Besides, let $\\tilde { \\epsilon } = \\sqrt { \\epsilon L _ { 2 } } / 2$ and $\\tilde { \\epsilon } _ { 1 } = \\tilde { \\epsilon } r ^ { 2 }$ in Algorithm 9. To find an $O ( \\epsilon ) { - } S O S P ~ w . p$ . at least $1 - \\delta$ , the runtime complexity is $\\tilde { \\mathcal { O } } ( d \\mathrm { m i n } \\{ n / \\epsilon ^ { 1 . 5 } + n ^ { 0 . 7 5 } / \\epsilon ^ { 1 . 7 5 } , 1 / \\epsilon ^ { 2 . 5 } + \\sqrt { n } / \\epsilon ^ { 2 } \\} \\mathrm { l o g } ( 1 / \\delta ) )$ . ", + "bbox": [ + 173, + 729, + 826, + 814 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Proof. The proof directly follows from that in Sec. D. ", + "bbox": [ + 174, + 832, + 529, + 847 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We compare the runtime complexity of $\\operatorname { S T R } _ { \\mathrm { f r e e } }$ and ${ \\mathrm { S T R } } _ { \\mathrm { f r e e } } +$ with existing Hessian free methods in Table 2. One can see that $\\mathrm { { S T R } _ { \\mathrm { { f r e e } } } }$ strictly outperforms Hessian-free Cubic. Besides, $\\mathrm { { S T R } _ { \\mathrm { { f r e e } } } }$ outperforms Fast-Cubic if $n \\ge \\Omega ( 1 / \\epsilon ^ { 4 / 3 } )$ , which is a mild condition for large-scale problems in the moderate accuracy case. ${ \\mathrm { S T R } } _ { \\mathrm { f r e e } ^ { + } }$ strictly outperforms both Hessian-free Cubic and Fast-Cubic. We note that the runtime analyses in (Tripuraneni et al., 2018; Zhou & Gu, 2019) rely on an additional assumption which states that for all $\\mathbf { x }$ , with probability 1, ", + "bbox": [ + 174, + 866, + 825, + 924 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/95ff5517569e6471795a7c5ec47d8ab253bfdb1a9cc4c1823c6362cb6ef03d6f.jpg", + "image_caption": [ + "Figure 4: Comparison of gradient norm on both the non-convex logistic regression and nonlinear least square problems. " + ], + "image_footnote": [], + "bbox": [ + 173, + 104, + 821, + 417 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 478, + 823, + 507 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/509f068284501f3903883673b0d3857c02310b9d5bdc531a8c42ef5487a8f873.jpg", + "text": "$$\n\\begin{array} { r } { \\| \\nabla f _ { i } ( \\mathbf { x } ) - \\nabla F ( \\mathbf { x } ) \\| \\leq \\sigma . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 411, + 513, + 586, + 531 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Under this additional assumption, one can use the same argument as in Section B.2 and B.3 to prove that ${ \\mathrm { { S T R } } } _ { \\mathrm { { f r e e } } }$ achieves a runtime complexity of $\\tilde { \\mathcal { O } } ( d \\operatorname* { m i n } \\{ n / \\epsilon ^ { 1 . 7 5 } , n ^ { 0 . 5 } / \\epsilon ^ { 2 } + 1 / \\epsilon ^ { 2 . 7 5 } , 1 / \\epsilon ^ { 3 } \\} ) ^ { 1 }$ . Similarly, the runtime complexity of ${ \\mathrm { S T R } } _ { \\mathrm { f r e e } ^ { + } }$ would be $\\tilde { \\mathcal { O } } ( d \\operatorname* { m i n } \\{ n / \\epsilon ^ { 1 . 5 } + n ^ { 0 . 7 5 } / \\epsilon ^ { 1 . 7 5 } , 1 / \\epsilon ^ { 2 . 5 } +$ $\\sqrt { n } / \\epsilon ^ { 2 } , \\dot { 1 } / \\epsilon ^ { 3 } \\} ,$ ). In this sense, both ${ \\mathrm { { S T R } } } _ { \\mathrm { { f r e e } } }$ and ${ \\mathrm { S T R } } _ { \\mathrm { f r e e } } +$ outperform Stochastic Cubic and $\\mathrm { S R V R C _ { \\mathrm { f r e e } } }$ . ", + "bbox": [ + 176, + 537, + 826, + 599 + ], + "page_idx": 20 + }, + { + "type": "table", + "img_path": "images/aaa7db88de47855f51ef9ff0d5360222271a70e204d05e9934103bd4e1614d5a.jpg", + "table_caption": [ + "Table 3: Descriptions of the five testing datasets. " + ], + "table_footnote": [], + "table_body": "
#sample#feature#sample#feature
a9a32.561123w8a49,749300
ijcnn49,99022phishing7,60468
codrna28.3058mnist60,000784
epsilon40,0002,000
", + "bbox": [ + 271, + 630, + 718, + 707 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "F ADDITIONAL EXPERIMENTAL RESULTS ", + "text_level": 1, + "bbox": [ + 173, + 728, + 535, + 744 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "F.1 MORE EXPERIMENTAL DETAILS ", + "text_level": 1, + "bbox": [ + 174, + 790, + 437, + 805 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Descriptions of Testing Datasets. We briefly introduce the seven testing datasets in the manuscript. Among them, six datasets are provided by the LibSVM website2, including (a9a, ijcnn, codrna, phishing, w8a and epsilon). The detailed information is summarized in Table 3. We can observe that these datasets are different from each other in feature dimension, training samples, etc. ", + "bbox": [ + 173, + 816, + 826, + 873 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Experimental Settings. In the manuscript, following SVRC (Zhou et al., 2018c) and LiteSVRC (Zhou et al., 2018b), we select hyper parameters from a set, namely $s _ { 1 }$ from $\\{ 0 . 2 n , 0 . 6 n , n \\}$ $s _ { 2 }$ from $\\lbrace 0 . 0 1 n , 0 . 1 n , 0 . 2 n \\rbrace$ , $p _ { 1 }$ and $p _ { 2 }$ from $\\{ 0 . 0 1 n ^ { 0 . 5 } , 0 . 0 5 n ^ { 0 . 5 } , 0 . 1 n ^ { 0 . 5 } \\}$ . For the Hessian estimation at the beginning of each $p _ { 2 }$ iterations, we use full Hessian. Similarly, for the gradient estimation at the beginning of each $p _ { 1 }$ iterations, we adopt the full gradient as the gradient estimation. ", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Memory Analysis. SVRC, Lite-SVRC, and our method need to store the previous and current gradient and Hessian and thus their memory complexity is $2 ( d ^ { 2 } + d )$ . TR, CR (ARC) and SCR need to compute current and Hessian and thus has complexity $d ^ { 2 } + d$ . So these memory is of the same order but our method is much faster than TR, CR and SCR both validated by theory and experiments. ", + "bbox": [ + 174, + 180, + 825, + 236 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "F.2 MORE EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 253, + 362, + 267 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Here we give more experimental results on the gradient norm v.s. the algorithm running time and the Hessian sample complexity. Due to the space limit, in the manuscript we only provide the gradient-norm related results on the codrna dataset. Here we provide results of a9a and ijcnn datasets in Figure 4. One can observe that on both the logistic regression with non-convex regularizer and the nonlinear least square problems, the proposed algorithm always shows sharper convergence behavior in terms of both the running time and the Hessian sample complexity. These observations are consistent with the results in Figure 2 in the manuscript. All these results demonstrate the high efficiency of our proposed algorithm and also confirm our theoretical implication. 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Experiments verify", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 315, + 421, + 327 + ], + "spans": [ + { + "bbox": [ + 142, + 315, + 421, + 327 + ], + "score": 1.0, + "content": "theoretical conclusions and the efficiency of the proposed algorithms.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 108, + 346, + 206, + 359 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 208, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 208, + 362 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 371, + 396, + 383 + ], + "lines": [ + { + "bbox": [ + 105, + 370, + 397, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 397, + 385 + ], + "score": 1.0, + "content": "We consider the following finite-sum non-convex minimization problem", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 243, + 387, + 366, + 413 + ], + "lines": [ + { + "bbox": [ + 243, + 387, + 366, + 413 + ], + "spans": [ + { + "bbox": [ + 243, + 387, + 366, + 413 + ], + "score": 0.94, + "content": "\\operatorname* { m i n } _ { \\mathbf { x } \\in \\mathbb { R } ^ { d } } F ( \\mathbf { x } ) = { \\frac { 1 } { n } } \\sum _ { i = 1 } ^ { n } f _ { i } ( \\mathbf { x } ) ,", + "type": "interline_equation", + "image_path": "5bc941f3b76f7588e96135223fb099d13cb31a550c3fcc643fd514ba8ede29cb.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 243, + 387, + 366, + 413 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 419, + 505, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 302, + 432 + ], + "score": 1.0, + "content": "where each (non-convex) component function", + "type": "text" + }, + { + "bbox": [ + 302, + 418, + 364, + 431 + ], + "score": 0.91, + "content": "f _ { i } : \\mathbb { R } ^ { d } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 418, + 450, + 432 + ], + "score": 1.0, + "content": "is assumed to have", + "type": "text" + }, + { + "bbox": [ + 450, + 420, + 462, + 430 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 418, + 506, + 432 + ], + "score": 1.0, + "content": "-Lipschitz", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 431, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 205, + 443 + ], + "score": 1.0, + "content": "continuous gradient and", + "type": "text" + }, + { + "bbox": [ + 205, + 431, + 218, + 442 + ], + "score": 0.89, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 431, + 505, + 443 + ], + "score": 1.0, + "content": "-Lipschitz continuous Hessian. Since first-order stationary points could", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "score": 1.0, + "content": "be saddle points with inferior generalization performance (Dauphin et al., 2014), in this work we are√", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 452, + 499, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 252, + 465 + ], + "score": 1.0, + "content": "particularly interested in computing", + "type": "text" + }, + { + "bbox": [ + 252, + 452, + 281, + 464 + ], + "score": 0.91, + "content": "( \\epsilon , \\sqrt { \\epsilon } )", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 452, + 461, + 465 + ], + "score": 1.0, + "content": "-approximate second-order stationary points,", + "type": "text" + }, + { + "bbox": [ + 461, + 454, + 466, + 462 + ], + "score": 0.63, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 452, + 499, + 465 + ], + "score": 1.0, + "content": "-SOSP:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 469, + 400, + 485 + ], + "lines": [ + { + "bbox": [ + 210, + 469, + 400, + 485 + ], + "spans": [ + { + "bbox": [ + 210, + 469, + 400, + 485 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\| \\nabla F ( { \\bf x } _ { \\epsilon } ) \\| \\le \\epsilon \\quad \\mathrm { a n d } \\quad \\nabla ^ { 2 } F ( { \\bf x } _ { \\epsilon } ) \\succcurlyeq - \\sqrt { L _ { 2 } \\epsilon } { \\bf I } . } \\end{array}", + "type": "interline_equation", + "image_path": "e51134a96f8117ef906583db50c1c35f049a30e8b883be7e1e2aa53e9c2854bd.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 210, + 469, + 400, + 485 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 490, + 506, + 613 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "score": 1.0, + "content": "To find a local minimum of problem (1), the cubic regularization approach (Nesterov & Polyak,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 501, + 507, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 507, + 513 + ], + "score": 1.0, + "content": "2006) and the trust region algorithm (Conn et al., 2000; Curtis et al., 2017) are two classical methods.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 433, + 525 + ], + "score": 1.0, + "content": "Specifically, cubic regularization forms a cubic surrogate function for the objective", + "type": "text" + }, + { + "bbox": [ + 433, + 512, + 455, + 524 + ], + "score": 0.92, + "content": "F ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 511, + 506, + 525 + ], + "score": 1.0, + "content": "by adding a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 524, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 506, + 535 + ], + "score": 1.0, + "content": "third-order regularization term to the second-order Taylor expansion, and minimizes it iteratively.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 260, + 547 + ], + "score": 1.0, + "content": "Such a method is proved to achieve an", + "type": "text" + }, + { + "bbox": [ + 260, + 534, + 304, + 547 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / k ^ { 2 / 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 533, + 462, + 547 + ], + "score": 1.0, + "content": "global convergence rate and thus needs", + "type": "text" + }, + { + "bbox": [ + 462, + 534, + 505, + 547 + ], + "score": 0.93, + "content": "\\mathcal { O } ( n / \\epsilon ^ { 1 . 5 } )", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 547, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 558 + ], + "score": 1.0, + "content": "stochastic first- and second-order oracle queries, namely the evaluation number of stochastic gradient", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 557, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 506, + 570 + ], + "score": 1.0, + "content": "and Hessian, to achieve a point that satisfies (2). On the other hand, trust region algorithms estimate", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "score": 1.0, + "content": "the objective with its second-order Taylor expansion but minimize it only within a local region.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 578, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 506, + 593 + ], + "score": 1.0, + "content": "Recently, Curtis et al. (2017) proposes a trust region variant to achieve the same convergence rate as", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "the cubic regularization approach. But both methods require computing full gradients and Hessians", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 600, + 420, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 117, + 613 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 601, + 140, + 613 + ], + "score": 0.92, + "content": "F ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 600, + 420, + 613 + ], + "score": 1.0, + "content": "and thus suffer from high computational cost in large-scale problems.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 617, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 617, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 505, + 629 + ], + "score": 1.0, + "content": "To avoid costly exact differential evaluations, many works explore the finite-sum structure of prob-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 628, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 506, + 641 + ], + "score": 1.0, + "content": "lem (1) and develop stochastic cubic regularization approaches. Both Kohler & Lucchi (2017b)", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "and Xu et al. 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By plugging a stochastic variance reduced estimator (Johnson & Zhang, 2013) and the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 673, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 505, + 686 + ], + "score": 1.0, + "content": "Hessian tracking technique (Gower et al., 2018) into the gradient and Hessian estimation, the approach", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 684, + 505, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 505, + 697 + ], + "score": 1.0, + "content": "in (Zhou et al., 2018a) improves both the stochastic first- and second-order oracle complexities to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 694, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 107, + 696, + 160, + 709 + ], + "score": 0.92, + "content": "\\tilde { \\mathcal { O } } ( n ^ { 0 . 8 } / \\epsilon ^ { 1 . 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 694, + 506, + 711 + ], + "score": 1.0, + "content": ". 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Towards this goal, we first prove that the trust region method with", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 234, + 470, + 246 + ], + "spans": [ + { + "bbox": [ + 141, + 234, + 470, + 246 + ], + "score": 1.0, + "content": "inexact gradient and Hessian estimation can achieve a convergence rate of order", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 142, + 244, + 471, + 259 + ], + "spans": [ + { + "bbox": [ + 142, + 244, + 187, + 258 + ], + "score": 0.92, + "content": "\\mathcal { O } ( 1 / k ^ { 2 / 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 244, + 471, + 259 + ], + "score": 1.0, + "content": "as long as those differential estimations are sufficiently accurate. 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This improves the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 291, + 470, + 306 + ], + "spans": [ + { + "bbox": [ + 141, + 291, + 281, + 306 + ], + "score": 1.0, + "content": "state-of-the-art result by a factor of", + "type": "text" + }, + { + "bbox": [ + 281, + 291, + 316, + 304 + ], + "score": 0.93, + "content": "\\mathcal { O } ( n ^ { 1 / 6 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 291, + 470, + 306 + ], + "score": 1.0, + "content": ". Finally, we also develop Hessian-free", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 303, + 469, + 316 + ], + "spans": [ + { + "bbox": [ + 141, + 303, + 469, + 316 + ], + "score": 1.0, + "content": "STR algorithms which achieve the lowest runtime complexity. Experiments verify", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 315, + 421, + 327 + ], + "spans": [ + { + "bbox": [ + 142, + 315, + 421, + 327 + ], + "score": 1.0, + "content": "theoretical conclusions and the efficiency of the proposed algorithms.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9.5, + "bbox_fs": [ + 141, + 212, + 471, + 327 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 346, + 206, + 359 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 208, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 208, + 362 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 371, + 396, + 383 + ], + "lines": [ + { + "bbox": [ + 105, + 370, + 397, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 397, + 385 + ], + "score": 1.0, + "content": "We consider the following finite-sum non-convex minimization problem", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 370, + 397, + 385 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 243, + 387, + 366, + 413 + ], + "lines": [ + { + "bbox": [ + 243, + 387, + 366, + 413 + ], + "spans": [ + { + "bbox": [ + 243, + 387, + 366, + 413 + ], + "score": 0.94, + "content": "\\operatorname* { m i n } _ { \\mathbf { x } \\in \\mathbb { R } ^ { d } } F ( \\mathbf { x } ) = { \\frac { 1 } { n } } \\sum _ { i = 1 } ^ { n } f _ { i } ( \\mathbf { x } ) ,", + "type": "interline_equation", + "image_path": "5bc941f3b76f7588e96135223fb099d13cb31a550c3fcc643fd514ba8ede29cb.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 243, + 387, + 366, + 413 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 419, + 505, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 302, + 432 + ], + "score": 1.0, + "content": "where each (non-convex) component function", + "type": "text" + }, + { + "bbox": [ + 302, + 418, + 364, + 431 + ], + "score": 0.91, + "content": "f _ { i } : \\mathbb { R } ^ { d } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 418, + 450, + 432 + ], + "score": 1.0, + "content": "is assumed to have", + "type": "text" + }, + { + "bbox": [ + 450, + 420, + 462, + 430 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 418, + 506, + 432 + ], + "score": 1.0, + "content": "-Lipschitz", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 431, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 205, + 443 + ], + "score": 1.0, + "content": "continuous gradient and", + "type": "text" + }, + { + "bbox": [ + 205, + 431, + 218, + 442 + ], + "score": 0.89, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 431, + 505, + 443 + ], + "score": 1.0, + "content": "-Lipschitz continuous Hessian. Since first-order stationary points could", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "score": 1.0, + "content": "be saddle points with inferior generalization performance (Dauphin et al., 2014), in this work we are√", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 452, + 499, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 252, + 465 + ], + "score": 1.0, + "content": "particularly interested in computing", + "type": "text" + }, + { + "bbox": [ + 252, + 452, + 281, + 464 + ], + "score": 0.91, + "content": "( \\epsilon , \\sqrt { \\epsilon } )", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 452, + 461, + 465 + ], + "score": 1.0, + "content": "-approximate second-order stationary points,", + "type": "text" + }, + { + "bbox": [ + 461, + 454, + 466, + 462 + ], + "score": 0.63, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 452, + 499, + 465 + ], + "score": 1.0, + "content": "-SOSP:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 418, + 506, + 465 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 469, + 400, + 485 + ], + "lines": [ + { + "bbox": [ + 210, + 469, + 400, + 485 + ], + "spans": [ + { + "bbox": [ + 210, + 469, + 400, + 485 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\| \\nabla F ( { \\bf x } _ { \\epsilon } ) \\| \\le \\epsilon \\quad \\mathrm { a n d } \\quad \\nabla ^ { 2 } F ( { \\bf x } _ { \\epsilon } ) \\succcurlyeq - \\sqrt { L _ { 2 } \\epsilon } { \\bf I } . } \\end{array}", + "type": "interline_equation", + "image_path": "e51134a96f8117ef906583db50c1c35f049a30e8b883be7e1e2aa53e9c2854bd.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 210, + 469, + 400, + 485 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 490, + 506, + 613 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "score": 1.0, + "content": "To find a local minimum of problem (1), the cubic regularization approach (Nesterov & Polyak,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 501, + 507, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 507, + 513 + ], + "score": 1.0, + "content": "2006) and the trust region algorithm (Conn et al., 2000; Curtis et al., 2017) are two classical methods.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 433, + 525 + ], + "score": 1.0, + "content": "Specifically, cubic regularization forms a cubic surrogate function for the objective", + "type": "text" + }, + { + "bbox": [ + 433, + 512, + 455, + 524 + ], + "score": 0.92, + "content": "F ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 511, + 506, + 525 + ], + "score": 1.0, + "content": "by adding a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 524, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 506, + 535 + ], + "score": 1.0, + "content": "third-order regularization term to the second-order Taylor expansion, and minimizes it iteratively.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 260, + 547 + ], + "score": 1.0, + "content": "Such a method is proved to achieve an", + "type": "text" + }, + { + "bbox": [ + 260, + 534, + 304, + 547 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / k ^ { 2 / 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 533, + 462, + 547 + ], + "score": 1.0, + "content": "global convergence rate and thus needs", + "type": "text" + }, + { + "bbox": [ + 462, + 534, + 505, + 547 + ], + "score": 0.93, + "content": "\\mathcal { O } ( n / \\epsilon ^ { 1 . 5 } )", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 547, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 558 + ], + "score": 1.0, + "content": "stochastic first- and second-order oracle queries, namely the evaluation number of stochastic gradient", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 557, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 506, + 570 + ], + "score": 1.0, + "content": "and Hessian, to achieve a point that satisfies (2). On the other hand, trust region algorithms estimate", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "score": 1.0, + "content": "the objective with its second-order Taylor expansion but minimize it only within a local region.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 578, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 506, + 593 + ], + "score": 1.0, + "content": "Recently, Curtis et al. (2017) proposes a trust region variant to achieve the same convergence rate as", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "the cubic regularization approach. But both methods require computing full gradients and Hessians", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 600, + 420, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 117, + 613 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 601, + 140, + 613 + ], + "score": 0.92, + "content": "F ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 600, + 420, + 613 + ], + "score": 1.0, + "content": "and thus suffer from high computational cost in large-scale problems.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 489, + 507, + 613 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 617, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 617, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 505, + 629 + ], + "score": 1.0, + "content": "To avoid costly exact differential evaluations, many works explore the finite-sum structure of prob-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 628, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 506, + 641 + ], + "score": 1.0, + "content": "lem (1) and develop stochastic cubic regularization approaches. 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By plugging a stochastic variance reduced estimator (Johnson & Zhang, 2013) and the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 673, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 505, + 686 + ], + "score": 1.0, + "content": "Hessian tracking technique (Gower et al., 2018) into the gradient and Hessian estimation, the approach", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 684, + 505, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 505, + 697 + ], + "score": 1.0, + "content": "in (Zhou et al., 2018a) improves both the stochastic first- and second-order oracle complexities to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 694, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 107, + 696, + 160, + 709 + ], + "score": 0.92, + "content": "\\tilde { \\mathcal { O } } ( n ^ { 0 . 8 } / \\epsilon ^ { 1 . 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 694, + 506, + 711 + ], + "score": 1.0, + "content": ". 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When SSO is prioritized, our", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 102, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 130, + 113 + ], + "score": 0.88, + "content": "\\mathrm { { S T R } _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 102, + 506, + 115 + ], + "score": 1.0, + "content": "has strictly better complexity than both SCR and Lite-SVRC. When SFO and SSO are treated", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 113, + 320, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 140, + 126 + ], + "score": 1.0, + "content": "equally,", + "type": "text" + }, + { + "bbox": [ + 140, + 113, + 164, + 124 + ], + "score": 0.88, + "content": "\\mathrm { { S T R _ { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 114, + 320, + 126 + ], + "score": 1.0, + "content": "improves the existing result in SVRC.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "table_body", + "bbox": [ + 128, + 129, + 483, + 235 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 128, + 129, + 483, + 235 + ], + "spans": [ + { + "bbox": [ + 128, + 129, + 483, + 235 + ], + "score": 0.984, + "html": "
AlgorithmSFOSSO
TR (Conn et al., 2000; Curtis et al., 2017)CR (Nesterov & Polyak, 2006)SCR (Kohler & Lucchi, 2017a)SVRC (Zhou et al., 2018c)Lite-SVRC (Zhou et al., 2018b)0(共n01.5013.52.5O n共0(1.50（1（.5）6国1.5
STRi (this paper)STR2 (this paper)O(min{n1.5，1.5O(min{1，1.5
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This idea is leveraged to develop two efficient stochastic", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 262, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 506, + 276 + ], + "score": 1.0, + "content": "trust region (STR) approaches. We tailor our methods to achieve state-of-the-art oracle complexities", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 274, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 506, + 286 + ], + "score": 1.0, + "content": "under the following two measurements: (i) the stochastic second-order oracle complexity is prioritized;", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "(ii) the stochastic first- and second-order oracle complexities are treated equally. Specifically, in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 296, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 205, + 309 + ], + "score": 1.0, + "content": "Setting (i), our method", + "type": "text" + }, + { + "bbox": [ + 206, + 296, + 230, + 307 + ], + "score": 0.89, + "content": "\\mathrm { { S T R } _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 296, + 506, + 309 + ], + "score": 1.0, + "content": "employs a newly proposed estimator to approximate the Hessian", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "and adopts the estimator in (Fang et al., 2018) for gradient approximation. Our novel Hessian", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 330 + ], + "score": 1.0, + "content": "estimator maintains an accurate second-order differential approximation with lower amortized oracle", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 328, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 104, + 328, + 211, + 344 + ], + "score": 1.0, + "content": "complexity. In this way,", + "type": "text" + }, + { + "bbox": [ + 211, + 330, + 235, + 342 + ], + "score": 0.89, + "content": "\\operatorname { S T R } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 328, + 275, + 344 + ], + "score": 1.0, + "content": "achieves", + "type": "text" + }, + { + "bbox": [ + 276, + 329, + 376, + 343 + ], + "score": 0.91, + "content": "\\tilde { \\mathcal { O } } ( \\operatorname* { m i n } \\{ 1 / \\epsilon ^ { 2 } , \\sqrt { n } / \\epsilon ^ { 1 . 5 } \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 328, + 506, + 344 + ], + "score": 1.0, + "content": "stochastic second-order oracle", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "score": 1.0, + "content": "complexity. This is lower than existing results for solving problem (1). In Setting (ii), our method", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 131, + 363 + ], + "score": 0.89, + "content": "\\mathrm { { S T R } _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 352, + 282, + 365 + ], + "score": 1.0, + "content": "substitutes the gradient estimator in", + "type": "text" + }, + { + "bbox": [ + 282, + 352, + 307, + 363 + ], + "score": 0.9, + "content": "\\mathrm { { S T R } _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "with one that integrates stochastic gradient and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 441, + 376 + ], + "score": 1.0, + "content": "Hessian together to maintain an accurate gradient approximation. As a result,", + "type": "text" + }, + { + "bbox": [ + 441, + 363, + 466, + 375 + ], + "score": 0.86, + "content": "\\mathrm { { S T R _ { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "achieves", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 374, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 104, + 374, + 168, + 389 + ], + "score": 1.0, + "content": "convergence in", + "type": "text" + }, + { + "bbox": [ + 169, + 374, + 223, + 388 + ], + "score": 0.94, + "content": "{ \\tilde { \\mathcal { O } } } ( n ^ { 3 / 4 } / \\epsilon ^ { 1 . 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 374, + 506, + 389 + ], + "score": 1.0, + "content": "overall stochastic first- and second-order oracle queries. Finally, based", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 384, + 399 + ], + "score": 1.0, + "content": "on STR, we further develop Hessian-free STR algorithms, namely", + "type": "text" + }, + { + "bbox": [ + 385, + 387, + 416, + 398 + ], + "score": 0.84, + "content": "\\mathrm { { S T R } _ { \\mathrm { { f r e e } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 387, + 436, + 399 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 436, + 387, + 473, + 398 + ], + "score": 0.89, + "content": "{ \\mathrm { S T R } } _ { \\mathrm { f r e e } } +", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 387, + 505, + 399 + ], + "score": 1.0, + "content": ", which", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 398, + 339, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 339, + 411 + ], + "score": 1.0, + "content": "outperform existing Hessian-free algorithms theoretically.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 108, + 419, + 203, + 430 + ], + "lines": [ + { + "bbox": [ + 106, + 419, + 204, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 204, + 431 + ], + "score": 1.0, + "content": "1.1 RELATED WORK", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 432, + 506, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 506, + 444 + ], + "score": 1.0, + "content": "Computing local minimum to a non-convex optimization problem is gaining considerable amount of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 443, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 455 + ], + "score": 1.0, + "content": "attentions in recent years. Both cubic regularization (CR) approaches (Nesterov & Polyak, 2006)", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "and trust region (TR) algorithms (Conn et al., 2000; Curtis et al., 2017) can escape saddle points", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "and find a local minimum by iterating the variable along the direction related to the eigenvector of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "the Hessian with the most negative eigenvalue. As the CR heavily depends on the regularization", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 487, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 506, + 499 + ], + "score": 1.0, + "content": "parameter for the cubic term, Cartis et al. (2011) propose an adaptive cubic regularization (ARC)", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "score": 1.0, + "content": "approach to boost the efficiency by adaptively tuning the regularization parameter according to the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 508, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 505, + 521 + ], + "score": 1.0, + "content": "current objective decrease. Noting the high cost of full gradient and Hessian computation in ARC,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 519, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 532 + ], + "score": 1.0, + "content": "sub-sampled cubic regularization (SCR) (Kohler & Lucchi, 2017a) is developed for sampling partial", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "data points to estimate the full gradient and Hessian. Recently, by exploring the finite-sum structure", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 542, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 506, + 554 + ], + "score": 1.0, + "content": "of the target problem, many works incorporate variance reduction technique (Johnson & Zhang, 2013)", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "into CR and propose stochastic variance-reduced methods. For example, Zhou et al. (2018c) propose", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 564, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 505, + 575 + ], + "score": 1.0, + "content": "stochastic variance-reduced cubic (SVRC) in which they integrate the stochastic variance-reduced", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "gradient estimator (Johnson & Zhang, 2013) and the Hessian tracking technique (Gower et al., 2018)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 303, + 598 + ], + "score": 1.0, + "content": "with CR. Such a method is proved to be at least", + "type": "text" + }, + { + "bbox": [ + 303, + 585, + 338, + 598 + ], + "score": 0.93, + "content": "\\mathcal { O } ( n ^ { 1 / 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 586, + 506, + 598 + ], + "score": 1.0, + "content": "faster than CR and TR. Then Zhou et al.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 597, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 506, + 610 + ], + "score": 1.0, + "content": "(2018b) use adaptive gradient batch size and constant Hessian batch size, and develop Lite-SVRC to", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 607, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 104, + 607, + 300, + 623 + ], + "score": 1.0, + "content": "further reduce the stochastic second-order oracle", + "type": "text" + }, + { + "bbox": [ + 301, + 608, + 355, + 622 + ], + "score": 0.94, + "content": "{ \\tilde { \\mathcal { O } } } ( n ^ { 4 / 5 } / \\epsilon ^ { 1 . 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 607, + 406, + 623 + ], + "score": 1.0, + "content": "of SVRC to", + "type": "text" + }, + { + "bbox": [ + 406, + 608, + 461, + 622 + ], + "score": 0.94, + "content": "\\tilde { \\mathcal { O } } ( n ^ { 2 / 3 } / \\epsilon ^ { 1 . 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 607, + 506, + 623 + ], + "score": 1.0, + "content": "at the cost", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "score": 1.0, + "content": "of higher gradient computation cost. Similarly, except turning the gradient batch size, Zhang et al.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "(2018) further adaptively sample a certain number of data points to estimate the Hessian and prove", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 643, + 492, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 492, + 654 + ], + "score": 1.0, + "content": "the proposed method to have the same stochastic second-order oracle complexity as Lite-SVRC.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 32.5 + }, + { + "type": "title", + "bbox": [ + 107, + 664, + 198, + 676 + ], + "lines": [ + { + "bbox": [ + 104, + 662, + 200, + 679 + ], + "spans": [ + { + "bbox": [ + 104, + 662, + 200, + 679 + ], + "score": 1.0, + "content": "2 PRELIMINARY", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 681, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 180, + 694 + ], + "score": 1.0, + "content": "Notation. We use", + "type": "text" + }, + { + "bbox": [ + 181, + 682, + 198, + 694 + ], + "score": 0.92, + "content": "\\| \\mathbf { v } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 681, + 356, + 694 + ], + "score": 1.0, + "content": "to denote the Euclidean norm of vector", + "type": "text" + }, + { + "bbox": [ + 356, + 684, + 363, + 692 + ], + "score": 0.52, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 681, + 397, + 694 + ], + "score": 1.0, + "content": "and use", + "type": "text" + }, + { + "bbox": [ + 397, + 682, + 417, + 694 + ], + "score": 0.91, + "content": "\\| \\mathbf { A } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "to denote the spectral", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 693, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 205, + 705 + ], + "score": 1.0, + "content": "norm of matrix A. Let", + "type": "text" + }, + { + "bbox": [ + 206, + 694, + 214, + 703 + ], + "score": 0.79, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 693, + 505, + 705 + ], + "score": 1.0, + "content": "be the set of component indices. We define the minibatch average of", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 103, + 700, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 103, + 700, + 210, + 723 + ], + "score": 1.0, + "content": "component functions by", + "type": "text" + }, + { + "bbox": [ + 210, + 705, + 325, + 722 + ], + "score": 0.92, + "content": "\\begin{array} { r } { f ( \\mathbf { x } ; \\mathcal { S } ) \\ \\stackrel { \\mathrm { d e f } } { = } \\ \\frac { 1 } { | \\mathcal { S } | } \\sum _ { i \\in \\mathcal { S } } f _ { i } ( \\mathbf { x } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 700, + 507, + 723 + ], + "score": 1.0, + "content": ". Then we specify the assumptions that are", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 720, + 273, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 273, + 733 + ], + "score": 1.0, + "content": "necessary to the analysis of our methods.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 128, + 129, + 483, + 235 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 505, + 124 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 79, + 506, + 93 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 506, + 93 + ], + "score": 1.0, + "content": "Table 1: Stochastic first- and second-order oracle complexities, SFO and SSO for short respectively,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 90, + 506, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 90, + 506, + 104 + ], + "score": 1.0, + "content": "of the proposed STR approaches and other state-of-the-art methods. When SSO is prioritized, our", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 102, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 130, + 113 + ], + "score": 0.88, + "content": "\\mathrm { { S T R } _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 102, + 506, + 115 + ], + "score": 1.0, + "content": "has strictly better complexity than both SCR and Lite-SVRC. When SFO and SSO are treated", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 113, + 320, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 140, + 126 + ], + "score": 1.0, + "content": "equally,", + "type": "text" + }, + { + "bbox": [ + 140, + 113, + 164, + 124 + ], + "score": 0.88, + "content": "\\mathrm { { S T R _ { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 114, + 320, + 126 + ], + "score": 1.0, + "content": "improves the existing result in SVRC.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "table_body", + "bbox": [ + 128, + 129, + 483, + 235 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 128, + 129, + 483, + 235 + ], + "spans": [ + { + "bbox": [ + 128, + 129, + 483, + 235 + ], + "score": 0.984, + "html": "
AlgorithmSFOSSO
TR (Conn et al., 2000; Curtis et al., 2017)CR (Nesterov & Polyak, 2006)SCR (Kohler & Lucchi, 2017a)SVRC (Zhou et al., 2018c)Lite-SVRC (Zhou et al., 2018b)0(共n01.5013.52.5O n共0(1.50（1（.5）6国1.5
STRi (this paper)STR2 (this paper)O(min{n1.5，1.5O(min{1，1.5
", + "type": "table", + "image_path": "641aadcafe1f9fc794a90c86245d00127cccfe259998bf5b8664df3dd144c614.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 128, + 129, + 483, + 164.33333333333334 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 128, + 164.33333333333334, + 483, + 199.66666666666669 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 128, + 199.66666666666669, + 483, + 235.00000000000003 + ], + "spans": [], + "index": 6 + } + ] + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 106, + 241, + 505, + 409 + ], + "lines": [ + { + "bbox": [ + 106, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "Contributions: In this paper we propose and exploit a formulation in which we make explicit control", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 253, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 264 + ], + "score": 1.0, + "content": "of the step size in the trust region method. This idea is leveraged to develop two efficient stochastic", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 262, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 506, + 276 + ], + "score": 1.0, + "content": "trust region (STR) approaches. We tailor our methods to achieve state-of-the-art oracle complexities", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 274, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 506, + 286 + ], + "score": 1.0, + "content": "under the following two measurements: (i) the stochastic second-order oracle complexity is prioritized;", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "(ii) the stochastic first- and second-order oracle complexities are treated equally. Specifically, in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 296, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 205, + 309 + ], + "score": 1.0, + "content": "Setting (i), our method", + "type": "text" + }, + { + "bbox": [ + 206, + 296, + 230, + 307 + ], + "score": 0.89, + "content": "\\mathrm { { S T R } _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 296, + 506, + 309 + ], + "score": 1.0, + "content": "employs a newly proposed estimator to approximate the Hessian", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "and adopts the estimator in (Fang et al., 2018) for gradient approximation. Our novel Hessian", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 330 + ], + "score": 1.0, + "content": "estimator maintains an accurate second-order differential approximation with lower amortized oracle", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 328, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 104, + 328, + 211, + 344 + ], + "score": 1.0, + "content": "complexity. In this way,", + "type": "text" + }, + { + "bbox": [ + 211, + 330, + 235, + 342 + ], + "score": 0.89, + "content": "\\operatorname { S T R } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 328, + 275, + 344 + ], + "score": 1.0, + "content": "achieves", + "type": "text" + }, + { + "bbox": [ + 276, + 329, + 376, + 343 + ], + "score": 0.91, + "content": "\\tilde { \\mathcal { O } } ( \\operatorname* { m i n } \\{ 1 / \\epsilon ^ { 2 } , \\sqrt { n } / \\epsilon ^ { 1 . 5 } \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 328, + 506, + 344 + ], + "score": 1.0, + "content": "stochastic second-order oracle", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "score": 1.0, + "content": "complexity. This is lower than existing results for solving problem (1). In Setting (ii), our method", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 131, + 363 + ], + "score": 0.89, + "content": "\\mathrm { { S T R } _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 352, + 282, + 365 + ], + "score": 1.0, + "content": "substitutes the gradient estimator in", + "type": "text" + }, + { + "bbox": [ + 282, + 352, + 307, + 363 + ], + "score": 0.9, + "content": "\\mathrm { { S T R } _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "with one that integrates stochastic gradient and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 441, + 376 + ], + "score": 1.0, + "content": "Hessian together to maintain an accurate gradient approximation. As a result,", + "type": "text" + }, + { + "bbox": [ + 441, + 363, + 466, + 375 + ], + "score": 0.86, + "content": "\\mathrm { { S T R _ { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "achieves", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 374, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 104, + 374, + 168, + 389 + ], + "score": 1.0, + "content": "convergence in", + "type": "text" + }, + { + "bbox": [ + 169, + 374, + 223, + 388 + ], + "score": 0.94, + "content": "{ \\tilde { \\mathcal { O } } } ( n ^ { 3 / 4 } / \\epsilon ^ { 1 . 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 374, + 506, + 389 + ], + "score": 1.0, + "content": "overall stochastic first- and second-order oracle queries. Finally, based", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 384, + 399 + ], + "score": 1.0, + "content": "on STR, we further develop Hessian-free STR algorithms, namely", + "type": "text" + }, + { + "bbox": [ + 385, + 387, + 416, + 398 + ], + "score": 0.84, + "content": "\\mathrm { { S T R } _ { \\mathrm { { f r e e } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 387, + 436, + 399 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 436, + 387, + 473, + 398 + ], + "score": 0.89, + "content": "{ \\mathrm { S T R } } _ { \\mathrm { f r e e } } +", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 387, + 505, + 399 + ], + "score": 1.0, + "content": ", which", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 398, + 339, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 339, + 411 + ], + "score": 1.0, + "content": "outperform existing Hessian-free algorithms theoretically.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 14, + "bbox_fs": [ + 104, + 241, + 506, + 411 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 419, + 203, + 430 + ], + "lines": [ + { + "bbox": [ + 106, + 419, + 204, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 204, + 431 + ], + "score": 1.0, + "content": "1.1 RELATED WORK", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 432, + 506, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 506, + 444 + ], + "score": 1.0, + "content": "Computing local minimum to a non-convex optimization problem is gaining considerable amount of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 443, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 455 + ], + "score": 1.0, + "content": "attentions in recent years. Both cubic regularization (CR) approaches (Nesterov & Polyak, 2006)", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "and trust region (TR) algorithms (Conn et al., 2000; Curtis et al., 2017) can escape saddle points", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "and find a local minimum by iterating the variable along the direction related to the eigenvector of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "the Hessian with the most negative eigenvalue. As the CR heavily depends on the regularization", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 487, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 506, + 499 + ], + "score": 1.0, + "content": "parameter for the cubic term, Cartis et al. (2011) propose an adaptive cubic regularization (ARC)", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "score": 1.0, + "content": "approach to boost the efficiency by adaptively tuning the regularization parameter according to the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 508, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 505, + 521 + ], + "score": 1.0, + "content": "current objective decrease. Noting the high cost of full gradient and Hessian computation in ARC,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 519, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 532 + ], + "score": 1.0, + "content": "sub-sampled cubic regularization (SCR) (Kohler & Lucchi, 2017a) is developed for sampling partial", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "data points to estimate the full gradient and Hessian. Recently, by exploring the finite-sum structure", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 542, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 506, + 554 + ], + "score": 1.0, + "content": "of the target problem, many works incorporate variance reduction technique (Johnson & Zhang, 2013)", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "into CR and propose stochastic variance-reduced methods. For example, Zhou et al. (2018c) propose", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 564, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 505, + 575 + ], + "score": 1.0, + "content": "stochastic variance-reduced cubic (SVRC) in which they integrate the stochastic variance-reduced", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "gradient estimator (Johnson & Zhang, 2013) and the Hessian tracking technique (Gower et al., 2018)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 303, + 598 + ], + "score": 1.0, + "content": "with CR. Such a method is proved to be at least", + "type": "text" + }, + { + "bbox": [ + 303, + 585, + 338, + 598 + ], + "score": 0.93, + "content": "\\mathcal { O } ( n ^ { 1 / 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 586, + 506, + 598 + ], + "score": 1.0, + "content": "faster than CR and TR. Then Zhou et al.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 597, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 506, + 610 + ], + "score": 1.0, + "content": "(2018b) use adaptive gradient batch size and constant Hessian batch size, and develop Lite-SVRC to", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 607, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 104, + 607, + 300, + 623 + ], + "score": 1.0, + "content": "further reduce the stochastic second-order oracle", + "type": "text" + }, + { + "bbox": [ + 301, + 608, + 355, + 622 + ], + "score": 0.94, + "content": "{ \\tilde { \\mathcal { O } } } ( n ^ { 4 / 5 } / \\epsilon ^ { 1 . 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 607, + 406, + 623 + ], + "score": 1.0, + "content": "of SVRC to", + "type": "text" + }, + { + "bbox": [ + 406, + 608, + 461, + 622 + ], + "score": 0.94, + "content": "\\tilde { \\mathcal { O } } ( n ^ { 2 / 3 } / \\epsilon ^ { 1 . 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 607, + 506, + 623 + ], + "score": 1.0, + "content": "at the cost", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "score": 1.0, + "content": "of higher gradient computation cost. Similarly, except turning the gradient batch size, Zhang et al.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "(2018) further adaptively sample a certain number of data points to estimate the Hessian and prove", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 643, + 492, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 492, + 654 + ], + "score": 1.0, + "content": "the proposed method to have the same stochastic second-order oracle complexity as Lite-SVRC.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 32.5, + "bbox_fs": [ + 104, + 432, + 506, + 654 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 664, + 198, + 676 + ], + "lines": [ + { + "bbox": [ + 104, + 662, + 200, + 679 + ], + "spans": [ + { + "bbox": [ + 104, + 662, + 200, + 679 + ], + "score": 1.0, + "content": "2 PRELIMINARY", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 681, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 180, + 694 + ], + "score": 1.0, + "content": "Notation. We use", + "type": "text" + }, + { + "bbox": [ + 181, + 682, + 198, + 694 + ], + "score": 0.92, + "content": "\\| \\mathbf { v } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 681, + 356, + 694 + ], + "score": 1.0, + "content": "to denote the Euclidean norm of vector", + "type": "text" + }, + { + "bbox": [ + 356, + 684, + 363, + 692 + ], + "score": 0.52, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 681, + 397, + 694 + ], + "score": 1.0, + "content": "and use", + "type": "text" + }, + { + "bbox": [ + 397, + 682, + 417, + 694 + ], + "score": 0.91, + "content": "\\| \\mathbf { A } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "to denote the spectral", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 693, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 205, + 705 + ], + "score": 1.0, + "content": "norm of matrix A. Let", + "type": "text" + }, + { + "bbox": [ + 206, + 694, + 214, + 703 + ], + "score": 0.79, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 693, + 505, + 705 + ], + "score": 1.0, + "content": "be the set of component indices. We define the minibatch average of", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 103, + 700, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 103, + 700, + 210, + 723 + ], + "score": 1.0, + "content": "component functions by", + "type": "text" + }, + { + "bbox": [ + 210, + 705, + 325, + 722 + ], + "score": 0.92, + "content": "\\begin{array} { r } { f ( \\mathbf { x } ; \\mathcal { S } ) \\ \\stackrel { \\mathrm { d e f } } { = } \\ \\frac { 1 } { | \\mathcal { S } | } \\sum _ { i \\in \\mathcal { S } } f _ { i } ( \\mathbf { x } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 700, + 507, + 723 + ], + "score": 1.0, + "content": ". Then we specify the assumptions that are", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 720, + 273, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 273, + 733 + ], + "score": 1.0, + "content": "necessary to the analysis of our methods.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45.5, + "bbox_fs": [ + 103, + 681, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 302, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 303, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 303, + 95 + ], + "score": 1.0, + "content": "MetaAlgorithm 1 Inexact Trust Region Method", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 99, + 504, + 200 + ], + "lines": [ + { + "bbox": [ + 105, + 97, + 504, + 111 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 186, + 111 + ], + "score": 1.0, + "content": "Input: initial point", + "type": "text" + }, + { + "bbox": [ + 186, + 99, + 198, + 109 + ], + "score": 0.8, + "content": "\\mathbf { x } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 97, + 237, + 111 + ], + "score": 1.0, + "content": ", step size", + "type": "text" + }, + { + "bbox": [ + 237, + 102, + 243, + 109 + ], + "score": 0.74, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 97, + 328, + 111 + ], + "score": 1.0, + "content": ", number of iterations", + "type": "text" + }, + { + "bbox": [ + 329, + 100, + 339, + 109 + ], + "score": 0.78, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 97, + 491, + 111 + ], + "score": 1.0, + "content": ", construction of differential estimators", + "type": "text" + }, + { + "bbox": [ + 492, + 98, + 504, + 111 + ], + "score": 0.72, + "content": "\\mathbf { g } ^ { k }", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 120, + 109, + 155, + 121 + ], + "spans": [ + { + "bbox": [ + 120, + 109, + 140, + 121 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 140, + 109, + 155, + 120 + ], + "score": 0.64, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 109, + 120, + 218, + 132 + ], + "spans": [ + { + "bbox": [ + 109, + 120, + 138, + 132 + ], + "score": 1.0, + "content": "1: for", + "type": "text" + }, + { + "bbox": [ + 138, + 122, + 164, + 131 + ], + "score": 0.89, + "content": "k = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 120, + 175, + 132 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 176, + 121, + 203, + 131 + ], + "score": 0.81, + "content": "K - 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We denote", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 218, + 198, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 193, + 233 + ], + "score": 0.92, + "content": "\\Delta = \\bar { F ( \\mathbf { x } ^ { 0 } ) } - F ( \\mathbf { x } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 218, + 198, + 234 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 233, + 486, + 246 + ], + "lines": [ + { + "bbox": [ + 105, + 232, + 484, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 202, + 247 + ], + "score": 1.0, + "content": "Assumption 2.2. Each", + "type": "text" + }, + { + "bbox": [ + 203, + 233, + 255, + 245 + ], + "score": 0.93, + "content": "f _ { i } : \\mathbb { R } ^ { d } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 232, + 272, + 247 + ], + "score": 1.0, + "content": "has", + "type": "text" + }, + { + "bbox": [ + 272, + 235, + 284, + 245 + ], + "score": 0.89, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 232, + 442, + 247 + ], + "score": 1.0, + "content": "-Lipschitz continuous gradient: for any", + "type": "text" + }, + { + "bbox": [ + 442, + 233, + 484, + 245 + ], + "score": 0.75, + "content": "\\mathbf { x } , \\mathbf { y } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 231, + 248, + 378, + 262 + ], + "lines": [ + { + "bbox": [ + 231, + 248, + 378, + 262 + ], + "spans": [ + { + "bbox": [ + 231, + 248, + 378, + 262 + ], + "score": 0.89, + "content": "\\| \\nabla f _ { i } ( \\mathbf { x } ) - \\nabla f _ { i } ( \\mathbf { y } ) \\| \\leq L _ { 1 } \\| \\mathbf { x } - \\mathbf { y } \\| .", + "type": "interline_equation", + "image_path": "fd6bffcd4ac329e3d2d08df3cf883bd01f854e8785247937cef9f572b0103e89.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 231, + 248, + 378, + 262 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 264, + 484, + 276 + ], + "lines": [ + { + "bbox": [ + 105, + 263, + 483, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 202, + 277 + ], + "score": 1.0, + "content": "Assumption 2.3. Each", + "type": "text" + }, + { + "bbox": [ + 203, + 263, + 255, + 276 + ], + "score": 0.92, + "content": "f _ { i } : \\mathbb { R } ^ { d } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 263, + 272, + 277 + ], + "score": 1.0, + "content": "has", + "type": "text" + }, + { + "bbox": [ + 272, + 264, + 284, + 275 + ], + "score": 0.88, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 263, + 440, + 277 + ], + "score": 1.0, + "content": "-Lipschitz continuous Hessian: for any", + "type": "text" + }, + { + "bbox": [ + 441, + 263, + 483, + 276 + ], + "score": 0.75, + "content": "\\mathbf { x } , \\mathbf { y } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 227, + 277, + 383, + 292 + ], + "lines": [ + { + "bbox": [ + 227, + 277, + 383, + 292 + ], + "spans": [ + { + "bbox": [ + 227, + 277, + 383, + 292 + ], + "score": 0.91, + "content": "\\| \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ) - \\nabla ^ { 2 } f _ { i } ( \\mathbf { y } ) \\| \\leq L _ { 2 } \\| \\mathbf { x } - \\mathbf { y } \\| .", + "type": "interline_equation", + "image_path": "f045f817af869389a938a90ea9e9cb9a5faa86aaab790f25f9fff951f42e8cbb.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 227, + 277, + 383, + 292 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 303, + 239, + 315 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 240, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 240, + 317 + ], + "score": 1.0, + "content": "2.1 TRUST REGION METHOD", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 324, + 504, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 323, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 505, + 337 + ], + "score": 1.0, + "content": "Here we briefly introduce the trust region method (Conn et al., 2000). In each step, it first solves the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 335, + 350, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 350, + 348 + ], + "score": 1.0, + "content": "Quadratic Constraint Quadratic Program (QCQP) defined as", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 197, + 349, + 412, + 377 + ], + "lines": [ + { + "bbox": [ + 197, + 349, + 412, + 377 + ], + "spans": [ + { + "bbox": [ + 197, + 349, + 412, + 377 + ], + "score": 0.91, + "content": "\\mathbf { h } ^ { k } : = \\operatorname * { a r g m i n } _ { \\mathbf { h } \\in \\mathbb { R } ^ { d } , \\| \\mathbf { h } \\| \\leq r } \\langle \\nabla F ( \\mathbf { x } ^ { k } ) , \\mathbf { h } \\rangle + \\frac { 1 } { 2 } \\langle \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) \\mathbf { h } , \\mathbf { h } \\rangle ,", + "type": "interline_equation", + "image_path": "0416e2f89ec5860a12a749b0d715f9cdd957cbbbf4133f5ba8eab08267fc6b24.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 197, + 349, + 412, + 377 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 379, + 386, + 391 + ], + "lines": [ + { + "bbox": [ + 106, + 378, + 386, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 133, + 393 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 382, + 139, + 389 + ], + "score": 0.77, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 378, + 386, + 393 + ], + "score": 1.0, + "content": "is the trust-region radius. Then it updates the new variable as", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 267, + 393, + 344, + 407 + ], + "lines": [ + { + "bbox": [ + 267, + 393, + 344, + 407 + ], + "spans": [ + { + "bbox": [ + 267, + 393, + 344, + 407 + ], + "score": 0.91, + "content": "\\mathbf { x } ^ { k + 1 } : = \\mathbf { x } ^ { k } + \\mathbf { h } ^ { k } .", + "type": "interline_equation", + "image_path": "c4ab9e77bd27ab82705bb5c6b81eae7b31e4a6605a63663107dc70ca7b75f4e0.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 267, + 393, + 344, + 407 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 411, + 505, + 434 + ], + "lines": [ + { + "bbox": [ + 106, + 408, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 131, + 424 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 410, + 171, + 423 + ], + "score": 0.93, + "content": "\\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 408, + 506, + 424 + ], + "score": 1.0, + "content": "is indefinite, the trust-region subproblem (5) is non-convex. But its global optimizer", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 421, + 447, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 447, + 434 + ], + "score": 1.0, + "content": "can be characterized by the following lemma (Corollary 7.2.2 in (Conn et al., 2000)).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 396, + 447 + ], + "lines": [ + { + "bbox": [ + 106, + 435, + 395, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 395, + 449 + ], + "score": 1.0, + "content": "Lemma 2.1. Any global minimizer of problem (5) satisfies the equation", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 450, + 377, + 465 + ], + "lines": [ + { + "bbox": [ + 233, + 450, + 377, + 465 + ], + "spans": [ + { + "bbox": [ + 233, + 450, + 377, + 465 + ], + "score": 0.92, + "content": "\\left( \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) + \\lambda \\mathbf { I } \\right) \\mathbf { h } ^ { k } = - \\nabla F ( \\mathbf { x } ^ { k } ) ,", + "type": "interline_equation", + "image_path": "4a076c3a0ce11ed5344c9d8696446694bae00258e95f126ea3728ffe56ebd0f5.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 233, + 450, + 377, + 465 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 468, + 461, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 462, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 203, + 483 + ], + "score": 1.0, + "content": "where the dual variable", + "type": "text" + }, + { + "bbox": [ + 204, + 469, + 229, + 480 + ], + "score": 0.9, + "content": "\\lambda \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 466, + 287, + 483 + ], + "score": 1.0, + "content": "should satisfy", + "type": "text" + }, + { + "bbox": [ + 288, + 468, + 368, + 482 + ], + "score": 0.93, + "content": "\\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) + \\lambda \\mathbf { I } \\succcurlyeq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 466, + 387, + 483 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 387, + 468, + 459, + 482 + ], + "score": 0.9, + "content": "\\lambda ( \\| \\mathbf { h } ^ { k } \\| - r ) = 0 .", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 466, + 462, + 483 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 488, + 506, + 547 + ], + "lines": [ + { + "bbox": [ + 104, + 486, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 104, + 486, + 375, + 502 + ], + "score": 1.0, + "content": "In particular, the standard QCQP solver returns both the minimizer", + "type": "text" + }, + { + "bbox": [ + 375, + 488, + 388, + 499 + ], + "score": 0.89, + "content": "\\mathbf { h } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 486, + 506, + 502 + ], + "score": 1.0, + "content": "as well as the corresponding", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 500, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 162, + 511 + ], + "score": 1.0, + "content": "dual variable", + "type": "text" + }, + { + "bbox": [ + 162, + 500, + 170, + 510 + ], + "score": 0.79, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 500, + 505, + 511 + ], + "score": 1.0, + "content": "of subproblem (5). In the following section, we first prove that the deterministic", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 510, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 339, + 525 + ], + "score": 1.0, + "content": "trust-region update (5) and (6) converges at the rate of √", + "type": "text" + }, + { + "bbox": [ + 340, + 510, + 384, + 524 + ], + "score": 0.94, + "content": "\\mathcal { O } ( 1 / k ^ { 2 / 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 510, + 506, + 525 + ], + "score": 1.0, + "content": ", much sharper than existing", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 523, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 211, + 537 + ], + "score": 1.0, + "content": "provable convergence rate", + "type": "text" + }, + { + "bbox": [ + 211, + 523, + 252, + 537 + ], + "score": 0.92, + "content": "\\mathcal { O } ( 1 / \\sqrt { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 524, + 506, + 537 + ], + "score": 1.0, + "content": "(Conn et al., 2000), and then develop a more efficient stochastic", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 535, + 196, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 196, + 547 + ], + "score": 1.0, + "content": "trust-region approach.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 108, + 562, + 209, + 575 + ], + "lines": [ + { + "bbox": [ + 104, + 560, + 211, + 578 + ], + "spans": [ + { + "bbox": [ + 104, + 560, + 211, + 578 + ], + "score": 1.0, + "content": "3 METHODOLOGY", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 105, + 586, + 504, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 587, + 507, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 507, + 600 + ], + "score": 1.0, + "content": "Here we first introduce a general inexact trust region method which is summarized in MetaAlgorithm 1.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 597, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 252, + 611 + ], + "score": 1.0, + "content": "It accepts inexact gradient estimation", + "type": "text" + }, + { + "bbox": [ + 252, + 597, + 264, + 609 + ], + "score": 0.88, + "content": "\\mathbf { g } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 597, + 357, + 611 + ], + "score": 1.0, + "content": "and Hessian estimation", + "type": "text" + }, + { + "bbox": [ + 357, + 598, + 371, + 608 + ], + "score": 0.89, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 597, + 505, + 611 + ], + "score": 1.0, + "content": "as input to the QCQP subproblem", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 616, + 388, + 644 + ], + "lines": [ + { + "bbox": [ + 222, + 616, + 388, + 644 + ], + "spans": [ + { + "bbox": [ + 222, + 616, + 388, + 644 + ], + "score": 0.94, + "content": "\\mathbf { h } ^ { k } : = \\operatorname * { a r g m i n } _ { \\mathbf { h } \\in \\mathbb { R } ^ { d } , \\| \\mathbf { h } \\| \\leq r } \\langle \\mathbf { g } ^ { k } , \\mathbf { h } \\rangle + \\frac { 1 } { 2 } \\langle \\mathbf { H } ^ { k } \\mathbf { h } , \\mathbf { h } \\rangle .", + "type": "interline_equation", + "image_path": "383fa79b49e3a62fed484af8e46b537ce0c10629f2ea7bb37f5efe898d0174e2.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 222, + 616, + 388, + 644 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 645, + 504, + 668 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "Similar to (5), Lemma 2.1 characterizes the global optimum to problem (8) which can be efficiently", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 654, + 502, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 468, + 668 + ], + "score": 1.0, + "content": "solved by Lanczos method (Gould et al., 1999). 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We denote", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 218, + 198, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 193, + 233 + ], + "score": 0.92, + "content": "\\Delta = \\bar { F ( \\mathbf { x } ^ { 0 } ) } - F ( \\mathbf { x } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 218, + 198, + 234 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 208, + 505, + 234 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 233, + 486, + 246 + ], + "lines": [ + { + "bbox": [ + 105, + 232, + 484, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 202, + 247 + ], + "score": 1.0, + "content": "Assumption 2.2. Each", + "type": "text" + }, + { + "bbox": [ + 203, + 233, + 255, + 245 + ], + "score": 0.93, + "content": "f _ { i } : \\mathbb { R } ^ { d } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 232, + 272, + 247 + ], + "score": 1.0, + "content": "has", + "type": "text" + }, + { + "bbox": [ + 272, + 235, + 284, + 245 + ], + "score": 0.89, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 232, + 442, + 247 + ], + "score": 1.0, + "content": "-Lipschitz continuous gradient: for any", + "type": "text" + }, + { + "bbox": [ + 442, + 233, + 484, + 245 + ], + "score": 0.75, + "content": "\\mathbf { x } , \\mathbf { y } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 232, + 484, + 247 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 231, + 248, + 378, + 262 + ], + "lines": [ + { + "bbox": [ + 231, + 248, + 378, + 262 + ], + "spans": [ + { + "bbox": [ + 231, + 248, + 378, + 262 + ], + "score": 0.89, + "content": "\\| \\nabla f _ { i } ( \\mathbf { x } ) - \\nabla f _ { i } ( \\mathbf { y } ) \\| \\leq L _ { 1 } \\| \\mathbf { x } - \\mathbf { y } \\| .", + "type": "interline_equation", + "image_path": "fd6bffcd4ac329e3d2d08df3cf883bd01f854e8785247937cef9f572b0103e89.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 231, + 248, + 378, + 262 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 264, + 484, + 276 + ], + "lines": [ + { + "bbox": [ + 105, + 263, + 483, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 202, + 277 + ], + "score": 1.0, + "content": "Assumption 2.3. Each", + "type": "text" + }, + { + "bbox": [ + 203, + 263, + 255, + 276 + ], + "score": 0.92, + "content": "f _ { i } : \\mathbb { R } ^ { d } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 263, + 272, + 277 + ], + "score": 1.0, + "content": "has", + "type": "text" + }, + { + "bbox": [ + 272, + 264, + 284, + 275 + ], + "score": 0.88, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 263, + 440, + 277 + ], + "score": 1.0, + "content": "-Lipschitz continuous Hessian: for any", + "type": "text" + }, + { + "bbox": [ + 441, + 263, + 483, + 276 + ], + "score": 0.75, + "content": "\\mathbf { x } , \\mathbf { y } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 263, + 483, + 277 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 227, + 277, + 383, + 292 + ], + "lines": [ + { + "bbox": [ + 227, + 277, + 383, + 292 + ], + "spans": [ + { + "bbox": [ + 227, + 277, + 383, + 292 + ], + "score": 0.91, + "content": "\\| \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ) - \\nabla ^ { 2 } f _ { i } ( \\mathbf { y } ) \\| \\leq L _ { 2 } \\| \\mathbf { x } - \\mathbf { y } \\| .", + "type": "interline_equation", + "image_path": "f045f817af869389a938a90ea9e9cb9a5faa86aaab790f25f9fff951f42e8cbb.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 227, + 277, + 383, + 292 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 303, + 239, + 315 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 240, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 240, + 317 + ], + "score": 1.0, + "content": "2.1 TRUST REGION METHOD", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 324, + 504, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 323, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 505, + 337 + ], + "score": 1.0, + "content": "Here we briefly introduce the trust region method (Conn et al., 2000). In each step, it first solves the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 335, + 350, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 350, + 348 + ], + "score": 1.0, + "content": "Quadratic Constraint Quadratic Program (QCQP) defined as", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 323, + 505, + 348 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 197, + 349, + 412, + 377 + ], + "lines": [ + { + "bbox": [ + 197, + 349, + 412, + 377 + ], + "spans": [ + { + "bbox": [ + 197, + 349, + 412, + 377 + ], + "score": 0.91, + "content": "\\mathbf { h } ^ { k } : = \\operatorname * { a r g m i n } _ { \\mathbf { h } \\in \\mathbb { R } ^ { d } , \\| \\mathbf { h } \\| \\leq r } \\langle \\nabla F ( \\mathbf { x } ^ { k } ) , \\mathbf { h } \\rangle + \\frac { 1 } { 2 } \\langle \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) \\mathbf { h } , \\mathbf { h } \\rangle ,", + "type": "interline_equation", + "image_path": "0416e2f89ec5860a12a749b0d715f9cdd957cbbbf4133f5ba8eab08267fc6b24.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 197, + 349, + 412, + 377 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 379, + 386, + 391 + ], + "lines": [ + { + "bbox": [ + 106, + 378, + 386, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 133, + 393 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 382, + 139, + 389 + ], + "score": 0.77, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 378, + 386, + 393 + ], + "score": 1.0, + "content": "is the trust-region radius. Then it updates the new variable as", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 106, + 378, + 386, + 393 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 267, + 393, + 344, + 407 + ], + "lines": [ + { + "bbox": [ + 267, + 393, + 344, + 407 + ], + "spans": [ + { + "bbox": [ + 267, + 393, + 344, + 407 + ], + "score": 0.91, + "content": "\\mathbf { x } ^ { k + 1 } : = \\mathbf { x } ^ { k } + \\mathbf { h } ^ { k } .", + "type": "interline_equation", + "image_path": "c4ab9e77bd27ab82705bb5c6b81eae7b31e4a6605a63663107dc70ca7b75f4e0.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 267, + 393, + 344, + 407 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 411, + 505, + 434 + ], + "lines": [ + { + "bbox": [ + 106, + 408, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 131, + 424 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 410, + 171, + 423 + ], + "score": 0.93, + "content": "\\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 408, + 506, + 424 + ], + "score": 1.0, + "content": "is indefinite, the trust-region subproblem (5) is non-convex. But its global optimizer", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 421, + 447, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 447, + 434 + ], + "score": 1.0, + "content": "can be characterized by the following lemma (Corollary 7.2.2 in (Conn et al., 2000)).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 106, + 408, + 506, + 434 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 396, + 447 + ], + "lines": [ + { + "bbox": [ + 106, + 435, + 395, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 395, + 449 + ], + "score": 1.0, + "content": "Lemma 2.1. Any global minimizer of problem (5) satisfies the equation", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 106, + 435, + 395, + 449 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 450, + 377, + 465 + ], + "lines": [ + { + "bbox": [ + 233, + 450, + 377, + 465 + ], + "spans": [ + { + "bbox": [ + 233, + 450, + 377, + 465 + ], + "score": 0.92, + "content": "\\left( \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) + \\lambda \\mathbf { I } \\right) \\mathbf { h } ^ { k } = - \\nabla F ( \\mathbf { x } ^ { k } ) ,", + "type": "interline_equation", + "image_path": "4a076c3a0ce11ed5344c9d8696446694bae00258e95f126ea3728ffe56ebd0f5.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 233, + 450, + 377, + 465 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 468, + 461, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 462, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 203, + 483 + ], + "score": 1.0, + "content": "where the dual variable", + "type": "text" + }, + { + "bbox": [ + 204, + 469, + 229, + 480 + ], + "score": 0.9, + "content": "\\lambda \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 466, + 287, + 483 + ], + "score": 1.0, + "content": "should satisfy", + "type": "text" + }, + { + "bbox": [ + 288, + 468, + 368, + 482 + ], + "score": 0.93, + "content": "\\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) + \\lambda \\mathbf { I } \\succcurlyeq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 466, + 387, + 483 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 387, + 468, + 459, + 482 + ], + "score": 0.9, + "content": "\\lambda ( \\| \\mathbf { h } ^ { k } \\| - r ) = 0 .", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 466, + 462, + 483 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 466, + 462, + 483 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 488, + 506, + 547 + ], + "lines": [ + { + "bbox": [ + 104, + 486, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 104, + 486, + 375, + 502 + ], + "score": 1.0, + "content": "In particular, the standard QCQP solver returns both the minimizer", + "type": "text" + }, + { + "bbox": [ + 375, + 488, + 388, + 499 + ], + "score": 0.89, + "content": "\\mathbf { h } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 486, + 506, + 502 + ], + "score": 1.0, + "content": "as well as the corresponding", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 500, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 162, + 511 + ], + "score": 1.0, + "content": "dual variable", + "type": "text" + }, + { + "bbox": [ + 162, + 500, + 170, + 510 + ], + "score": 0.79, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 500, + 505, + 511 + ], + "score": 1.0, + "content": "of subproblem (5). In the following section, we first prove that the deterministic", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 510, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 339, + 525 + ], + "score": 1.0, + "content": "trust-region update (5) and (6) converges at the rate of √", + "type": "text" + }, + { + "bbox": [ + 340, + 510, + 384, + 524 + ], + "score": 0.94, + "content": "\\mathcal { O } ( 1 / k ^ { 2 / 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 510, + 506, + 525 + ], + "score": 1.0, + "content": ", much sharper than existing", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 523, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 211, + 537 + ], + "score": 1.0, + "content": "provable convergence rate", + "type": "text" + }, + { + "bbox": [ + 211, + 523, + 252, + 537 + ], + "score": 0.92, + "content": "\\mathcal { O } ( 1 / \\sqrt { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 524, + 506, + 537 + ], + "score": 1.0, + "content": "(Conn et al., 2000), and then develop a more efficient stochastic", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 535, + 196, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 196, + 547 + ], + "score": 1.0, + "content": "trust-region approach.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29, + "bbox_fs": [ + 104, + 486, + 506, + 547 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 562, + 209, + 575 + ], + "lines": [ + { + "bbox": [ + 104, + 560, + 211, + 578 + ], + "spans": [ + { + "bbox": [ + 104, + 560, + 211, + 578 + ], + "score": 1.0, + "content": "3 METHODOLOGY", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 105, + 586, + 504, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 587, + 507, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 507, + 600 + ], + "score": 1.0, + "content": "Here we first introduce a general inexact trust region method which is summarized in MetaAlgorithm 1.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 597, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 252, + 611 + ], + "score": 1.0, + "content": "It accepts inexact gradient estimation", + "type": "text" + }, + { + "bbox": [ + 252, + 597, + 264, + 609 + ], + "score": 0.88, + "content": "\\mathbf { g } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 597, + 357, + 611 + ], + "score": 1.0, + "content": "and Hessian estimation", + "type": "text" + }, + { + "bbox": [ + 357, + 598, + 371, + 608 + ], + "score": 0.89, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 597, + 505, + 611 + ], + "score": 1.0, + "content": "as input to the QCQP subproblem", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 587, + 507, + 611 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 616, + 388, + 644 + ], + "lines": [ + { + "bbox": [ + 222, + 616, + 388, + 644 + ], + "spans": [ + { + "bbox": [ + 222, + 616, + 388, + 644 + ], + "score": 0.94, + "content": "\\mathbf { h } ^ { k } : = \\operatorname * { a r g m i n } _ { \\mathbf { h } \\in \\mathbb { R } ^ { d } , \\| \\mathbf { h } \\| \\leq r } \\langle \\mathbf { g } ^ { k } , \\mathbf { h } \\rangle + \\frac { 1 } { 2 } \\langle \\mathbf { H } ^ { k } \\mathbf { h } , \\mathbf { h } \\rangle .", + "type": "interline_equation", + "image_path": "383fa79b49e3a62fed484af8e46b537ce0c10629f2ea7bb37f5efe898d0174e2.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 222, + 616, + 388, + 644 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 645, + 504, + 668 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "Similar to (5), Lemma 2.1 characterizes the global optimum to problem (8) which can be efficiently", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 654, + 502, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 468, + 668 + ], + "score": 1.0, + "content": "solved by Lanczos method (Gould et al., 1999). Assume the dual variable of the minimizer", + "type": "text" + }, + { + "bbox": [ + 468, + 655, + 480, + 666 + ], + "score": 0.87, + "content": "\\mathbf { h } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 654, + 491, + 668 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 491, + 655, + 502, + 666 + ], + "score": 0.86, + "content": "\\lambda ^ { \\dot { k } }", + "type": "inline_equation" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 644, + 505, + 668 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 673, + 506, + 708 + ], + "lines": [ + { + "bbox": [ + 105, + 672, + 506, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 388, + 687 + ], + "score": 1.0, + "content": "We prove that such inexact trust region method achieves the optimal", + "type": "text" + }, + { + "bbox": [ + 388, + 673, + 432, + 686 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / k ^ { 2 / 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 672, + 506, + 687 + ], + "score": 1.0, + "content": "convergence rate", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 683, + 506, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 186, + 698 + ], + "score": 1.0, + "content": "when the estimation", + "type": "text" + }, + { + "bbox": [ + 187, + 684, + 199, + 696 + ], + "score": 0.87, + "content": "\\mathbf { g } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 683, + 216, + 698 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 216, + 684, + 231, + 695 + ], + "score": 0.89, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 683, + 506, + 698 + ], + "score": 1.0, + "content": "at each iteration are sufficiently close to their full (exact) counterparts", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 107, + 694, + 256, + 709 + ], + "spans": [ + { + "bbox": [ + 107, + 695, + 142, + 708 + ], + "score": 0.93, + "content": "\\nabla F ( \\mathbf { x } ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 694, + 161, + 709 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 161, + 695, + 201, + 708 + ], + "score": 0.93, + "content": "\\nabla ^ { 2 } F ( \\bar { \\mathbf { x } } ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 694, + 256, + 709 + ], + "score": 1.0, + "content": "respectively:", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 672, + 506, + 709 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 710, + 413, + 736 + ], + "lines": [ + { + "bbox": [ + 198, + 710, + 413, + 736 + ], + "spans": [ + { + "bbox": [ + 198, + 710, + 413, + 736 + ], + "score": 0.91, + "content": "\\| \\mathbf { g } ^ { k } - \\nabla F ( \\mathbf { x } ^ { k } ) \\| \\leq \\frac { \\epsilon } { 6 } , \\| \\mathbf { H } ^ { k } - \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) \\| \\leq \\frac { \\sqrt { \\epsilon L _ { 2 } } } { 3 } .", + "type": "interline_equation", + "image_path": "f963400f2268043dadd66165d09d23c7c2c691ca8d928a843ba29b60a2d5e794.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 198, + 710, + 413, + 736 + ], + "spans": [], + "index": 41 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 183, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 185, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 185, + 97 + ], + "score": 1.0, + "content": "Algorithm 2 STR1", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table", + "bbox": [ + 108, + 98, + 342, + 212 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 98, + 342, + 212 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 98, + 342, + 212 + ], + "spans": [ + { + "bbox": [ + 108, + 98, + 342, + 212 + ], + "score": 0.517, + "html": "
1:for k= 1 to K do 2: 3: Compute hk and Xk by solving (8);Input:initial point x°,step size r,number of iterations K Construct gradient estimator gk by Estimator 4; Construct Hessian estimator Hk by Estimator 3;
", + "type": "table", + "image_path": "ef2c9b6f7a41269eb6b756ffff844c0d4332196f5ba7452e1276d9c1a2b52c00.jpg" + } + ] + } + ], + "index": 4.5, + "virtual_lines": [ + { + "bbox": [ + 108, + 98, + 342, + 112.25 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 112.25, + 342, + 126.5 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 108, + 126.5, + 342, + 140.75 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 108, + 140.75, + 342, + 155.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 108, + 155.0, + 342, + 169.25 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 108, + 169.25, + 342, + 183.5 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 108, + 183.5, + 342, + 197.75 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 108, + 197.75, + 342, + 212.0 + ], + "spans": [], + "index": 8 + } + ] + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 108, + 220, + 504, + 253 + ], + "lines": [ + { + "bbox": [ + 106, + 221, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 505, + 231 + ], + "score": 1.0, + "content": "Such result allows us to derive stochastic trust-region variants with novel differential estimators that", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 231, + 507, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 507, + 243 + ], + "score": 1.0, + "content": "are tailored to ensure the optimal convergence rate. We state our formal results in Theorem 3.1,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 241, + 362, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 362, + 254 + ], + "score": 1.0, + "content": "whose proof is deferred to Appendix B.1 due to the space limit.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 256, + 506, + 293 + ], + "lines": [ + { + "bbox": [ + 105, + 256, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 506, + 270 + ], + "score": 1.0, + "content": "Theorem 3.1 (Main Result). Consider problem (1) under Assumption 2.1-2.3. If the differential", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 267, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 151, + 281 + ], + "score": 1.0, + "content": "estimators", + "type": "text" + }, + { + "bbox": [ + 151, + 267, + 163, + 279 + ], + "score": 0.88, + "content": "\\mathbf { g } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 267, + 182, + 281 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 182, + 267, + 197, + 279 + ], + "score": 0.88, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 267, + 290, + 281 + ], + "score": 1.0, + "content": "satisfy Eqn. (9) for all", + "type": "text" + }, + { + "bbox": [ + 290, + 268, + 297, + 278 + ], + "score": 0.68, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 267, + 407, + 281 + ], + "score": 1.0, + "content": ", MetaAlgorithm 1 finds an", + "type": "text" + }, + { + "bbox": [ + 407, + 268, + 428, + 280 + ], + "score": 0.82, + "content": "\\mathcal { O } ( \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 267, + 506, + 281 + ], + "score": 1.0, + "content": "-SOSP in less than", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 279, + 433, + 295 + ], + "spans": [ + { + "bbox": [ + 107, + 280, + 194, + 293 + ], + "score": 0.92, + "content": "K = \\mathcal { O } ( \\sqrt { L _ { 2 } } \\Delta / \\epsilon ^ { 1 . 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 279, + 379, + 295 + ], + "score": 1.0, + "content": "iterations by setting the trust-region radius as", + "type": "text" + }, + { + "bbox": [ + 380, + 280, + 429, + 294 + ], + "score": 0.93, + "content": "r = \\sqrt { \\epsilon / L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 279, + 433, + 295 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 295, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 106, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "Remark 3.1. We emphasize that MetaAlgorithm 1 degenerates to the exact trust region method by", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 305, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 135, + 320 + ], + "score": 1.0, + "content": "taking", + "type": "text" + }, + { + "bbox": [ + 135, + 306, + 197, + 319 + ], + "score": 0.94, + "content": "\\mathbf { g } ^ { k } = \\nabla F ( \\mathbf { x } ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 305, + 217, + 320 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 217, + 306, + 286, + 319 + ], + "score": 0.93, + "content": "\\mathbf { H } ^ { k } = \\nabla ^ { 2 } F ( \\mathbf { \\bar { x } } ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 305, + 506, + 320 + ], + "score": 1.0, + "content": ". Such result is of its own interest because this is the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 317, + 507, + 333 + ], + "spans": [ + { + "bbox": [ + 104, + 317, + 388, + 333 + ], + "score": 1.0, + "content": "first proof to show that the vanilla trust region method has the optimal", + "type": "text" + }, + { + "bbox": [ + 388, + 318, + 433, + 331 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / k ^ { 2 / 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 317, + 507, + 333 + ], + "score": 1.0, + "content": "convergence rate.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "Similar rate is achieved by Curtis et al. (2017) but with a much more complicated trust region variant.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 106, + 351, + 505, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 350, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 506, + 363 + ], + "score": 1.0, + "content": "Theorem 3.1 shows the explicit step size control of the trust region method: Since the dual variable√", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 359, + 507, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 141, + 375 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 141, + 361, + 229, + 374 + ], + "score": 0.93, + "content": "\\lambda ^ { k } > 3 \\epsilon ^ { 0 . 5 } / \\sqrt { L _ { 2 } } ^ { \\bullet } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 359, + 507, + 375 + ], + "score": 1.0, + "content": "for all but the last iteration, we always find the solution to the trust-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 372, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 285, + 386 + ], + "score": 1.0, + "content": "region subproblem (8) in the boundary, i.e.", + "type": "text" + }, + { + "bbox": [ + 285, + 372, + 326, + 385 + ], + "score": 0.93, + "content": "\\| \\mathbf { h } ^ { k } \\| = r", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 372, + 505, + 386 + ], + "score": 1.0, + "content": ", according to the complementary condition", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 383, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 397 + ], + "score": 1.0, + "content": "(15) in Appendix B.1. Such exact step size control property is missing in the cubic-regularization", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 394, + 454, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 454, + 408 + ], + "score": 1.0, + "content": "method where the step size is implicitly decided by the cubic regularization parameter.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 412, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 426 + ], + "score": 1.0, + "content": "More importantly, we emphasize that such explicit step size control is crucial to the sample efficiency", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 423, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 434 + ], + "score": 1.0, + "content": "of our variance reduced differential estimators. The essence of variance reduction is to exploit the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 433, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 505, + 447 + ], + "score": 1.0, + "content": "correlations between the differentials in consecutive iterations. Intuitively, when two neighboring", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "iterates are close, so are their differentials due to the Lipschitz continuity, and hence a smaller number", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "score": 1.0, + "content": "of samples suffice to maintain the accuracy of the estimators. On the other hand, smaller step size", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "reduces the per-iteration objective decrease which harms the convergence rate of the algorithm (see", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "score": 1.0, + "content": "proof of Theorem 3.1). Therefore, the explicit step size control in trust region method allows us to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "well trade-off the per-iteration sample complexity and convergence rate, from which we can derive", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 500, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "stochastic trust region approaches with the state-of-the-art sample efficiency. In contrast, existing trust", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "region methods change the step size at every iteration according to progress made, which requires", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 521, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 506, + 535 + ], + "score": 1.0, + "content": "loss evaluations that can be as expensive as gradient computations (e.g. the non-convex linear model", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 532, + 353, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 353, + 545 + ], + "score": 1.0, + "content": "in Section 7) and is thus prohibitive for large-scale problems.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 29.5 + }, + { + "type": "title", + "bbox": [ + 107, + 560, + 367, + 573 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 368, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 368, + 574 + ], + "score": 1.0, + "content": "4 STOCHASTIC TRUST REGION METHOD: TYPE I", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 585, + 505, + 665 + ], + "lines": [ + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "score": 1.0, + "content": "Having the inexact trust region method as prototype, we now present our first sample-efficient", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 263, + 609 + ], + "score": 1.0, + "content": "stochastic trust region method, namely", + "type": "text" + }, + { + "bbox": [ + 264, + 596, + 288, + 608 + ], + "score": 0.89, + "content": "\\mathrm { { S T R } _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 596, + 505, + 609 + ], + "score": 1.0, + "content": ", in Algorithm 2 which emphasizes cheaper stochastic", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 606, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 621 + ], + "score": 1.0, + "content": "second-order oracle complexity. As Theorem 3.1 already guarantees the optimal convergence", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 327, + 631 + ], + "score": 1.0, + "content": "rate of MetaAlgorithm 1 when the gradient estimator", + "type": "text" + }, + { + "bbox": [ + 327, + 618, + 340, + 630 + ], + "score": 0.88, + "content": "\\mathbf { g } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 618, + 450, + 631 + ], + "score": 1.0, + "content": "and the Hessian estimator", + "type": "text" + }, + { + "bbox": [ + 451, + 618, + 465, + 628 + ], + "score": 0.88, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "meet the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "score": 1.0, + "content": "requirement (9), here we focus on constructing such novel differential estimators. Specifically, we", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 640, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 652 + ], + "score": 1.0, + "content": "first present our Hessian estimator in Estimator 3 and our first gradient estimator in Estimator 4, both", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 651, + 417, + 665 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 263, + 665 + ], + "score": 1.0, + "content": "of which exploit the trust region radius", + "type": "text" + }, + { + "bbox": [ + 264, + 651, + 313, + 665 + ], + "score": 0.94, + "content": "r = \\sqrt { \\epsilon / L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 653, + 417, + 665 + ], + "score": 1.0, + "content": "to reduce their variances.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40 + }, + { + "type": "title", + "bbox": [ + 108, + 678, + 223, + 689 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 224, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 224, + 690 + ], + "score": 1.0, + "content": "4.1 HESSIAN ESTIMATOR", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 108, + 698, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 249, + 711 + ], + "score": 1.0, + "content": "Our epoch-wise Hessian estimator", + "type": "text" + }, + { + "bbox": [ + 249, + 698, + 263, + 709 + ], + "score": 0.87, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 698, + 388, + 711 + ], + "score": 1.0, + "content": "is given in Estimator 3, where", + "type": "text" + }, + { + "bbox": [ + 388, + 701, + 399, + 711 + ], + "score": 0.8, + "content": "p _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "controls the epoch length", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 123, + 722 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 711, + 134, + 721 + ], + "score": 0.83, + "content": "s _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 709, + 198, + 722 + ], + "score": 1.0, + "content": "(and optionally", + "type": "text" + }, + { + "bbox": [ + 199, + 710, + 209, + 721 + ], + "score": 0.86, + "content": "s _ { 2 } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 709, + 506, + 722 + ], + "score": 1.0, + "content": ") controls the minibatch size. At the beginning of each epoch, Estimator 3", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "has two options, designed for different target accuracy: Option I is preferable for the high accuracy", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 183, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 185, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 185, + 97 + ], + "score": 1.0, + "content": "Algorithm 2 STR1", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table", + "bbox": [ + 108, + 98, + 342, + 212 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 98, + 342, + 212 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 98, + 342, + 212 + ], + "spans": [ + { + "bbox": [ + 108, + 98, + 342, + 212 + ], + "score": 0.517, + "html": "
1:for k= 1 to K do 2: 3: Compute hk and Xk by solving (8);Input:initial point x°,step size r,number of iterations K Construct gradient estimator gk by Estimator 4; Construct Hessian estimator Hk by Estimator 3;
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We state our formal results in Theorem 3.1,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 241, + 362, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 362, + 254 + ], + "score": 1.0, + "content": "whose proof is deferred to Appendix B.1 due to the space limit.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10, + "bbox_fs": [ + 106, + 221, + 507, + 254 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 256, + 506, + 293 + ], + "lines": [ + { + "bbox": [ + 105, + 256, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 506, + 270 + ], + "score": 1.0, + "content": "Theorem 3.1 (Main Result). Consider problem (1) under Assumption 2.1-2.3. If the differential", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 267, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 151, + 281 + ], + "score": 1.0, + "content": "estimators", + "type": "text" + }, + { + "bbox": [ + 151, + 267, + 163, + 279 + ], + "score": 0.88, + "content": "\\mathbf { g } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 267, + 182, + 281 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 182, + 267, + 197, + 279 + ], + "score": 0.88, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 267, + 290, + 281 + ], + "score": 1.0, + "content": "satisfy Eqn. (9) for all", + "type": "text" + }, + { + "bbox": [ + 290, + 268, + 297, + 278 + ], + "score": 0.68, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 267, + 407, + 281 + ], + "score": 1.0, + "content": ", MetaAlgorithm 1 finds an", + "type": "text" + }, + { + "bbox": [ + 407, + 268, + 428, + 280 + ], + "score": 0.82, + "content": "\\mathcal { O } ( \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 267, + 506, + 281 + ], + "score": 1.0, + "content": "-SOSP in less than", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 279, + 433, + 295 + ], + "spans": [ + { + "bbox": [ + 107, + 280, + 194, + 293 + ], + "score": 0.92, + "content": "K = \\mathcal { O } ( \\sqrt { L _ { 2 } } \\Delta / \\epsilon ^ { 1 . 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 279, + 379, + 295 + ], + "score": 1.0, + "content": "iterations by setting the trust-region radius as", + "type": "text" + }, + { + "bbox": [ + 380, + 280, + 429, + 294 + ], + "score": 0.93, + "content": "r = \\sqrt { \\epsilon / L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 279, + 433, + 295 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 256, + 506, + 295 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 295, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 106, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "Remark 3.1. We emphasize that MetaAlgorithm 1 degenerates to the exact trust region method by", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 305, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 135, + 320 + ], + "score": 1.0, + "content": "taking", + "type": "text" + }, + { + "bbox": [ + 135, + 306, + 197, + 319 + ], + "score": 0.94, + "content": "\\mathbf { g } ^ { k } = \\nabla F ( \\mathbf { x } ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 305, + 217, + 320 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 217, + 306, + 286, + 319 + ], + "score": 0.93, + "content": "\\mathbf { H } ^ { k } = \\nabla ^ { 2 } F ( \\mathbf { \\bar { x } } ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 305, + 506, + 320 + ], + "score": 1.0, + "content": ". Such result is of its own interest because this is the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 317, + 507, + 333 + ], + "spans": [ + { + "bbox": [ + 104, + 317, + 388, + 333 + ], + "score": 1.0, + "content": "first proof to show that the vanilla trust region method has the optimal", + "type": "text" + }, + { + "bbox": [ + 388, + 318, + 433, + 331 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / k ^ { 2 / 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 317, + 507, + 333 + ], + "score": 1.0, + "content": "convergence rate.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "Similar rate is achieved by Curtis et al. (2017) but with a much more complicated trust region variant.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 104, + 295, + 507, + 343 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 351, + 505, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 350, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 506, + 363 + ], + "score": 1.0, + "content": "Theorem 3.1 shows the explicit step size control of the trust region method: Since the dual variable√", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 359, + 507, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 141, + 375 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 141, + 361, + 229, + 374 + ], + "score": 0.93, + "content": "\\lambda ^ { k } > 3 \\epsilon ^ { 0 . 5 } / \\sqrt { L _ { 2 } } ^ { \\bullet } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 359, + 507, + 375 + ], + "score": 1.0, + "content": "for all but the last iteration, we always find the solution to the trust-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 372, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 285, + 386 + ], + "score": 1.0, + "content": "region subproblem (8) in the boundary, i.e.", + "type": "text" + }, + { + "bbox": [ + 285, + 372, + 326, + 385 + ], + "score": 0.93, + "content": "\\| \\mathbf { h } ^ { k } \\| = r", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 372, + 505, + 386 + ], + "score": 1.0, + "content": ", according to the complementary condition", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 383, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 397 + ], + "score": 1.0, + "content": "(15) in Appendix B.1. Such exact step size control property is missing in the cubic-regularization", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 394, + 454, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 454, + 408 + ], + "score": 1.0, + "content": "method where the step size is implicitly decided by the cubic regularization parameter.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 350, + 507, + 408 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 412, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 426 + ], + "score": 1.0, + "content": "More importantly, we emphasize that such explicit step size control is crucial to the sample efficiency", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 423, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 434 + ], + "score": 1.0, + "content": "of our variance reduced differential estimators. The essence of variance reduction is to exploit the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 433, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 505, + 447 + ], + "score": 1.0, + "content": "correlations between the differentials in consecutive iterations. Intuitively, when two neighboring", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "iterates are close, so are their differentials due to the Lipschitz continuity, and hence a smaller number", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "score": 1.0, + "content": "of samples suffice to maintain the accuracy of the estimators. On the other hand, smaller step size", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "reduces the per-iteration objective decrease which harms the convergence rate of the algorithm (see", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "score": 1.0, + "content": "proof of Theorem 3.1). Therefore, the explicit step size control in trust region method allows us to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "well trade-off the per-iteration sample complexity and convergence rate, from which we can derive", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 500, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "stochastic trust region approaches with the state-of-the-art sample efficiency. In contrast, existing trust", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "region methods change the step size at every iteration according to progress made, which requires", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 521, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 506, + 535 + ], + "score": 1.0, + "content": "loss evaluations that can be as expensive as gradient computations (e.g. the non-convex linear model", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 532, + 353, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 353, + 545 + ], + "score": 1.0, + "content": "in Section 7) and is thus prohibitive for large-scale problems.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 411, + 506, + 545 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 560, + 367, + 573 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 368, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 368, + 574 + ], + "score": 1.0, + "content": "4 STOCHASTIC TRUST REGION METHOD: TYPE I", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 585, + 505, + 665 + ], + "lines": [ + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "score": 1.0, + "content": "Having the inexact trust region method as prototype, we now present our first sample-efficient", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 263, + 609 + ], + "score": 1.0, + "content": "stochastic trust region method, namely", + "type": "text" + }, + { + "bbox": [ + 264, + 596, + 288, + 608 + ], + "score": 0.89, + "content": "\\mathrm { { S T R } _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 596, + 505, + 609 + ], + "score": 1.0, + "content": ", in Algorithm 2 which emphasizes cheaper stochastic", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 606, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 621 + ], + "score": 1.0, + "content": "second-order oracle complexity. As Theorem 3.1 already guarantees the optimal convergence", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 327, + 631 + ], + "score": 1.0, + "content": "rate of MetaAlgorithm 1 when the gradient estimator", + "type": "text" + }, + { + "bbox": [ + 327, + 618, + 340, + 630 + ], + "score": 0.88, + "content": "\\mathbf { g } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 618, + 450, + 631 + ], + "score": 1.0, + "content": "and the Hessian estimator", + "type": "text" + }, + { + "bbox": [ + 451, + 618, + 465, + 628 + ], + "score": 0.88, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "meet the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "score": 1.0, + "content": "requirement (9), here we focus on constructing such novel differential estimators. 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"index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 284, + 505, + 351 + ], + "lines": [ + { + "bbox": [ + 106, + 284, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 128, + 297 + ], + "score": 1.0, + "content": "case", + "type": "text" + }, + { + "bbox": [ + 129, + 284, + 185, + 297 + ], + "score": 0.91, + "content": "( \\epsilon < \\mathcal { O } ( 1 / n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 284, + 505, + 297 + ], + "score": 1.0, + "content": "where we compute the full Hessian to avoid approximation error, and Option", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 295, + 308 + ], + "score": 1.0, + "content": "II is designed for the moderate accuracy case", + "type": "text" + }, + { + "bbox": [ + 295, + 295, + 352, + 307 + ], + "score": 0.9, + "content": "( \\epsilon > \\mathcal { O } ( 1 / n ) )", + "type": "inline_equation" + }, + { + "bbox": [ 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Then,", + "type": "text" + }, + { + "bbox": [ + 206, + 308, + 217, + 318 + ], + "score": 0.85, + "content": "p _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 306, + 304, + 318 + ], + "score": 1.0, + "content": "iterations follow with", + "type": "text" + }, + { + "bbox": [ + 305, + 306, + 320, + 317 + ], + "score": 0.86, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 306, + 506, + 318 + ], + "score": 1.0, + "content": "defined in a recurrent manner. These recurrent", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "score": 1.0, + "content": "estimators exist for the first-order case (Nguyen et al., 2017; Fang et al., 2018), but their bound only", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 328, + 506, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 195, + 341 + ], + "score": 1.0, + "content": "holds under the vector", + "type": "text" + }, + { + "bbox": [ + 195, + 329, + 205, + 339 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 328, + 506, + 341 + ], + "score": 1.0, + "content": "norm. Here we generalize them into Hessian estimation with matrix spectral", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 339, + 160, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 160, + 350 + ], + "score": 1.0, + "content": "norm bound.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 356, + 505, + 400 + ], + "lines": [ + { + "bbox": [ + 104, + 355, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 104, + 355, + 505, + 369 + ], + "score": 1.0, + "content": "The following lemma analyzes the amortized stochastic second-order oracle (Hessian) complexity", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "score": 1.0, + "content": "for Algorithm 3 to meet the requirement in Theorem 3.1. As we need an error bound under the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 378, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 390 + ], + "score": 1.0, + "content": "spectral norm, we will appeal to the matrix Azuma’s inequality (Tropp, 2012). The proof is deferred", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 388, + 178, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 178, + 401 + ], + "score": 1.0, + "content": "to Appendix B.2.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 403, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 371, + 417 + ], + "score": 1.0, + "content": "Lemma 4.1. Assume Algorithm 2 takes the trust region radius", + "type": "text" + }, + { + "bbox": [ + 371, + 403, + 424, + 417 + ], + "score": 0.94, + "content": "r = \\sqrt { \\epsilon / L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 403, + 506, + 417 + ], + "score": 1.0, + "content": "as in Theorem 3.1.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 141, + 429 + ], + "score": 1.0, + "content": "For any", + "type": "text" + }, + { + "bbox": [ + 142, + 417, + 171, + 428 + ], + "score": 0.89, + "content": "k \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 416, + 309, + 429 + ], + "score": 1.0, + "content": ", Estimator 3 produces estimator", + "type": "text" + }, + { + "bbox": [ + 310, + 416, + 325, + 428 + ], + "score": 0.88, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 416, + 464, + 429 + ], + "score": 1.0, + "content": "for the second order differential", + "type": "text" + }, + { + "bbox": [ + 465, + 416, + 505, + 429 + ], + "score": 0.91, + "content": "\\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } )", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 427, + 504, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 146, + 441 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 147, + 428, + 270, + 441 + ], + "score": 0.9, + "content": "\\| \\mathbf { H } ^ { k } - \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) \\| \\leq \\sqrt { \\epsilon L _ { 2 } } / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 427, + 371, + 441 + ], + "score": 1.0, + "content": "with probability at least", + "type": "text" + }, + { + "bbox": [ + 372, + 429, + 413, + 441 + ], + "score": 0.91, + "content": "1 - \\delta / K _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 427, + 462, + 441 + ], + "score": 1.0, + "content": "if we set (1)", + "type": "text" + }, + { + "bbox": [ + 462, + 429, + 504, + 441 + ], + "score": 0.89, + "content": "\\boldsymbol { p } _ { 2 } = \\sqrt { n }", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 124, + 453 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 441, + 222, + 453 + ], + "score": 0.88, + "content": "s _ { 2 } = 3 2 \\sqrt { n } \\log ( d K _ { 0 } / \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 440, + 263, + 453 + ], + "score": 1.0, + "content": "in option", + "type": "text" + }, + { + "bbox": [ + 263, + 442, + 270, + 451 + ], + "score": 0.43, + "content": "I ,", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 440, + 298, + 453 + ], + "score": 1.0, + "content": "or (2)", + "type": "text" + }, + { + "bbox": [ + 298, + 441, + 375, + 453 + ], + "score": 0.9, + "content": "p _ { 2 } = L _ { 1 } / ( 2 \\sqrt { \\epsilon L _ { 2 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 440, + 379, + 453 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 380, + 441, + 503, + 453 + ], + "score": 0.87, + "content": "s _ { 2 } ^ { \\prime } = 1 6 L _ { 1 } ^ { 2 } / ( \\epsilon L _ { 2 } ) \\log ( d K _ { 0 } / \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 440, + 505, + 453 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 453, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 125, + 465 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 125, + 453, + 259, + 465 + ], + "score": 0.89, + "content": "s _ { 2 } = 3 2 L _ { 1 } / ( \\sqrt { \\epsilon L _ { 2 } } ) \\log ( d K _ { 0 } / \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 453, + 301, + 465 + ], + "score": 1.0, + "content": "in option", + "type": "text" + }, + { + "bbox": [ + 302, + 454, + 310, + 463 + ], + "score": 0.47, + "content": "\\boldsymbol { { I I } }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 453, + 338, + 465 + ], + "score": 1.0, + "content": ". Here", + "type": "text" + }, + { + "bbox": [ + 338, + 453, + 352, + 464 + ], + "score": 0.87, + "content": "K _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 453, + 506, + 465 + ], + "score": 1.0, + "content": "is a constant to be determined later.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 463, + 504, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 488, + 476 + ], + "score": 1.0, + "content": "Consequently the amortized per-iteration stochastic second-order oracle complexity to construct", + "type": "text" + }, + { + "bbox": [ + 105, + 468, + 171, + 493 + ], + "score": 1.0, + "content": "is no more than", + "type": "text" + }, + { + "bbox": [ + 171, + 474, + 279, + 490 + ], + "score": 0.85, + "content": "\\begin{array} { r } { 2 s _ { 2 } = \\operatorname* { m i n } \\{ 6 4 \\sqrt { n } \\log \\frac { d K _ { 0 } } { \\delta } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 468, + 282, + 493 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 283, + 474, + 343, + 491 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\frac { 6 4 L _ { 1 } } { \\sqrt { \\epsilon L _ { 2 } } } \\log \\frac { d K _ { 0 } } { \\delta } \\biggr \\} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 463, + 504, + 474 + ], + "score": 0.88, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + } + ], + "index": 31 + } + ], + "index": 28.5 + }, + { + "type": "title", + "bbox": [ + 108, + 502, + 276, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 277, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 277, + 516 + ], + "score": 1.0, + "content": "4.2 GRADIENT ESTIMATOR: CASE (1)", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 523, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "When the stochastic second-order oracle complexity is prioritized, we directly employ the SPIDER", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 226, + 546 + ], + "score": 1.0, + "content": "gradient estimator to construct", + "type": "text" + }, + { + "bbox": [ + 226, + 533, + 238, + 546 + ], + "score": 0.87, + "content": "\\mathbf { g } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 534, + 434, + 546 + ], + "score": 1.0, + "content": "(Fang et al., 2018). Similar to the construction for", + "type": "text" + }, + { + "bbox": [ + 434, + 534, + 449, + 545 + ], + "score": 0.89, + "content": "\\bar { \\mathbf { H } } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 534, + 506, + 546 + ], + "score": 1.0, + "content": ", the estimator", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 544, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 118, + 557 + ], + "score": 0.87, + "content": "\\bar { \\mathbf { g } } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 544, + 443, + 558 + ], + "score": 1.0, + "content": "is also constructed in an epoch-wise manner as presented in Estimator 4, where", + "type": "text" + }, + { + "bbox": [ + 443, + 547, + 453, + 556 + ], + "score": 0.86, + "content": "p _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 544, + 506, + 558 + ], + "score": 1.0, + "content": "controls the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 556, + 301, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 177, + 568 + ], + "score": 1.0, + "content": "epoch length and", + "type": "text" + }, + { + "bbox": [ + 177, + 557, + 187, + 567 + ], + "score": 0.85, + "content": "s _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 556, + 301, + 568 + ], + "score": 1.0, + "content": "controls the minibatch size.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 105, + 573, + 506, + 670 + ], + "lines": [ + { + "bbox": [ + 106, + 573, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 506, + 586 + ], + "score": 1.0, + "content": "We now analyze the stochastic first-order oracle complexity to meet the requirement in Theorem 3.1.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 373, + 601 + ], + "score": 1.0, + "content": "Lemma 4.2. 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These recurrent", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "score": 1.0, + "content": "estimators exist for the first-order case (Nguyen et al., 2017; Fang et al., 2018), but their bound only", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 328, + 506, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 195, + 341 + ], + "score": 1.0, + "content": "holds under the vector", + "type": "text" + }, + { + "bbox": [ + 195, + 329, + 205, + 339 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 328, + 506, + 341 + ], + "score": 1.0, + "content": "norm. Here we generalize them into Hessian estimation with matrix spectral", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 339, + 160, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 160, + 350 + ], + "score": 1.0, + "content": "norm bound.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 284, + 506, + 350 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 356, + 505, + 400 + ], + "lines": [ + { + "bbox": [ + 104, + 355, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 104, + 355, + 505, + 369 + ], + "score": 1.0, + "content": "The following lemma analyzes the amortized stochastic second-order oracle (Hessian) complexity", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "score": 1.0, + "content": "for Algorithm 3 to meet the requirement in Theorem 3.1. As we need an error bound under the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 378, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 390 + ], + "score": 1.0, + "content": "spectral norm, we will appeal to the matrix Azuma’s inequality (Tropp, 2012). The proof is deferred", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 388, + 178, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 178, + 401 + ], + "score": 1.0, + "content": "to Appendix B.2.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 104, + 355, + 506, + 401 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 403, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 371, + 417 + ], + "score": 1.0, + "content": "Lemma 4.1. Assume Algorithm 2 takes the trust region radius", + "type": "text" + }, + { + "bbox": [ + 371, + 403, + 424, + 417 + ], + "score": 0.94, + "content": "r = \\sqrt { \\epsilon / L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 403, + 506, + 417 + ], + "score": 1.0, + "content": "as in Theorem 3.1.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 141, + 429 + ], + "score": 1.0, + "content": "For any", + "type": "text" + }, + { + "bbox": [ + 142, + 417, + 171, + 428 + ], + "score": 0.89, + "content": "k \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 416, + 309, + 429 + ], + "score": 1.0, + "content": ", Estimator 3 produces estimator", + "type": "text" + }, + { + "bbox": [ + 310, + 416, + 325, + 428 + ], + "score": 0.88, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 416, + 464, + 429 + ], + "score": 1.0, + "content": "for the second order differential", + "type": "text" + }, + { + "bbox": [ + 465, + 416, + 505, + 429 + ], + "score": 0.91, + "content": "\\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } )", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 427, + 504, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 146, + 441 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 147, + 428, + 270, + 441 + ], + "score": 0.9, + "content": "\\| \\mathbf { H } ^ { k } - \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) \\| \\leq \\sqrt { \\epsilon L _ { 2 } } / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 427, + 371, + 441 + ], + "score": 1.0, + "content": "with probability at least", + "type": "text" + }, + { + "bbox": [ + 372, + 429, + 413, + 441 + ], + "score": 0.91, + "content": "1 - \\delta / K _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 427, + 462, + 441 + ], + "score": 1.0, + "content": "if we set (1)", + "type": "text" + }, + { + "bbox": [ + 462, + 429, + 504, + 441 + ], + "score": 0.89, + "content": "\\boldsymbol { p } _ { 2 } = \\sqrt { n }", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 124, + 453 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 441, + 222, + 453 + ], + "score": 0.88, + "content": "s _ { 2 } = 3 2 \\sqrt { n } \\log ( d K _ { 0 } / \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 440, + 263, + 453 + ], + "score": 1.0, + "content": "in option", + "type": "text" + }, + { + "bbox": [ + 263, + 442, + 270, + 451 + ], + "score": 0.43, + "content": "I ,", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 440, + 298, + 453 + ], + "score": 1.0, + "content": "or (2)", + "type": "text" + }, + { + "bbox": [ + 298, + 441, + 375, + 453 + ], + "score": 0.9, + "content": "p _ { 2 } = L _ { 1 } / ( 2 \\sqrt { \\epsilon L _ { 2 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 440, + 379, + 453 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 380, + 441, + 503, + 453 + ], + "score": 0.87, + "content": "s _ { 2 } ^ { \\prime } = 1 6 L _ { 1 } ^ { 2 } / ( \\epsilon L _ { 2 } ) \\log ( d K _ { 0 } / \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 440, + 505, + 453 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 453, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 125, + 465 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 125, + 453, + 259, + 465 + ], + "score": 0.89, + "content": "s _ { 2 } = 3 2 L _ { 1 } / ( \\sqrt { \\epsilon L _ { 2 } } ) \\log ( d K _ { 0 } / \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 453, + 301, + 465 + ], + "score": 1.0, + "content": "in option", + "type": "text" + }, + { + "bbox": [ + 302, + 454, + 310, + 463 + ], + "score": 0.47, + "content": "\\boldsymbol { { I I } }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 453, + 338, + 465 + ], + "score": 1.0, + "content": ". Here", + "type": "text" + }, + { + "bbox": [ + 338, + 453, + 352, + 464 + ], + "score": 0.87, + "content": "K _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 453, + 506, + 465 + ], + "score": 1.0, + "content": "is a constant to be determined later.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 463, + 504, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 488, + 476 + ], + "score": 1.0, + "content": "Consequently the amortized per-iteration stochastic second-order oracle complexity to construct", + "type": "text" + }, + { + "bbox": [ + 105, + 468, + 171, + 493 + ], + "score": 1.0, + "content": "is no more than", + "type": "text" + }, + { + "bbox": [ + 171, + 474, + 279, + 490 + ], + "score": 0.85, + "content": "\\begin{array} { r } { 2 s _ { 2 } = \\operatorname* { m i n } \\{ 6 4 \\sqrt { n } \\log \\frac { d K _ { 0 } } { \\delta } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 468, + 282, + 493 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 283, + 474, + 343, + 491 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\frac { 6 4 L _ { 1 } } { \\sqrt { \\epsilon L _ { 2 } } } \\log \\frac { d K _ { 0 } } { \\delta } \\biggr \\} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 463, + 504, + 474 + ], + "score": 0.88, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 403, + 506, + 493 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 502, + 276, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 277, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 277, + 516 + ], + "score": 1.0, + "content": "4.2 GRADIENT ESTIMATOR: CASE (1)", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 523, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "When the stochastic second-order oracle complexity is prioritized, we directly employ the SPIDER", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 226, + 546 + ], + "score": 1.0, + "content": "gradient estimator to construct", + "type": "text" + }, + { + "bbox": [ + 226, + 533, + 238, + 546 + ], + "score": 0.87, + "content": "\\mathbf { g } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 534, + 434, + 546 + ], + "score": 1.0, + "content": "(Fang et al., 2018). Similar to the construction for", + "type": "text" + }, + { + "bbox": [ + 434, + 534, + 449, + 545 + ], + "score": 0.89, + "content": "\\bar { \\mathbf { H } } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 534, + 506, + 546 + ], + "score": 1.0, + "content": ", the estimator", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 544, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 118, + 557 + ], + "score": 0.87, + "content": "\\bar { \\mathbf { g } } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 544, + 443, + 558 + ], + "score": 1.0, + "content": "is also constructed in an epoch-wise manner as presented in Estimator 4, where", + "type": "text" + }, + { + "bbox": [ + 443, + 547, + 453, + 556 + ], + "score": 0.86, + "content": "p _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 544, + 506, + 558 + ], + "score": 1.0, + "content": "controls the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 556, + 301, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 177, + 568 + ], + "score": 1.0, + "content": "epoch length and", + "type": "text" + }, + { + "bbox": [ + 177, + 557, + 187, + 567 + ], + "score": 0.85, + "content": "s _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 556, + 301, + 568 + ], + "score": 1.0, + "content": "controls the minibatch size.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 522, + 506, + 568 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 573, + 506, + 670 + ], + "lines": [ + { + "bbox": [ + 106, + 573, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 506, + 586 + ], + "score": 1.0, + "content": "We now analyze the stochastic first-order oracle complexity to meet the requirement in Theorem 3.1.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 373, + 601 + ], + "score": 1.0, + "content": "Lemma 4.2. Assume Algorithm 2 takes the trust region radius", + "type": "text" + }, + { + "bbox": [ + 373, + 587, + 426, + 600 + ], + "score": 0.92, + "content": "r = \\sqrt { \\epsilon / L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 588, + 478, + 601 + ], + "score": 1.0, + "content": ". 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Consequently, the amortized per-iteration stochastic first-order oracle complexity", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 655, + 331, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 156, + 670 + ], + "score": 1.0, + "content": "to construct", + "type": "text" + }, + { + "bbox": [ + 156, + 655, + 328, + 669 + ], + "score": 0.91, + "content": "\\mathbf { g } ^ { k } i s \\operatorname* { m i n } \\{ n , \\sqrt { 4 c n L _ { 1 } ^ { 2 } \\log { ( K _ { 0 } / \\delta ) } / ( \\epsilon L _ { 2 } ) } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 656, + 331, + 670 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40, + "bbox_fs": [ + 104, + 573, + 506, + 670 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "The proof of Lemma 4.2 is similar to the one of Lemma 4.1 and is deferred to Appendix B.3. These", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 686, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 505, + 700 + ], + "score": 1.0, + "content": "two lemmas only guarantee that the differential estimators satisfy the requirement (9) in a single", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 699, + 504, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 286, + 711 + ], + "score": 1.0, + "content": "iteration and can be extended to hold for all", + "type": "text" + }, + { + "bbox": [ + 286, + 699, + 293, + 709 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 699, + 420, + 711 + ], + "score": 1.0, + "content": "by using the union bound with", + "type": "text" + }, + { + "bbox": [ + 420, + 699, + 462, + 710 + ], + "score": 0.92, + "content": "K _ { 0 } = 2 K", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 699, + 493, + 711 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 493, + 699, + 504, + 709 + ], + "score": 0.8, + "content": "K", + "type": "inline_equation" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "score": 1.0, + "content": "denotes the number of iterations. 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With", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 466, + 507, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 253, + 481 + ], + "score": 1.0, + "content": "this extra assumption, STR1 enjoys", + "type": "text" + }, + { + "bbox": [ + 253, + 466, + 342, + 480 + ], + "score": 0.91, + "content": "\\tilde { \\mathcal { O } } ( n / \\epsilon ^ { 1 . 5 } , n / \\epsilon ^ { 2 } , 1 / \\epsilon ^ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 466, + 383, + 481 + ], + "score": 1.0, + "content": "SFO and", + "type": "text" + }, + { + "bbox": [ + 383, + 468, + 456, + 480 + ], + "score": 0.89, + "content": "\\tilde { \\mathcal { O } } ( \\sqrt { n } / \\epsilon ^ { 1 . 5 } , 1 / \\epsilon ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 466, + 507, + 481 + ], + "score": 1.0, + "content": "SSO. 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In this section, we consider a different complexity measure where first-order", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "and second-order oracle complexities are treated equally and our goal is to minimize the maximum", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 578, + 507, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 309, + 593 + ], + "score": 1.0, + "content": "of them. 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Denote", + "type": "text" + }, + { + "bbox": [ + 235, + 689, + 331, + 701 + ], + "score": 0.94, + "content": "\\mathbf { x } ( a ) = a \\bar { \\mathbf { x } } ^ { t } + ( 1 - a ) \\tilde { \\mathbf { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 687, + 464, + 702 + ], + "score": 1.0, + "content": ". 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Specifically, define the correction", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 217, + 99, + 393, + 114 + ], + "lines": [ + { + "bbox": [ + 217, + 99, + 393, + 114 + ], + "spans": [ + { + "bbox": [ + 217, + 99, + 393, + 114 + ], + "score": 0.91, + "content": "\\mathbf { c } ^ { k } = [ \\nabla ^ { 2 } F ( \\tilde { \\mathbf { x } } ) - \\nabla ^ { 2 } f ( \\tilde { \\mathbf { x } } ; \\mathcal { G } ) ] ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) ,", + "type": "interline_equation", + "image_path": "374f1a04ace9edfc2eff14647f0696751e7030cb0adc31d0cd8e315d6054623a.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 217, + 99, + 393, + 114 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 119, + 505, + 163 + ], + "lines": [ + { + "bbox": [ + 105, + 118, + 505, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 134, + 131 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 120, + 142, + 129 + ], + "score": 0.64, + "content": "\\tilde { \\bf x }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 118, + 465, + 131 + ], + "score": 1.0, + "content": "is some reference point updated in an epoch-wise manner. Estimator 6 adds", + "type": "text" + }, + { + "bbox": [ + 465, + 118, + 477, + 129 + ], + "score": 0.87, + "content": "\\mathbf { c } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 118, + 505, + 131 + ], + "score": 1.0, + "content": "to the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 129, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 505, + 142 + ], + "score": 1.0, + "content": "estimator in Estimator 4. Note that in Estimator 6, the first- and second-order oracle complexities are", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 141, + 505, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 505, + 154 + ], + "score": 1.0, + "content": "the same. We now analyze the first-order (and second-order) oracle complexity to meet requirement", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 151, + 124, + 165 + ], + "spans": [ + { + "bbox": [ + 104, + 151, + 124, + 165 + ], + "score": 1.0, + "content": "(9).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 167, + 505, + 226 + ], + "lines": [ + { + "bbox": [ + 105, + 167, + 506, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 237, + 180 + ], + "score": 1.0, + "content": "Lemma 5.1. Assume Algorithm", + "type": "text" + }, + { + "bbox": [ + 238, + 169, + 245, + 178 + ], + "score": 0.43, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 167, + 360, + 180 + ], + "score": 1.0, + "content": "takes the trust region radius", + "type": "text" + }, + { + "bbox": [ + 360, + 167, + 409, + 181 + ], + "score": 0.93, + "content": "r = \\sqrt { \\epsilon / L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 167, + 506, + 180 + ], + "score": 1.0, + "content": "as in Theorem 3.1. For", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 179, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 123, + 192 + ], + "score": 1.0, + "content": "any", + "type": "text" + }, + { + "bbox": [ + 123, + 180, + 148, + 191 + ], + "score": 0.88, + "content": "k \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 179, + 276, + 192 + ], + "score": 1.0, + "content": ", Estimator 6 produces estimator", + "type": "text" + }, + { + "bbox": [ + 276, + 180, + 288, + 192 + ], + "score": 0.87, + "content": "\\mathbf { g } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 179, + 403, + 192 + ], + "score": 1.0, + "content": "for the first order differential", + "type": "text" + }, + { + "bbox": [ + 403, + 180, + 439, + 192 + ], + "score": 0.91, + "content": "\\nabla F ( \\mathbf { x } ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 179, + 478, + 192 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 478, + 180, + 505, + 192 + ], + "score": 0.88, + "content": "\\| \\mathbf { g } ^ { k } - \\mathbf { \\mu }", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 189, + 503, + 205 + ], + "spans": [ + { + "bbox": [ + 107, + 191, + 175, + 203 + ], + "score": 0.91, + "content": "\\nabla F ( \\mathbf { x } ^ { k } ) \\| \\leq \\epsilon / 6", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 189, + 271, + 205 + ], + "score": 1.0, + "content": "with probability at least", + "type": "text" + }, + { + "bbox": [ + 271, + 192, + 309, + 204 + ], + "score": 0.92, + "content": "1 - \\delta / K _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 189, + 346, + 205 + ], + "score": 1.0, + "content": ", if we set", + "type": "text" + }, + { + "bbox": [ + 347, + 191, + 391, + 203 + ], + "score": 0.91, + "content": "p _ { 1 } = n ^ { 0 . 2 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 189, + 410, + 205 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 410, + 191, + 503, + 203 + ], + "score": 0.88, + "content": "s _ { 1 } \\doteq \\dot { n } ^ { 0 . 7 5 } c \\log ( \\ddot { K _ { 0 } } / \\delta )", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 203, + 506, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 133, + 214 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 204, + 173, + 213 + ], + "score": 0.83, + "content": "c = 1 1 5 2", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 203, + 191, + 214 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 191, + 203, + 205, + 214 + ], + "score": 0.89, + "content": "K _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 203, + 506, + 214 + ], + "score": 1.0, + "content": "is a constant to be determined. Consequently, the amortized per-iteration", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 212, + 445, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 315, + 227 + ], + "score": 1.0, + "content": "stochastic first-order oracle complexity to construct", + "type": "text" + }, + { + "bbox": [ + 315, + 213, + 327, + 225 + ], + "score": 0.89, + "content": "\\mathbf { g } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 212, + 337, + 227 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 337, + 213, + 441, + 226 + ], + "score": 0.92, + "content": "2 s _ { 1 } = 2 n ^ { 0 . \\dot { 7 } 5 } c \\log ( K _ { 0 } / \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 212, + 445, + 227 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 234, + 505, + 279 + ], + "lines": [ + { + "bbox": [ + 106, + 234, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 506, + 246 + ], + "score": 1.0, + "content": "The proof of Lemma 5.1 is similar to the one of Lemma 4.1 and is deferred to Appendix B.4.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 244, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 505, + 258 + ], + "score": 1.0, + "content": "Similar to the previous section, Lemma 5.1 only guarantees that the gradient estimator satisfies the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 257, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 421, + 268 + ], + "score": 1.0, + "content": "requirement (9) in a single iteration. Such result can be extended to hold for all", + "type": "text" + }, + { + "bbox": [ + 421, + 257, + 428, + 266 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 257, + 505, + 268 + ], + "score": 1.0, + "content": "by using the union", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 266, + 454, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 154, + 280 + ], + "score": 1.0, + "content": "bound with", + "type": "text" + }, + { + "bbox": [ + 154, + 267, + 195, + 279 + ], + "score": 0.91, + "content": "K _ { 0 } = 2 K", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 266, + 454, + 280 + ], + "score": 1.0, + "content": ", which together with Theorem 3.1 gives the following corollary.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 106, + 281, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 106, + 281, + 504, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 288, + 293 + ], + "score": 1.0, + "content": "Corollary 5.1 (Computational Complexity of", + "type": "text" + }, + { + "bbox": [ + 288, + 282, + 311, + 293 + ], + "score": 0.83, + "content": "\\mathrm { { S T R } _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 281, + 398, + 293 + ], + "score": 1.0, + "content": "). Algorithm 5 finds a", + "type": "text" + }, + { + "bbox": [ + 398, + 282, + 413, + 292 + ], + "score": 0.71, + "content": "1 2 \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 281, + 504, + 293 + ], + "score": 1.0, + "content": "-SOSP with probability", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 288, + 507, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 140, + 311 + ], + "score": 1.0, + "content": "at least", + "type": "text" + }, + { + "bbox": [ + 140, + 295, + 164, + 307 + ], + "score": 0.8, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 288, + 198, + 311 + ], + "score": 1.0, + "content": ", within", + "type": "text" + }, + { + "bbox": [ + 198, + 293, + 302, + 309 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathcal { O } ( \\frac { n ^ { 0 . 7 5 } \\sqrt { L _ { 2 } } \\Delta } { \\epsilon ^ { 1 . 5 } } \\log ( \\frac { \\sqrt { L _ { 2 } } \\Delta } { \\delta \\epsilon ^ { 1 . 5 } } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 288, + 507, + 311 + ], + "score": 1.0, + "content": ") overall stochastic first-order oracle queries and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 300, + 408, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 217, + 324 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathcal { O } ( \\frac { n ^ { 0 . 7 5 } \\sqrt { L _ { 2 } } \\Delta } { \\epsilon ^ { 1 . 5 } } \\log ( \\frac { d \\sqrt { L _ { 2 } } \\Delta } { \\delta \\epsilon ^ { 1 . 5 } } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 300, + 408, + 329 + ], + "score": 1.0, + "content": "\u000f1.5 δ\u000f1.5 overall stochastic second-order oracle queries.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 333, + 504, + 359 + ], + "lines": [ + { + "bbox": [ + 106, + 331, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 251, + 347 + ], + "score": 1.0, + "content": "Corollary 5.1 shows that to find an", + "type": "text" + }, + { + "bbox": [ + 252, + 336, + 257, + 344 + ], + "score": 0.7, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 331, + 379, + 347 + ], + "score": 1.0, + "content": "-SOSP, both SFO and SSO of", + "type": "text" + }, + { + "bbox": [ + 380, + 334, + 404, + 345 + ], + "score": 0.87, + "content": "\\mathrm { { S T R _ { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 331, + 421, + 347 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 421, + 332, + 476, + 347 + ], + "score": 0.92, + "content": "\\tilde { \\mathcal { O } } ( n ^ { 3 / 4 } / \\epsilon ^ { 1 . 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 331, + 506, + 347 + ], + "score": 1.0, + "content": "which", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 345, + 382, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 231, + 360 + ], + "score": 1.0, + "content": "surpasses the best existing one", + "type": "text" + }, + { + "bbox": [ + 231, + 345, + 286, + 359 + ], + "score": 0.93, + "content": "{ \\tilde { \\mathcal { O } } } ( n ^ { 4 / 5 } / \\epsilon ^ { 1 . 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 345, + 382, + 360 + ], + "score": 1.0, + "content": "in (Zhou et al., 2018c).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 108, + 374, + 393, + 388 + ], + "lines": [ + { + "bbox": [ + 105, + 373, + 394, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 394, + 389 + ], + "score": 1.0, + "content": "6 PRACTICAL STOCHASTIC TRUST REGION VARIANTS", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 108, + 398, + 304, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 398, + 306, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 306, + 412 + ], + "score": 1.0, + "content": "6.1 HANDLING INEXACT QCQP SOLUTIONS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 419, + 506, + 520 + ], + "lines": [ + { + "bbox": [ + 106, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "One drawback of MetaAlgorithm 1 is that it requires the exact solution to the QCQP subproblem (8)", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "score": 1.0, + "content": "and uses the dual variable as stopping criterion. We address this problem by developing a practical", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 441, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 506, + 453 + ], + "score": 1.0, + "content": "variant, MetaAlgorithm 7, which admits inexact QCQP solutions without access to the dual variable.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 452, + 507, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 384, + 466 + ], + "score": 1.0, + "content": "This algorithm repeatedly invokes a procedure called INEXACTTR", + "type": "text" + }, + { + "bbox": [ + 384, + 453, + 403, + 464 + ], + "score": 0.26, + "content": "\\mathbf { W E A K }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 452, + 507, + 466 + ], + "score": 1.0, + "content": ", which, as we shall see,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 462, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 152, + 477 + ], + "score": 1.0, + "content": "outputs an", + "type": "text" + }, + { + "bbox": [ + 153, + 464, + 173, + 476 + ], + "score": 0.86, + "content": "\\mathcal { O } ( \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 462, + 327, + 477 + ], + "score": 1.0, + "content": "-SOSP with a constant probability of", + "type": "text" + }, + { + "bbox": [ + 328, + 464, + 344, + 476 + ], + "score": 0.5, + "content": "2 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 462, + 357, + 477 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 357, + 464, + 398, + 476 + ], + "score": 0.91, + "content": "\\mathcal { O } ( 1 / \\epsilon ^ { 1 . 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 462, + 506, + 477 + ], + "score": 1.0, + "content": "iterations. By repeatedly", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 474, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 194, + 488 + ], + "score": 1.0, + "content": "invoking INEXACTTR", + "type": "text" + }, + { + "bbox": [ + 195, + 475, + 214, + 486 + ], + "score": 0.4, + "content": "\\mathbf { \\partial } \\cdot \\mathbf { W E A K }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 474, + 229, + 488 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 230, + 474, + 280, + 487 + ], + "score": 0.91, + "content": "\\Theta ( \\log ( 1 / \\bar { \\delta } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 474, + 475, + 488 + ], + "score": 1.0, + "content": "times, MetaAlgorithm 7 boosts the probability to", + "type": "text" + }, + { + "bbox": [ + 476, + 475, + 505, + 487 + ], + "score": 0.9, + "content": "( 1 - \\delta )", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 484, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 104, + 484, + 168, + 499 + ], + "score": 1.0, + "content": "for any desired", + "type": "text" + }, + { + "bbox": [ + 169, + 486, + 175, + 496 + ], + "score": 0.71, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 484, + 505, + 499 + ], + "score": 1.0, + "content": ". This repeating technique has been studied by, e.g., (Allen-Zhu & Li, 2018; Allen-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 495, + 504, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 244, + 510 + ], + "score": 1.0, + "content": "Zhu, 2018b). To test whether the", + "type": "text" + }, + { + "bbox": [ + 244, + 497, + 249, + 506 + ], + "score": 0.8, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 495, + 322, + 510 + ], + "score": 1.0, + "content": "-th run outputs an", + "type": "text" + }, + { + "bbox": [ + 323, + 496, + 344, + 509 + ], + "score": 0.9, + "content": "\\mathcal { O } ( \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 495, + 460, + 510 + ], + "score": 1.0, + "content": "-SOSP, we need to compute", + "type": "text" + }, + { + "bbox": [ + 460, + 497, + 504, + 509 + ], + "score": 0.92, + "content": "\\| \\nabla F ( \\mathbf { x } ^ { t } ) \\|", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 507, + 502, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 228, + 520 + ], + "score": 1.0, + "content": "and the smallest eigenvalue of", + "type": "text" + }, + { + "bbox": [ + 229, + 507, + 267, + 520 + ], + "score": 0.93, + "content": "\\nabla ^ { 2 } F ( \\mathbf { x } ^ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 507, + 502, + 520 + ], + "score": 1.0, + "content": ". The latter one can be approximated by solving the QCQP", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26 + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 523, + 374, + 546 + ], + "lines": [ + { + "bbox": [ + 236, + 523, + 374, + 546 + ], + "spans": [ + { + "bbox": [ + 236, + 523, + 374, + 546 + ], + "score": 0.94, + "content": "\\mathbf { v } ^ { t } : = \\operatorname * { a r g m i n } _ { \\| \\mathbf { v } \\| \\leq 1 } \\psi _ { t } ( \\mathbf { v } ) = \\langle \\mathbf { H } ^ { t } \\mathbf { v } , \\mathbf { v } \\rangle ,", + "type": "interline_equation", + "image_path": "fecab025599a069ccd8344e4f1adcf4e617058a069250ece8202a60bfb7555b8.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 236, + 523, + 374, + 546 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 552, + 505, + 578 + ], + "lines": [ + { + "bbox": [ + 106, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 132, + 565 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 552, + 146, + 563 + ], + "score": 0.88, + "content": "\\mathbf { H } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 552, + 219, + 565 + ], + "score": 1.0, + "content": "is the full Hessian", + "type": "text" + }, + { + "bbox": [ + 219, + 552, + 258, + 564 + ], + "score": 0.93, + "content": "\\nabla ^ { 2 } F ( \\mathbf { x } ^ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "or its estimation. One can show that MetaAlgorithm 7 finds an", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 563, + 507, + 580 + ], + "spans": [ + { + "bbox": [ + 107, + 565, + 127, + 578 + ], + "score": 0.87, + "content": "\\mathcal { O } ( \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 563, + 203, + 580 + ], + "score": 1.0, + "content": "-SOSP w.p. at least", + "type": "text" + }, + { + "bbox": [ + 204, + 565, + 231, + 578 + ], + "score": 0.92, + "content": "( 1 - \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 563, + 243, + 580 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 243, + 564, + 284, + 578 + ], + "score": 0.93, + "content": "\\tilde { \\mathcal { O } } ( 1 / \\epsilon ^ { 1 . 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 563, + 507, + 580 + ], + "score": 1.0, + "content": "iterations. We defer the detailed analysis to Appendix C.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "title", + "bbox": [ + 108, + 590, + 277, + 601 + ], + "lines": [ + { + "bbox": [ + 106, + 590, + 278, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 278, + 603 + ], + "score": 1.0, + "content": "6.2 HESSIAN-FREE IMPLEMENTATION", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 385, + 624 + ], + "score": 1.0, + "content": "Based on MetaAlgorithm 7, we propose a Hessian-free method named", + "type": "text" + }, + { + "bbox": [ + 385, + 611, + 415, + 622 + ], + "score": 0.87, + "content": "{ \\mathrm { { S T R } } } _ { \\mathrm { { f r e e } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 610, + 506, + 624 + ], + "score": 1.0, + "content": ", which is summarized", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 620, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 104, + 620, + 170, + 634 + ], + "score": 1.0, + "content": "in Algorithm 8.", + "type": "text" + }, + { + "bbox": [ + 170, + 622, + 200, + 633 + ], + "score": 0.87, + "content": "\\mathrm { S T R } _ { \\mathrm { f r e e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 620, + 457, + 634 + ], + "score": 1.0, + "content": "leverages the full/stochastic Hessian and Estimator 4 to construct", + "type": "text" + }, + { + "bbox": [ + 458, + 621, + 473, + 632 + ], + "score": 0.88, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 620, + 490, + 634 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 490, + 621, + 502, + 633 + ], + "score": 0.88, + "content": "\\mathbf { g } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 620, + 506, + 634 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "respectively. Besides, it uses Lanczos method (Gould et al., 1999; Carmon & Duchi, 2018) as the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "QCQP solver, which can be implemented in a Hessian-free manner (i.e., using only Hessian-vector", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 351, + 667 + ], + "score": 1.0, + "content": "products without explicit Hessian matrix evaluations). Thus,", + "type": "text" + }, + { + "bbox": [ + 352, + 654, + 367, + 665 + ], + "score": 0.87, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "is only accessed through Hessian-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "vector products and is never explicitly constructed. Since Hessian-vector products can be computed", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 268, + 689 + ], + "score": 1.0, + "content": "in linear time (in terms of the dimension", + "type": "text" + }, + { + "bbox": [ + 268, + 677, + 275, + 687 + ], + "score": 0.66, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 677, + 506, + 689 + ], + "score": 1.0, + "content": ") for many machine learning problems (Allen-Zhu, 2018b;", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Agarwal et al., 2017), Hessian-free methods are usually more practical than Hessian based ones", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "for high dimensional problems. The following theorem, whose proof can be found in Appendix D,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "establishes the runtime complexity (i.e., the total complexity of stochastic gradient and Hessian-vector", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 717, + 316, + 735 + ], + "spans": [ + { + "bbox": [ + 104, + 717, + 281, + 735 + ], + "score": 1.0, + "content": "product evaluations (Zhou & Gu, 2019)) of", + "type": "text" + }, + { + "bbox": [ + 281, + 721, + 311, + 732 + ], + "score": 0.87, + "content": "{ \\bf S T R } _ { \\mathrm { f r e e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 717, + 316, + 735 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 40 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "Such property can be used to improve Lemma 4.2 of Estimator 4. Specifically, define the correction", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 81, + 505, + 96 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 217, + 99, + 393, + 114 + ], + "lines": [ + { + "bbox": [ + 217, + 99, + 393, + 114 + ], + "spans": [ + { + "bbox": [ + 217, + 99, + 393, + 114 + ], + "score": 0.91, + "content": "\\mathbf { c } ^ { k } = [ \\nabla ^ { 2 } F ( \\tilde { \\mathbf { x } } ) - \\nabla ^ { 2 } f ( \\tilde { \\mathbf { x } } ; \\mathcal { G } ) ] ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) ,", + "type": "interline_equation", + "image_path": "374f1a04ace9edfc2eff14647f0696751e7030cb0adc31d0cd8e315d6054623a.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 217, + 99, + 393, + 114 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 119, + 505, + 163 + ], + "lines": [ + { + "bbox": [ + 105, + 118, + 505, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 134, + 131 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 120, + 142, + 129 + ], + "score": 0.64, + "content": "\\tilde { \\bf x }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 118, + 465, + 131 + ], + "score": 1.0, + "content": "is some reference point updated in an epoch-wise manner. Estimator 6 adds", + "type": "text" + }, + { + "bbox": [ + 465, + 118, + 477, + 129 + ], + "score": 0.87, + "content": "\\mathbf { c } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 118, + 505, + 131 + ], + "score": 1.0, + "content": "to the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 129, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 505, + 142 + ], + "score": 1.0, + "content": "estimator in Estimator 4. Note that in Estimator 6, the first- and second-order oracle complexities are", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 141, + 505, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 505, + 154 + ], + "score": 1.0, + "content": "the same. We now analyze the first-order (and second-order) oracle complexity to meet requirement", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 151, + 124, + 165 + ], + "spans": [ + { + "bbox": [ + 104, + 151, + 124, + 165 + ], + "score": 1.0, + "content": "(9).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5, + "bbox_fs": [ + 104, + 118, + 505, + 165 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 167, + 505, + 226 + ], + "lines": [ + { + "bbox": [ + 105, + 167, + 506, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 237, + 180 + ], + "score": 1.0, + "content": "Lemma 5.1. Assume Algorithm", + "type": "text" + }, + { + "bbox": [ + 238, + 169, + 245, + 178 + ], + "score": 0.43, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 167, + 360, + 180 + ], + "score": 1.0, + "content": "takes the trust region radius", + "type": "text" + }, + { + "bbox": [ + 360, + 167, + 409, + 181 + ], + "score": 0.93, + "content": "r = \\sqrt { \\epsilon / L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 167, + 506, + 180 + ], + "score": 1.0, + "content": "as in Theorem 3.1. For", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 179, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 123, + 192 + ], + "score": 1.0, + "content": "any", + "type": "text" + }, + { + "bbox": [ + 123, + 180, + 148, + 191 + ], + "score": 0.88, + "content": "k \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 179, + 276, + 192 + ], + "score": 1.0, + "content": ", Estimator 6 produces estimator", + "type": "text" + }, + { + "bbox": [ + 276, + 180, + 288, + 192 + ], + "score": 0.87, + "content": "\\mathbf { g } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 179, + 403, + 192 + ], + "score": 1.0, + "content": "for the first order differential", + "type": "text" + }, + { + "bbox": [ + 403, + 180, + 439, + 192 + ], + "score": 0.91, + "content": "\\nabla F ( \\mathbf { x } ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 179, + 478, + 192 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 478, + 180, + 505, + 192 + ], + "score": 0.88, + "content": "\\| \\mathbf { g } ^ { k } - \\mathbf { \\mu }", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 189, + 503, + 205 + ], + "spans": [ + { + "bbox": [ + 107, + 191, + 175, + 203 + ], + "score": 0.91, + "content": "\\nabla F ( \\mathbf { x } ^ { k } ) \\| \\leq \\epsilon / 6", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 189, + 271, + 205 + ], + "score": 1.0, + "content": "with probability at least", + "type": "text" + }, + { + "bbox": [ + 271, + 192, + 309, + 204 + ], + "score": 0.92, + "content": "1 - \\delta / K _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 189, + 346, + 205 + ], + "score": 1.0, + "content": ", if we set", + "type": "text" + }, + { + "bbox": [ + 347, + 191, + 391, + 203 + ], + "score": 0.91, + "content": "p _ { 1 } = n ^ { 0 . 2 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 189, + 410, + 205 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 410, + 191, + 503, + 203 + ], + "score": 0.88, + "content": "s _ { 1 } \\doteq \\dot { n } ^ { 0 . 7 5 } c \\log ( \\ddot { K _ { 0 } } / \\delta )", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 203, + 506, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 133, + 214 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 204, + 173, + 213 + ], + "score": 0.83, + "content": "c = 1 1 5 2", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 203, + 191, + 214 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 191, + 203, + 205, + 214 + ], + "score": 0.89, + "content": "K _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 203, + 506, + 214 + ], + "score": 1.0, + "content": "is a constant to be determined. Consequently, the amortized per-iteration", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 212, + 445, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 315, + 227 + ], + "score": 1.0, + "content": "stochastic first-order oracle complexity to construct", + "type": "text" + }, + { + "bbox": [ + 315, + 213, + 327, + 225 + ], + "score": 0.89, + "content": "\\mathbf { g } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 212, + 337, + 227 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 337, + 213, + 441, + 226 + ], + "score": 0.92, + "content": "2 s _ { 1 } = 2 n ^ { 0 . \\dot { 7 } 5 } c \\log ( K _ { 0 } / \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 212, + 445, + 227 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 167, + 506, + 227 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 234, + 505, + 279 + ], + "lines": [ + { + "bbox": [ + 106, + 234, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 506, + 246 + ], + "score": 1.0, + "content": "The proof of Lemma 5.1 is similar to the one of Lemma 4.1 and is deferred to Appendix B.4.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 244, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 505, + 258 + ], + "score": 1.0, + "content": "Similar to the previous section, Lemma 5.1 only guarantees that the gradient estimator satisfies the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 257, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 421, + 268 + ], + "score": 1.0, + "content": "requirement (9) in a single iteration. Such result can be extended to hold for all", + "type": "text" + }, + { + "bbox": [ + 421, + 257, + 428, + 266 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 257, + 505, + 268 + ], + "score": 1.0, + "content": "by using the union", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 266, + 454, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 154, + 280 + ], + "score": 1.0, + "content": "bound with", + "type": "text" + }, + { + "bbox": [ + 154, + 267, + 195, + 279 + ], + "score": 0.91, + "content": "K _ { 0 } = 2 K", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 266, + 454, + 280 + ], + "score": 1.0, + "content": ", which together with Theorem 3.1 gives the following corollary.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 234, + 506, + 280 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 281, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 106, + 281, + 504, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 288, + 293 + ], + "score": 1.0, + "content": "Corollary 5.1 (Computational Complexity of", + "type": "text" + }, + { + "bbox": [ + 288, + 282, + 311, + 293 + ], + "score": 0.83, + "content": "\\mathrm { { S T R } _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 281, + 398, + 293 + ], + "score": 1.0, + "content": "). Algorithm 5 finds a", + "type": "text" + }, + { + "bbox": [ + 398, + 282, + 413, + 292 + ], + "score": 0.71, + "content": "1 2 \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 281, + 504, + 293 + ], + "score": 1.0, + "content": "-SOSP with probability", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 288, + 507, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 140, + 311 + ], + "score": 1.0, + "content": "at least", + "type": "text" + }, + { + "bbox": [ + 140, + 295, + 164, + 307 + ], + "score": 0.8, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 288, + 198, + 311 + ], + "score": 1.0, + "content": ", within", + "type": "text" + }, + { + "bbox": [ + 198, + 293, + 302, + 309 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathcal { O } ( \\frac { n ^ { 0 . 7 5 } \\sqrt { L _ { 2 } } \\Delta } { \\epsilon ^ { 1 . 5 } } \\log ( \\frac { \\sqrt { L _ { 2 } } \\Delta } { \\delta \\epsilon ^ { 1 . 5 } } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 288, + 507, + 311 + ], + "score": 1.0, + "content": ") overall stochastic first-order oracle queries and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 300, + 408, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 217, + 324 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathcal { O } ( \\frac { n ^ { 0 . 7 5 } \\sqrt { L _ { 2 } } \\Delta } { \\epsilon ^ { 1 . 5 } } \\log ( \\frac { d \\sqrt { L _ { 2 } } \\Delta } { \\delta \\epsilon ^ { 1 . 5 } } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 300, + 408, + 329 + ], + "score": 1.0, + "content": "\u000f1.5 δ\u000f1.5 overall stochastic second-order oracle queries.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 281, + 507, + 329 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 333, + 504, + 359 + ], + "lines": [ + { + "bbox": [ + 106, + 331, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 251, + 347 + ], + "score": 1.0, + "content": "Corollary 5.1 shows that to find an", + "type": "text" + }, + { + "bbox": [ + 252, + 336, + 257, + 344 + ], + "score": 0.7, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 331, + 379, + 347 + ], + "score": 1.0, + "content": "-SOSP, both SFO and SSO of", + "type": "text" + }, + { + "bbox": [ + 380, + 334, + 404, + 345 + ], + "score": 0.87, + "content": "\\mathrm { { S T R _ { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 331, + 421, + 347 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 421, + 332, + 476, + 347 + ], + "score": 0.92, + "content": "\\tilde { \\mathcal { O } } ( n ^ { 3 / 4 } / \\epsilon ^ { 1 . 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 331, + 506, + 347 + ], + "score": 1.0, + "content": "which", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 345, + 382, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 231, + 360 + ], + "score": 1.0, + "content": "surpasses the best existing one", + "type": "text" + }, + { + "bbox": [ + 231, + 345, + 286, + 359 + ], + "score": 0.93, + "content": "{ \\tilde { \\mathcal { O } } } ( n ^ { 4 / 5 } / \\epsilon ^ { 1 . 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 345, + 382, + 360 + ], + "score": 1.0, + "content": "in (Zhou et al., 2018c).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 331, + 506, + 360 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 374, + 393, + 388 + ], + "lines": [ + { + "bbox": [ + 105, + 373, + 394, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 394, + 389 + ], + "score": 1.0, + "content": "6 PRACTICAL STOCHASTIC TRUST REGION VARIANTS", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 108, + 398, + 304, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 398, + 306, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 306, + 412 + ], + "score": 1.0, + "content": "6.1 HANDLING INEXACT QCQP SOLUTIONS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 419, + 506, + 520 + ], + "lines": [ + { + "bbox": [ + 106, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "One drawback of MetaAlgorithm 1 is that it requires the exact solution to the QCQP subproblem (8)", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "score": 1.0, + "content": "and uses the dual variable as stopping criterion. We address this problem by developing a practical", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 441, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 506, + 453 + ], + "score": 1.0, + "content": "variant, MetaAlgorithm 7, which admits inexact QCQP solutions without access to the dual variable.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 452, + 507, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 384, + 466 + ], + "score": 1.0, + "content": "This algorithm repeatedly invokes a procedure called INEXACTTR", + "type": "text" + }, + { + "bbox": [ + 384, + 453, + 403, + 464 + ], + "score": 0.26, + "content": "\\mathbf { W E A K }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 452, + 507, + 466 + ], + "score": 1.0, + "content": ", which, as we shall see,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 462, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 152, + 477 + ], + "score": 1.0, + "content": "outputs an", + "type": "text" + }, + { + "bbox": [ + 153, + 464, + 173, + 476 + ], + "score": 0.86, + "content": "\\mathcal { O } ( \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 462, + 327, + 477 + ], + "score": 1.0, + "content": "-SOSP with a constant probability of", + "type": "text" + }, + { + "bbox": [ + 328, + 464, + 344, + 476 + ], + "score": 0.5, + "content": "2 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 462, + 357, + 477 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 357, + 464, + 398, + 476 + ], + "score": 0.91, + "content": "\\mathcal { O } ( 1 / \\epsilon ^ { 1 . 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 462, + 506, + 477 + ], + "score": 1.0, + "content": "iterations. By repeatedly", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 474, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 194, + 488 + ], + "score": 1.0, + "content": "invoking INEXACTTR", + "type": "text" + }, + { + "bbox": [ + 195, + 475, + 214, + 486 + ], + "score": 0.4, + "content": "\\mathbf { \\partial } \\cdot \\mathbf { W E A K }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 474, + 229, + 488 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 230, + 474, + 280, + 487 + ], + "score": 0.91, + "content": "\\Theta ( \\log ( 1 / \\bar { \\delta } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 474, + 475, + 488 + ], + "score": 1.0, + "content": "times, MetaAlgorithm 7 boosts the probability to", + "type": "text" + }, + { + "bbox": [ + 476, + 475, + 505, + 487 + ], + "score": 0.9, + "content": "( 1 - \\delta )", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 484, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 104, + 484, + 168, + 499 + ], + "score": 1.0, + "content": "for any desired", + "type": "text" + }, + { + "bbox": [ + 169, + 486, + 175, + 496 + ], + "score": 0.71, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 484, + 505, + 499 + ], + "score": 1.0, + "content": ". This repeating technique has been studied by, e.g., (Allen-Zhu & Li, 2018; Allen-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 495, + 504, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 244, + 510 + ], + "score": 1.0, + "content": "Zhu, 2018b). To test whether the", + "type": "text" + }, + { + "bbox": [ + 244, + 497, + 249, + 506 + ], + "score": 0.8, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 495, + 322, + 510 + ], + "score": 1.0, + "content": "-th run outputs an", + "type": "text" + }, + { + "bbox": [ + 323, + 496, + 344, + 509 + ], + "score": 0.9, + "content": "\\mathcal { O } ( \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 495, + 460, + 510 + ], + "score": 1.0, + "content": "-SOSP, we need to compute", + "type": "text" + }, + { + "bbox": [ + 460, + 497, + 504, + 509 + ], + "score": 0.92, + "content": "\\| \\nabla F ( \\mathbf { x } ^ { t } ) \\|", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 507, + 502, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 228, + 520 + ], + "score": 1.0, + "content": "and the smallest eigenvalue of", + "type": "text" + }, + { + "bbox": [ + 229, + 507, + 267, + 520 + ], + "score": 0.93, + "content": "\\nabla ^ { 2 } F ( \\mathbf { x } ^ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 507, + 502, + 520 + ], + "score": 1.0, + "content": ". The latter one can be approximated by solving the QCQP", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26, + "bbox_fs": [ + 104, + 419, + 507, + 520 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 523, + 374, + 546 + ], + "lines": [ + { + "bbox": [ + 236, + 523, + 374, + 546 + ], + "spans": [ + { + "bbox": [ + 236, + 523, + 374, + 546 + ], + "score": 0.94, + "content": "\\mathbf { v } ^ { t } : = \\operatorname * { a r g m i n } _ { \\| \\mathbf { v } \\| \\leq 1 } \\psi _ { t } ( \\mathbf { v } ) = \\langle \\mathbf { H } ^ { t } \\mathbf { v } , \\mathbf { v } \\rangle ,", + "type": "interline_equation", + "image_path": "fecab025599a069ccd8344e4f1adcf4e617058a069250ece8202a60bfb7555b8.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 236, + 523, + 374, + 546 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 552, + 505, + 578 + ], + "lines": [ + { + "bbox": [ + 106, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 132, + 565 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 552, + 146, + 563 + ], + "score": 0.88, + "content": "\\mathbf { H } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 552, + 219, + 565 + ], + "score": 1.0, + "content": "is the full Hessian", + "type": "text" + }, + { + "bbox": [ + 219, + 552, + 258, + 564 + ], + "score": 0.93, + "content": "\\nabla ^ { 2 } F ( \\mathbf { x } ^ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "or its estimation. One can show that MetaAlgorithm 7 finds an", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 563, + 507, + 580 + ], + "spans": [ + { + "bbox": [ + 107, + 565, + 127, + 578 + ], + "score": 0.87, + "content": "\\mathcal { O } ( \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 563, + 203, + 580 + ], + "score": 1.0, + "content": "-SOSP w.p. at least", + "type": "text" + }, + { + "bbox": [ + 204, + 565, + 231, + 578 + ], + "score": 0.92, + "content": "( 1 - \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 563, + 243, + 580 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 243, + 564, + 284, + 578 + ], + "score": 0.93, + "content": "\\tilde { \\mathcal { O } } ( 1 / \\epsilon ^ { 1 . 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 563, + 507, + 580 + ], + "score": 1.0, + "content": "iterations. We defer the detailed analysis to Appendix C.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 106, + 552, + 507, + 580 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 590, + 277, + 601 + ], + "lines": [ + { + "bbox": [ + 106, + 590, + 278, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 278, + 603 + ], + "score": 1.0, + "content": "6.2 HESSIAN-FREE IMPLEMENTATION", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 385, + 624 + ], + "score": 1.0, + "content": "Based on MetaAlgorithm 7, we propose a Hessian-free method named", + "type": "text" + }, + { + "bbox": [ + 385, + 611, + 415, + 622 + ], + "score": 0.87, + "content": "{ \\mathrm { { S T R } } } _ { \\mathrm { { f r e e } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 610, + 506, + 624 + ], + "score": 1.0, + "content": ", which is summarized", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 620, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 104, + 620, + 170, + 634 + ], + "score": 1.0, + "content": "in Algorithm 8.", + "type": "text" + }, + { + "bbox": [ + 170, + 622, + 200, + 633 + ], + "score": 0.87, + "content": "\\mathrm { S T R } _ { \\mathrm { f r e e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 620, + 457, + 634 + ], + "score": 1.0, + "content": "leverages the full/stochastic Hessian and Estimator 4 to construct", + "type": "text" + }, + { + "bbox": [ + 458, + 621, + 473, + 632 + ], + "score": 0.88, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 620, + 490, + 634 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 490, + 621, + 502, + 633 + ], + "score": 0.88, + "content": "\\mathbf { g } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 620, + 506, + 634 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "respectively. Besides, it uses Lanczos method (Gould et al., 1999; Carmon & Duchi, 2018) as the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "QCQP solver, which can be implemented in a Hessian-free manner (i.e., using only Hessian-vector", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 351, + 667 + ], + "score": 1.0, + "content": "products without explicit Hessian matrix evaluations). Thus,", + "type": "text" + }, + { + "bbox": [ + 352, + 654, + 367, + 665 + ], + "score": 0.87, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "is only accessed through Hessian-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "vector products and is never explicitly constructed. 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Consider Algorithm 8 for solving problem (1). Let", + "type": "text" + }, + { + "bbox": [ + 382, + 465, + 419, + 477 + ], + "score": 0.87, + "content": "\\zeta = 1 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 463, + 424, + 478 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 424, + 463, + 476, + 478 + ], + "score": 0.82, + "content": "r = \\sqrt { \\epsilon / L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 463, + 481, + 478 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 481, + 464, + 505, + 477 + ], + "score": 0.76, + "content": "K =", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 474, + 507, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 159, + 492 + ], + "score": 0.88, + "content": "4 \\sqrt { L _ { 2 } } \\Delta / \\epsilon ^ { 1 . 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 478, + 163, + 494 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 164, + 479, + 227, + 493 + ], + "score": 0.93, + "content": "\\begin{array} { r } { T = \\frac { 3 } { 2 } \\mathrm { { l o g } ( 2 / \\delta ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 478, + 230, + 494 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 231, + 480, + 269, + 492 + ], + "score": 0.82, + "content": "c _ { 1 } = 6 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 478, + 272, + 494 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 273, + 480, + 311, + 492 + ], + "score": 0.84, + "content": "c _ { 2 } = 5 0 0 \\mathrm { \\Omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 478, + 332, + 494 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 347, + 474, + 507, + 498 + ], + "score": 1.0, + "content": "32L21\u000fL2 log(4d/δ). The hyper-parameters", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 493, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 157, + 504 + ], + "score": 1.0, + "content": "in Estimator", + "type": "text" + }, + { + "bbox": [ + 158, + 494, + 164, + 502 + ], + "score": 0.3, + "content": "^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 493, + 505, + 504 + ], + "score": 1.0, + "content": "are set to the same values as those in Lemma 4.2. The number of iterations of Lanczos", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 503, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 170, + 518 + ], + "score": 1.0, + "content": "method is set to", + "type": "text" + }, + { + "bbox": [ + 170, + 503, + 234, + 517 + ], + "score": 0.91, + "content": "\\tilde { \\mathcal { O } } ( 1 / ( L _ { 2 } \\epsilon ) ^ { 0 . 2 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 503, + 280, + 518 + ], + "score": 1.0, + "content": ". To find an", + "type": "text" + }, + { + "bbox": [ + 281, + 505, + 301, + 517 + ], + "score": 0.58, + "content": "\\mathcal { O } ( \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 503, + 378, + 518 + ], + "score": 1.0, + "content": "-SOSP w.p. at least", + "type": "text" + }, + { + "bbox": [ + 378, + 505, + 401, + 515 + ], + "score": 0.82, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 503, + 506, + 518 + ], + "score": 1.0, + "content": ", the runtime complexity is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 514, + 300, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 294, + 530 + ], + "score": 0.89, + "content": "\\tilde { \\mathcal { O } } ( d \\operatorname* { m i n } \\{ n / \\epsilon ^ { 1 . 7 5 } , 1 / \\epsilon ^ { 2 . 7 5 } + \\sqrt { n } / \\epsilon ^ { 2 } \\} \\log ( 1 / \\delta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 514, + 300, + 531 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "score": 1.0, + "content": "To solve the QCQP more efficiently, here we develop a faster solver which is based on the AppxPCA", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 544, + 507, + 561 + ], + "spans": [ + { + "bbox": [ + 104, + 544, + 507, + 561 + ], + "score": 1.0, + "content": "method (Allen-Zhu & Li, 2016) and KatyushaXW (Allen-Zhu, 2018a). See details in Appendix E. By", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 557, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 292, + 572 + ], + "score": 1.0, + "content": "replacing Lanczos method with this solver in", + "type": "text" + }, + { + "bbox": [ + 292, + 559, + 323, + 570 + ], + "score": 0.89, + "content": "{ \\bf S T R } _ { \\mathrm { f r e e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 557, + 505, + 572 + ], + "score": 1.0, + "content": ", we further improve the runtime complexity", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 568, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 117, + 585 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 570, + 316, + 583 + ], + "score": 0.77, + "content": "\\tilde { \\mathcal { O } } ( d \\operatorname* { m i n } \\{ n / \\epsilon ^ { 1 . 5 } + n ^ { 0 . 7 5 } / \\epsilon ^ { 1 . 7 5 } , 1 / \\epsilon ^ { 2 . 5 } + \\sqrt { n } / \\epsilon ^ { 2 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 568, + 438, + 585 + ], + "score": 1.0, + "content": "). We call this new algorithm", + "type": "text" + }, + { + "bbox": [ + 438, + 571, + 475, + 582 + ], + "score": 0.88, + "content": "{ \\mathrm { S T R } } _ { \\mathrm { f r e e } } +", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 568, + 506, + 585 + ], + "score": 1.0, + "content": "whose", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 580, + 504, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 472, + 596 + ], + "score": 1.0, + "content": "details can be found in Appendix E. Table 2 shows that for the runtime complexities, both", + "type": "text" + }, + { + "bbox": [ + 473, + 582, + 504, + 593 + ], + "score": 0.7, + "content": "\\mathrm { { S T R } _ { \\mathrm { { f r e e } } } }", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 592, + 502, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 123, + 605 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 593, + 160, + 604 + ], + "score": 0.89, + "content": "{ \\mathrm { S T R } } _ { \\mathrm { f r e e } } +", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 592, + 502, + 605 + ], + "score": 1.0, + "content": "outperform existing methods. 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For STR, we estimate the gradient as the way in case", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "(1). This is because such a method enjoys lower Hessian computational complexity over the way in", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "case (2) and for most problems, computing their Hessian matrices is much more time-consuming", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "than computing their gradients. 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To find an", + "type": "text" + }, + { + "bbox": [ + 281, + 505, + 301, + 517 + ], + "score": 0.58, + "content": "\\mathcal { O } ( \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 503, + 378, + 518 + ], + "score": 1.0, + "content": "-SOSP w.p. at least", + "type": "text" + }, + { + "bbox": [ + 378, + 505, + 401, + 515 + ], + "score": 0.82, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 503, + 506, + 518 + ], + "score": 1.0, + "content": ", the runtime complexity is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 514, + 300, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 294, + 530 + ], + "score": 0.89, + "content": "\\tilde { \\mathcal { O } } ( d \\operatorname* { m i n } \\{ n / \\epsilon ^ { 1 . 7 5 } , 1 / \\epsilon ^ { 2 . 7 5 } + \\sqrt { n } / \\epsilon ^ { 2 } \\} \\log ( 1 / \\delta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 514, + 300, + 531 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32, + "bbox_fs": [ + 104, + 463, + 507, + 531 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "score": 1.0, + "content": "To solve the QCQP more efficiently, here we develop a faster solver which is based on the AppxPCA", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 544, + 507, + 561 + ], + "spans": [ + { + "bbox": [ + 104, + 544, + 507, + 561 + ], + "score": 1.0, + "content": "method (Allen-Zhu & Li, 2016) and KatyushaXW (Allen-Zhu, 2018a). See details in Appendix E. By", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 557, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 292, + 572 + ], + "score": 1.0, + "content": "replacing Lanczos method with this solver in", + "type": "text" + }, + { + "bbox": [ + 292, + 559, + 323, + 570 + ], + "score": 0.89, + "content": "{ \\bf S T R } _ { \\mathrm { f r e e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 557, + 505, + 572 + ], + "score": 1.0, + "content": ", we further improve the runtime complexity", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 568, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 117, + 585 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 570, + 316, + 583 + ], + "score": 0.77, + "content": "\\tilde { \\mathcal { O } } ( d \\operatorname* { m i n } \\{ n / \\epsilon ^ { 1 . 5 } + n ^ { 0 . 7 5 } / \\epsilon ^ { 1 . 7 5 } , 1 / \\epsilon ^ { 2 . 5 } + \\sqrt { n } / \\epsilon ^ { 2 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 568, + 438, + 585 + ], + "score": 1.0, + "content": "). We call this new algorithm", + "type": "text" + }, + { + "bbox": [ + 438, + 571, + 475, + 582 + ], + "score": 0.88, + "content": "{ \\mathrm { S T R } } _ { \\mathrm { f r e e } } +", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 568, + 506, + 585 + ], + "score": 1.0, + "content": "whose", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 580, + 504, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 472, + 596 + ], + "score": 1.0, + "content": "details can be found in Appendix E. Table 2 shows that for the runtime complexities, both", + "type": "text" + }, + { + "bbox": [ + 473, + 582, + 504, + 593 + ], + "score": 0.7, + "content": "\\mathrm { { S T R } _ { \\mathrm { { f r e e } } } }", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 592, + 502, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 123, + 605 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 593, + 160, + 604 + ], + "score": 0.89, + "content": "{ \\mathrm { S T R } } _ { \\mathrm { f r e e } } +", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 592, + 502, + 605 + ], + "score": 1.0, + "content": "outperform existing methods. See more comparison and discussion in Appendix E.4.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5, + "bbox_fs": [ + 104, + 536, + 507, + 605 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 620, + 200, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 618, + 201, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 201, + 635 + ], + "score": 1.0, + "content": "7 EXPERIMENTS", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 507, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 507, + 657 + ], + "score": 1.0, + "content": "Here we compare the proposed STR with several state-of-the-art (stochastic) cubic regularized algo-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "rithms and trust region approaches, including trust region (TR) algorithm (Conn et al., 2000), adaptive", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "cubic regularization (ARC) (Cartis et al., 2011), sub-sampled cubic regularization (SCR) (Kohler &", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "Lucchi, 2017a), stochastic variance-reduced cubic (SVRC) (Zhou et al., 2018c), Lite-SVRC (Zhou", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "et al., 2018b), and SRVRC (Zhou & Gu, 2019). For STR, we estimate the gradient as the way in case", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "(1). This is because such a method enjoys lower Hessian computational complexity over the way in", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "case (2) and for most problems, computing their Hessian matrices is much more time-consuming", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "than computing their gradients. For the subproblems in these compared methods, we use Lanczos", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "score": 1.0, + "content": "method (Gould et al., 1999; Kohler & Lucchi, 2017a) to solve the subproblem approximately in a", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 462, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 476 + ], + "score": 1.0, + "content": "Hessian-related Krylov subspace. We run simulations on seven datasets from LibSVM (a9a, ijcnn,", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 473, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 506, + 487 + ], + "score": 1.0, + "content": "codrna, phishing, w8a, epsilon and mnist). 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AlgorithmRuntime
Hessian-free Cubic (Carmon & Duchi, 2016)Fast-Cubic (Agarwal et al., 2017)Stochastic Cubic (Tripuraneni et al., 2018)SRVRCfree (Zhou & Gu, 2019)（）0dn dn0.75+151.756d *35O(min{ d}）*
STRfree (this paper)STRfree+ (this paper)O(min{dn d dn0.5e1.75，2.752O(min{ dn0.75dn d dn0.51.5+e1.75，2.5+20)
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AlgorithmRuntime
Hessian-free Cubic (Carmon & Duchi, 2016)Fast-Cubic (Agarwal et al., 2017)Stochastic Cubic (Tripuraneni et al., 2018)SRVRCfree (Zhou & Gu, 2019)（）0dn dn0.75+151.756d *35O(min{ d}）*
STRfree (this paper)STRfree+ (this paper)O(min{dn d dn0.5e1.75，2.752O(min{ dn0.75dn d dn0.51.5+e1.75，2.5+20)
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STR", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 404, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 506, + 417 + ], + "score": 1.0, + "content": "shows very similar behaviors as those in Figure 1. Specifically, STR achieves fastest convergence rate", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "in terms of both algorithm running time and Hessian sample complexity. On the codrna dataset we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "further plot the gradient norm versus running time and Hessian sample complexity. 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All these results confirm the superiority of the proposed STR.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 476, + 297, + 608 + ], + "lines": [ + { + "bbox": [ + 106, + 475, + 298, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 298, + 488 + ], + "score": 1.0, + "content": "Comparison of Hessian-free algorithms.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 486, + 298, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 298, + 499 + ], + "score": 1.0, + "content": "Here we compare our proposed Hessian-free", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 497, + 298, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 170, + 510 + ], + "score": 1.0, + "content": "STR, namely", + "type": "text" + }, + { + "bbox": [ + 171, + 498, + 201, + 509 + ], + "score": 0.85, + "content": "\\mathrm { { S T R } _ { \\mathrm { { f r e e } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 497, + 298, + 510 + ], + "score": 1.0, + "content": ", with other state-of-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 508, + 297, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 297, + 520 + ], + "score": 1.0, + "content": "the-art Hessian-free algorithms on the two", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 520, + 297, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 297, + 532 + ], + "score": 1.0, + "content": "high-dimensional datasets, including epsilon", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 531, + 297, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 297, + 542 + ], + "score": 1.0, + "content": "and mnist (see details in Appendix F). 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STR", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 404, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 506, + 417 + ], + "score": 1.0, + "content": "shows very similar behaviors as those in Figure 1. Specifically, STR achieves fastest convergence rate", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "in terms of both algorithm running time and Hessian sample complexity. On the codrna dataset we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "further plot the gradient norm versus running time and Hessian sample complexity. One can obverse", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "that the gradient in STR vanishes significantly faster than other algorithms which means that STR can", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 448, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 506, + 461 + ], + "score": 1.0, + "content": "find the stationary point with high efficiency. See Figure 4 in Appendix F.2 for more experimental", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 459, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 471 + ], + "score": 1.0, + "content": "results on gradient norm comparison. All these results confirm the superiority of the proposed STR.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 393, + 506, + 471 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 476, + 297, + 608 + ], + "lines": [ + { + "bbox": [ + 106, + 475, + 298, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 298, + 488 + ], + "score": 1.0, + "content": "Comparison of Hessian-free algorithms.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 486, + 298, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 298, + 499 + ], + "score": 1.0, + "content": "Here we compare our proposed Hessian-free", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 497, + 298, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 170, + 510 + ], + "score": 1.0, + "content": "STR, namely", + "type": "text" + }, + { + "bbox": [ + 171, + 498, + 201, + 509 + ], + "score": 0.85, + "content": "\\mathrm { { S T R } _ { \\mathrm { { f r e e } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 497, + 298, + 510 + ], + "score": 1.0, + "content": ", with other state-of-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 508, + 297, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 297, + 520 + ], + "score": 1.0, + "content": "the-art Hessian-free algorithms on the two", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 520, + 297, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 297, + 532 + ], + "score": 1.0, + "content": "high-dimensional datasets, including epsilon", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 531, + 297, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 297, + 542 + ], + "score": 1.0, + "content": "and mnist (see details in Appendix F). Here", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 541, + 298, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 194, + 554 + ], + "score": 1.0, + "content": "we do not compare", + "type": "text" + }, + { + "bbox": [ + 194, + 542, + 231, + 553 + ], + "score": 0.88, + "content": "{ \\mathrm { S T R } } _ { \\mathrm { f r e e } } +", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 541, + 298, + 554 + ], + "score": 1.0, + "content": ", as it is based", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 552, + 298, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 298, + 565 + ], + "score": 1.0, + "content": "on AppxPCA method (Allen-Zhu & Li, 2016)", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 562, + 298, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 298, + 576 + ], + "score": 1.0, + "content": "and KatyushaXW (Allen-Zhu, 2018a) which", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 575, + 297, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 297, + 586 + ], + "score": 1.0, + "content": "require tuning a lot of hyper-parameters. From", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 585, + 297, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 297, + 597 + ], + "score": 1.0, + "content": "the results in Figure 3, one can observe that", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 595, + 296, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 264, + 610 + ], + "score": 1.0, + "content": "compared with other algorithms, our", + "type": "text" + }, + { + "bbox": [ + 265, + 596, + 296, + 608 + ], + "score": 0.82, + "content": "{ \\mathrm { { S T R } } } _ { \\mathrm { { f r e e } } }", + "type": "inline_equation" + } + ], + "index": 29 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 475, + 298, + 610 + ] + }, + { + "type": "image", + "bbox": [ + 308, + 478, + 506, + 576 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 308, + 478, + 506, + 576 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 308, + 478, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 308, + 478, + 506, + 576 + ], + "score": 0.942, + "type": "image", + "image_path": "6ae2d6fe5712438b857d2a33bd755443d66d2c7c1e1911c6e91971831b384a67.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 308, + 478, + 506, + 492.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 308, + 492.0, + 506, + 506.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 308, + 506.0, + 506, + 520.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 308, + 520.0, + 506, + 534.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 308, + 534.0, + 506, + 548.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 308, + 548.0, + 506, + 562.0 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 308, + 562.0, + 506, + 576.0 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 304, + 581, + 504, + 604 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 304, + 581, + 504, + 593 + ], + "spans": [ + { + "bbox": [ + 304, + 581, + 504, + 593 + ], + "score": 1.0, + "content": "Figure 3: Comparison of Hessian-free algorithms", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 304, + 592, + 502, + 605 + ], + "spans": [ + { + "bbox": [ + 304, + 592, + 502, + 605 + ], + "score": 1.0, + "content": "on non-convex logistic and soft-max regressions.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + } + ], + "index": 35.25 + }, + { + "type": "text", + "bbox": [ + 108, + 608, + 505, + 641 + ], + "lines": [ + { + "bbox": [ + 106, + 607, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 506, + 620 + ], + "score": 1.0, + "content": "achieves the best convergence speed which demonstrates its high efficiency in realistic applications.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 619, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 230, + 631 + ], + "score": 1.0, + "content": "Besides, one also can find that", + "type": "text" + }, + { + "bbox": [ + 230, + 619, + 261, + 630 + ], + "score": 0.87, + "content": "\\operatorname { S T R } _ { \\mathrm { f r e e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 619, + 505, + 631 + ], + "score": 1.0, + "content": "is much faster than Hessian based STR since computing full", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 629, + 492, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 492, + 641 + ], + "score": 1.0, + "content": "Hessian is actually much computationally expensive than the computation of the Hessian vector.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 607, + 506, + 641 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 662, + 195, + 675 + ], + "lines": [ + { + "bbox": [ + 104, + 660, + 197, + 678 + ], + "spans": [ + { + "bbox": [ + 104, + 660, + 197, + 678 + ], + "score": 1.0, + "content": "8 CONCLUSION", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "We proposed two stochastic trust region variants. Under two settings (whether stochastic first- and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 507, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 507, + 712 + ], + "score": 1.0, + "content": "second-order oracle complexities are treated equally), the proposed methods achieve state-of-the-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 711, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 722 + ], + "score": 1.0, + "content": "art oracle complexities. 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For simplicity of notation, we denote", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 201, + 227, + 406, + 245 + ], + "lines": [ + { + "bbox": [ + 201, + 227, + 406, + 245 + ], + "spans": [ + { + "bbox": [ + 201, + 227, + 406, + 245 + ], + "score": 0.82, + "content": "\\nabla _ { k } \\overset { \\mathrm { d e f } } { = } \\nabla F ( { \\mathbf x } ^ { k } ) - { \\mathbf g } ^ { k } \\mathrm { a n d } \\nabla _ { k } ^ { 2 } \\overset { \\mathrm { d e f } } { = } \\nabla ^ { 2 } F ( { \\mathbf x } ^ { k } ) - { \\mathbf H } ^ { k } .", + "type": "interline_equation", + "image_path": "1616005f39564fe9d71d3519e3e67722604349daf5642951c666821e901afad5.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 201, + 227, + 406, + 245 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 249, + 231, + 261 + ], + "lines": [ + { + "bbox": [ + 105, + 247, + 231, + 262 + ], + 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k } \\| ^ { 2 } \\leq \\frac { 1 } { 3 } \\cdot \\frac { \\epsilon ^ { 1 . 5 } } { \\sqrt { L _ { 2 } } } .", + "type": "interline_equation", + "image_path": "20f59b80b78f28c39cababf76e097dc8503b4bdd19e4adab58f76d072d24133d.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 223, + 379, + 388, + 407 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 412, + 505, + 435 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 363, + 425 + ], + "score": 1.0, + "content": "The optimality of (5) indicates that there exists a dual variable", + "type": "text" + }, + { + "bbox": [ + 364, + 411, + 395, + 424 + ], + "score": 0.92, + "content": "\\lambda ^ { k } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 411, + 506, + 425 + ], + "score": 1.0, + "content": "so that (Corollary 7.2.2 in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 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have", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 528, + 377, + 554 + ], + "lines": [ + { + "bbox": [ + 234, + 528, + 377, + 554 + ], + "spans": [ + { + "bbox": [ + 234, + 528, + 377, + 554 + ], + "score": 0.94, + "content": "\\langle \\mathbf { g } ^ { k } + \\mathbf { H } ^ { k } \\mathbf { h } ^ { k } + \\frac { \\lambda ^ { k } L _ { 2 } } { 2 } \\mathbf { h } ^ { k } , \\mathbf { h } ^ { k } \\rangle = 0 .", + "type": "interline_equation", + "image_path": "3ca4b496905c8f950c02fddce5bfead8062c1e2d6890f6ca671ead521c898670.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 234, + 528, + 377, + 554 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 557, + 239, + 569 + ], + "lines": [ + { + "bbox": [ + 106, + 555, + 239, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 239, + 570 + ], + "score": 1.0, + "content": "Additionally, using (14) we have", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 572, + 365, + 599 + ], + "lines": [ + { + "bbox": [ + 245, + 572, + 365, + 599 + ], + "spans": [ + { + "bbox": [ + 245, + 572, + 365, + 599 + ], + "score": 0.94, + "content": "\\langle ( \\mathbf { H } ^ { k } + \\frac { \\lambda ^ { k } L _ { 2 } } { 2 } \\mathbf { I } ) \\mathbf { h } ^ { k } , \\mathbf { h } ^ { k } \\rangle \\geq 0 ,", + "type": "interline_equation", + "image_path": "9fe5336125356da135765759de8b320f4b0b0bb83a80b24651b692e574126573.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 245, + 572, + 365, + 599 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 602, + 231, + 614 + ], + "lines": [ + { + "bbox": [ + 106, + 601, + 230, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 230, + 615 + ], + "score": 1.0, + "content": "which together with (16) gives", + 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"content": "\\| \\mathbf { h } ^ { k } \\| = \\sqrt { \\epsilon / L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 628, + 425, + 643 + ], + "score": 1.0, + "content": "as we have", + "type": "text" + }, + { + "bbox": [ + 425, + 628, + 504, + 642 + ], + "score": 0.92, + "content": "\\lambda ^ { k } > 3 \\sqrt { \\epsilon / L _ { 2 } } > 0", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 642, + 493, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 424, + 657 + ], + "score": 1.0, + "content": "before MetaAlgorithm 1 terminates. Plug (12), (16), and (17) into (11) and use", + "type": "text" + }, + { + "bbox": [ + 424, + 642, + 489, + 656 + ], + "score": 0.94, + "content": "\\| \\mathbf { h } ^ { k } \\| = \\sqrt { \\epsilon / L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 642, + 493, + 657 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 660, + 400, + 688 + ], + "lines": [ + { + "bbox": [ + 211, + 660, + 400, + 688 + ], + "spans": [ + { + "bbox": [ + 211, + 660, + 400, + 688 + ], + "score": 0.94, + "content": "F ( \\mathbf { x } ^ { k + 1 } ) \\leq F ( \\mathbf { x } ^ { k } ) - \\frac { L _ { 2 } \\lambda ^ { k } } { 4 } \\cdot \\frac { \\epsilon } { L _ { 2 } } + \\frac { 1 } { 2 } \\cdot \\frac { \\epsilon ^ { 1 . 5 } } { \\sqrt { L _ { 2 } } } .", + "type": "interline_equation", + "image_path": "5d9f43775d126fd33c19908719dde4c8f9bdab9615d428224acaa9852c3a134b.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 211, + 660, + 400, + 688 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 693, + 287, + 705 + ], + "lines": [ + { + "bbox": [ + 105, + 691, + 288, + 707 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 194, + 707 + ], + "score": 1.0, + "content": "Therefore, if we have", + "type": "text" + }, + { + "bbox": [ + 194, + 693, + 263, + 706 + ], + "score": 0.94, + "content": "\\lambda ^ { k } > 3 \\epsilon ^ { 0 . 5 } / \\sqrt { L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 691, + 288, + 707 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 709, + 377, + 736 + ], + "lines": [ + { + "bbox": [ + 234, + 709, + 377, + 736 + ], + "spans": [ + { + "bbox": [ + 234, + 709, + 377, + 736 + ], + "score": 0.94, + "content": "F ( \\mathbf { x } ^ { k + 1 } ) \\leq F ( \\mathbf { x } ^ { k } ) - { \\frac { 1 } { 4 { \\sqrt { L _ { 2 } } } } } \\cdot \\epsilon ^ { 1 . 5 } .", + "type": "interline_equation", + "image_path": "d2678914ca47a96601f43b408625494bc06fbe5a3e6fe963e2b298c694e0db95.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 234, + 709, + 377, + 736 + ], + "spans": [], + "index": 33 + } + ] + } + ], + "page_idx": 12, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "13", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 183, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 184, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 184, + 96 + ], + "score": 1.0, + "content": "A APPENDIX", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 105, + 505, + 150 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "In this appendix, Sec. B first provides the proofs for the results in the manuscript. Then, we analyze", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 195, + 129 + ], + "score": 1.0, + "content": "MetaAlgorithm 7 and", + "type": "text" + }, + { + "bbox": [ + 196, + 117, + 227, + 128 + ], + "score": 0.91, + "content": "\\mathrm { { S T R } _ { \\mathrm { { f r e e } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "in Sec. C and Sec. D, respectively. Next, in Sec. E, we develop a fast", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 128, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 363, + 141 + ], + "score": 1.0, + "content": "QCQP solver to further improve the computational complexity of", + "type": "text" + }, + { + "bbox": [ + 363, + 128, + 393, + 139 + ], + "score": 0.84, + "content": "\\operatorname { S T R } _ { \\mathrm { f r e e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 128, + 505, + 141 + ], + "score": 1.0, + "content": ". Finally, more experimental", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 139, + 276, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 139, + 276, + 151 + ], + "score": 1.0, + "content": "details and results are presented in Sec. F.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 105, + 505, + 151 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 166, + 228, + 179 + ], + "lines": [ + { + "bbox": [ + 104, + 164, + 229, + 181 + ], + "spans": [ + { + "bbox": [ + 104, + 164, + 229, + 181 + ], + "score": 1.0, + "content": "B DEFERRED PROOFS", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "title", + "bbox": [ + 107, + 191, + 238, + 202 + ], + "lines": [ + { + "bbox": [ + 105, + 190, + 239, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 239, + 204 + ], + "score": 1.0, + "content": "B.1 PROOF OF THEOREM 3.1", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 108, + 211, + 286, + 223 + ], + "lines": [ + { + "bbox": [ + 106, + 211, + 286, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 211, + 286, + 225 + ], + "score": 1.0, + "content": "Proof. For simplicity of notation, we denote", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 106, + 211, + 286, + 225 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 201, + 227, + 406, + 245 + ], + "lines": [ + { + "bbox": [ + 201, + 227, + 406, + 245 + ], + "spans": [ + { + "bbox": [ + 201, + 227, + 406, + 245 + ], + "score": 0.82, + "content": "\\nabla _ { k } \\overset { \\mathrm { d e f } } { = } \\nabla F ( { \\mathbf x } ^ { k } ) - { \\mathbf g } ^ { k } \\mathrm { a n d } \\nabla _ { k } ^ { 2 } \\overset { \\mathrm { d e f } } { = } \\nabla ^ { 2 } F ( { \\mathbf x } ^ { k } ) - { \\mathbf H } ^ { k } .", + "type": "interline_equation", + "image_path": "1616005f39564fe9d71d3519e3e67722604349daf5642951c666821e901afad5.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 201, + 227, + 406, + 245 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 249, + 231, + 261 + ], + "lines": [ + 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^ { k } , \\mathbf { h } ^ { k } \\rangle + \\displaystyle \\frac { 1 } { 2 } \\langle [ \\nabla _ { k } ^ { 2 } + \\mathbf { H } ^ { k } ] { \\mathbf h } ^ { k } , \\mathbf { h } ^ { k } \\rangle + \\displaystyle \\frac { L _ { 2 } } { 6 } \\| { \\mathbf h } ^ { k } \\| ^ { 3 } . } \\end{array}", + "type": "interline_equation", + "image_path": "60776325382d53ffc379302cc2bd007615bafade13a29997a5454b14ac1cb486.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 159, + 265, + 451, + 281.6666666666667 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 159, + 281.6666666666667, + 451, + 298.33333333333337 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 159, + 298.33333333333337, + 451, + 315.00000000000006 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 286, + 329 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 286, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 286, + 331 + ], + "score": 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} L _ { 2 } } { 2 } \\cdot { \\bf I } \\approx 0 , } } \\\\ { { \\displaystyle \\mathrm { C o m p l e m e n t a r y : ~ } \\lambda ^ { k } \\cdot ( \\| { \\bf h } ^ { k } \\| - r ) = 0 . } } \\end{array}", + "type": "interline_equation", + "image_path": "76c1bfa5079d2ca5ab495230ca27f14a0fb325aa2f4fbe19b88aeb68dd1ab56f.jpg" + } + ] + } + ], + "index": 20.5, + "virtual_lines": [ + { + "bbox": [ + 210, + 438, + 401, + 455.25 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 210, + 455.25, + 401, + 472.5 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 210, + 472.5, + 401, + 489.75 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 210, + 489.75, + 401, + 507.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 511, + 240, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 240, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 188, + 524 + ], + "score": 1.0, + "content": "Multiplying (13) by", + "type": "text" + }, + { 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"type": "text", + "bbox": [ + 107, + 630, + 507, + 656 + ], + "lines": [ + { + "bbox": [ + 105, + 628, + 504, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 316, + 643 + ], + "score": 1.0, + "content": "Moreover, the complementary property (15) indicates", + "type": "text" + }, + { + "bbox": [ + 316, + 629, + 381, + 643 + ], + "score": 0.94, + "content": "\\| \\mathbf { h } ^ { k } \\| = \\sqrt { \\epsilon / L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 628, + 425, + 643 + ], + "score": 1.0, + "content": "as we have", + "type": "text" + }, + { + "bbox": [ + 425, + 628, + 504, + 642 + ], + "score": 0.92, + "content": "\\lambda ^ { k } > 3 \\sqrt { \\epsilon / L _ { 2 } } > 0", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 642, + 493, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 424, + 657 + ], + "score": 1.0, + "content": "before MetaAlgorithm 1 terminates. Plug (12), (16), and (17) into (11) and use", + "type": "text" + }, + { + "bbox": [ + 424, + 642, + 489, + 656 + ], + "score": 0.94, + "content": "\\| \\mathbf { h } ^ { k } \\| = \\sqrt { \\epsilon / L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 642, + 493, + 657 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 628, + 504, + 657 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 660, + 400, + 688 + ], + "lines": [ + { + "bbox": [ + 211, + 660, + 400, + 688 + ], + "spans": [ + { + "bbox": [ + 211, + 660, + 400, + 688 + ], + "score": 0.94, + "content": "F ( \\mathbf { x } ^ { k + 1 } ) \\leq F ( \\mathbf { x } ^ { k } ) - \\frac { L _ { 2 } \\lambda ^ { k } } { 4 } \\cdot \\frac { \\epsilon } { L _ { 2 } } + \\frac { 1 } { 2 } \\cdot \\frac { \\epsilon ^ { 1 . 5 } } { \\sqrt { L _ { 2 } } } .", + "type": "interline_equation", + "image_path": "5d9f43775d126fd33c19908719dde4c8f9bdab9615d428224acaa9852c3a134b.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 211, + 660, + 400, + 688 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 693, + 287, + 705 + ], + "lines": [ + { + "bbox": [ + 105, + 691, + 288, + 707 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 194, + 707 + ], + "score": 1.0, + "content": "Therefore, if we have", + "type": "text" + }, + { + "bbox": [ + 194, + 693, + 263, + 706 + ], + "score": 0.94, + "content": "\\lambda ^ { k } > 3 \\epsilon ^ { 0 . 5 } / \\sqrt { L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 691, + 288, + 707 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 691, + 288, + 707 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 709, + 377, + 736 + ], + "lines": [ + { + "bbox": [ + 234, + 709, + 377, + 736 + ], + "spans": [ + { + "bbox": [ + 234, + 709, + 377, + 736 + ], + "score": 0.94, + "content": "F ( \\mathbf { x } ^ { k + 1 } ) \\leq F ( \\mathbf { x } ^ { k } ) - { \\frac { 1 } { 4 { \\sqrt { L _ { 2 } } } } } \\cdot \\epsilon ^ { 1 . 5 } .", + "type": "interline_equation", + "image_path": "d2678914ca47a96601f43b408625494bc06fbe5a3e6fe963e2b298c694e0db95.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 234, + 709, + 377, + 736 + ], + "spans": [], + "index": 33 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 504, + 106 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 504, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 237, + 96 + ], + "score": 1.0, + "content": "Using Assumption 2.1, we find", + "type": "text" + }, + { + "bbox": [ + 238, + 81, + 309, + 95 + ], + "score": 0.93, + "content": "\\lambda ^ { k } \\le 3 \\epsilon ^ { 0 . 5 } / \\sqrt { L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 79, + 379, + 96 + ], + "score": 1.0, + "content": "in no more than", + "type": "text" + }, + { + "bbox": [ + 379, + 81, + 504, + 95 + ], + "score": 0.9, + "content": "4 \\sqrt { L _ { 2 } } \\cdot ( F ( \\mathbf { x } ^ { 0 } ) - F ( \\mathbf { x } ^ { * } ) ) / \\epsilon ^ { 1 . 5 }", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 150, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 150, + 107 + ], + "score": 1.0, + "content": "iterations.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 108, + 109, + 501, + 123 + ], + "lines": [ + { + "bbox": [ + 105, + 107, + 503, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 107, + 202, + 124 + ], + "score": 1.0, + "content": "We now show that once", + "type": "text" + }, + { + "bbox": [ + 202, + 109, + 271, + 123 + ], + "score": 0.94, + "content": "\\lambda ^ { k } \\le 3 \\epsilon ^ { 0 . 5 } / \\sqrt { L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 107, + 294, + 124 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 295, + 110, + 317, + 121 + ], + "score": 0.91, + "content": "\\mathbf { x } ^ { k + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 107, + 370, + 124 + ], + "score": 1.0, + "content": "is already an", + "type": "text" + }, + { + "bbox": [ + 371, + 110, + 392, + 123 + ], + "score": 0.89, + "content": "\\mathcal { O } ( \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 107, + 503, + 124 + ], + "score": 1.0, + "content": "-SOSP: From (13), we have", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 126, + 381, + 152 + ], + "lines": [ + { + "bbox": [ + 230, + 126, + 381, + 152 + ], + "spans": [ + { + "bbox": [ + 230, + 126, + 381, + 152 + ], + "score": 0.94, + "content": "\\| \\mathbf { g } ^ { k } + \\mathbf { H } ^ { k } \\mathbf { h } ^ { k } \\| = \\frac { L _ { 2 } \\lambda ^ { k } } { 2 } \\cdot \\| \\mathbf { h } ^ { k } \\| \\leq 2 \\epsilon .", + "type": "interline_equation", + "image_path": "fe5176f9e4fc73913bf17cea59355d763b3609c4969c2274a129d80ca54486ee.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 230, + 126, + 381, + 152 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 156, + 506, + 182 + ], + "lines": [ + { + "bbox": [ + 105, + 155, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 178, + 172 + ], + "score": 1.0, + "content": "The assumptions", + "type": "text" + }, + { + "bbox": [ + 178, + 157, + 231, + 170 + ], + "score": 0.92, + "content": "\\lVert \\nabla _ { k } \\rVert \\leq \\epsilon / 6", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 155, + 251, + 172 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 251, + 156, + 324, + 170 + ], + "score": 0.93, + "content": "\\| \\nabla _ { k } ^ { 2 } \\| \\le \\sqrt { \\epsilon L _ { 2 } } / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 155, + 475, + 172 + ], + "score": 1.0, + "content": "together with the trust region radius", + "type": "text" + }, + { + "bbox": [ + 475, + 158, + 505, + 170 + ], + "score": 0.89, + "content": "\\| \\mathbf { h } \\| \\leq", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 168, + 165, + 185 + ], + "spans": [ + { + "bbox": [ + 107, + 169, + 138, + 183 + ], + "score": 0.92, + "content": "\\sqrt { \\epsilon / L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 168, + 165, + 185 + ], + "score": 1.0, + "content": "imply", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 187, + 457, + 203 + ], + "lines": [ + { + "bbox": [ + 153, + 187, + 457, + 203 + ], + "spans": [ + { + "bbox": [ + 153, + 187, + 457, + 203 + ], + "score": 0.88, + "content": "\\| \\nabla F ( { \\mathbf x } ^ { k } ) + \\nabla ^ { 2 } F ( { \\mathbf x } ^ { k } ) { \\mathbf h } ^ { k } \\| \\leq \\| { \\mathbf g } ^ { k } + { \\mathbf H } ^ { k } { \\mathbf h } ^ { k } \\| + \\| \\nabla _ { k } \\| + \\| \\nabla _ { k } ^ { 2 } \\cdot { \\mathbf h } ^ { k } \\| \\leq 2 . 5 \\epsilon .", + "type": "interline_equation", + "image_path": "361b3602948ad49869427833b2f7dc2ffec583dca112dd5db25b66e21207d4d3.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 153, + 187, + 457, + 203 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 207, + 299, + 218 + ], + "lines": [ + { + "bbox": [ + 106, + 206, + 299, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 299, + 219 + ], + "score": 1.0, + "content": "On the other hand use Assumption 2.3 to bound", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 183, + 222, + 426, + 246 + ], + "lines": [ + { + "bbox": [ + 183, + 222, + 426, + 246 + ], + "spans": [ + { + "bbox": [ + 183, + 222, + 426, + 246 + ], + "score": 0.92, + "content": "\\| \\nabla F ( { \\mathbf { x } } ^ { k + 1 } ) - \\nabla F ( { \\mathbf { x } } ^ { k } ) - \\nabla ^ { 2 } F ( { \\mathbf { x } } ^ { k } ) { \\mathbf { h } } ^ { k } \\| \\leq \\frac { L _ { 2 } } { 2 } \\| { \\mathbf { h } } ^ { k } \\| ^ { 2 } \\leq \\frac { \\epsilon } { 2 } .", + "type": "interline_equation", + "image_path": "48e7f410cff77d58233676d4808238ec882e3f54bf80340cf3f8c99867ddc477.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 183, + 222, + 426, + 246 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 251, + 326, + 263 + ], + "lines": [ + { + "bbox": [ + 106, + 250, + 324, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 246, + 265 + ], + "score": 1.0, + "content": "Combining these two results gives", + "type": "text" + }, + { + "bbox": [ + 246, + 251, + 324, + 264 + ], + "score": 0.91, + "content": "\\| \\nabla F ( \\mathbf { x } ^ { k + 1 } ) \\| \\leq 3 \\epsilon", + "type": "inline_equation" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 108, + 263, + 489, + 276 + ], + "lines": [ + { + "bbox": [ + 106, + 261, + 490, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 236, + 277 + ], + "score": 1.0, + "content": "Besides, using Assumption 2.3,", + "type": "text" + }, + { + "bbox": [ + 236, + 263, + 306, + 276 + ], + "score": 0.9, + "content": "\\| \\nabla _ { k } ^ { 2 } \\| \\le \\sqrt { \\epsilon L _ { 2 } } / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 261, + 490, + 277 + ], + "score": 1.0, + "content": ", and (14), we derive the Hessian lower bound", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 129, + 280, + 471, + 297 + ], + "lines": [ + { + "bbox": [ + 129, + 280, + 471, + 297 + ], + "spans": [ + { + "bbox": [ + 129, + 280, + 471, + 297 + ], + "score": 0.89, + "content": "\\nabla ^ { 2 } F ( \\mathbf { x } ^ { k + 1 } ) \\approx \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) - L _ { 2 } \\cdot \\Vert \\mathbf { h } ^ { k } \\Vert \\mathbf { I } \\approx \\mathbf { H } ^ { k } - \\sqrt { \\epsilon L _ { 2 } } / 3 \\mathbf { I } - L _ { 2 } \\Vert \\mathbf { h } ^ { k } \\Vert \\mathbf { I } \\approx - \\sqrt { 1 2 \\epsilon L _ { 2 } } \\mathbf { I } .", + "type": "interline_equation", + "image_path": "e36cc1c7ae4617bfd606174d203031bbdf0f504a87031e63a7bc9653d9b34b71.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 129, + 280, + 471, + 297 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 302, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 300, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 134, + 316 + ], + "score": 1.0, + "content": "Hence", + "type": "text" + }, + { + "bbox": [ + 134, + 301, + 156, + 313 + ], + "score": 0.9, + "content": "\\mathbf { x } ^ { k + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 300, + 173, + 316 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 174, + 303, + 189, + 313 + ], + "score": 0.78, + "content": "1 2 \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 300, + 347, + 316 + ], + "score": 1.0, + "content": "-stationary point. Additionally, we have", + "type": "text" + }, + { + "bbox": [ + 347, + 302, + 388, + 315 + ], + "score": 0.93, + "content": "\\| \\mathbf { h } ^ { k } \\| = r", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 300, + 506, + 316 + ], + "score": 1.0, + "content": "according to the complemen-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 313, + 504, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 294, + 326 + ], + "score": 1.0, + "content": "tary condition (15) for all but the last iteration.", + "type": "text" + }, + { + "bbox": [ + 495, + 316, + 504, + 322 + ], + "score": 0.989, + "content": "□", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 107, + 338, + 228, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 336, + 229, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 229, + 351 + ], + "score": 1.0, + "content": "B.2 PROOF OF LEMMA 4.1", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 105, + 358, + 505, + 381 + ], + "lines": [ + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 332, + 371 + ], + "score": 1.0, + "content": "Proof. Without loss of generality, we analyze the case", + "type": "text" + }, + { + "bbox": [ + 332, + 359, + 382, + 370 + ], + "score": 0.92, + "content": "0 \\leq k < q _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "for ease of notation. We first", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 369, + 396, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 396, + 383 + ], + "score": 1.0, + "content": "focus on Option II. The proof for Option I follows the similar argument.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 108, + 381, + 271, + 392 + ], + "lines": [ + { + "bbox": [ + 106, + 378, + 270, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 195, + 395 + ], + "score": 1.0, + "content": "Option II: Define for", + "type": "text" + }, + { + "bbox": [ + 196, + 381, + 221, + 390 + ], + "score": 0.9, + "content": "k = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 378, + 239, + 395 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 239, + 380, + 270, + 393 + ], + "score": 0.91, + "content": "i \\in [ s _ { 2 } ^ { \\prime } ]", + "type": "inline_equation" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 242, + 397, + 367, + 415 + ], + "lines": [ + { + "bbox": [ + 242, + 397, + 367, + 415 + ], + "spans": [ + { + "bbox": [ + 242, + 397, + 367, + 415 + ], + "score": 0.91, + "content": "\\mathbf { B } _ { i } ^ { 0 } \\overset { \\mathrm { d e f } } { = } \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { 0 } ) - \\nabla ^ { 2 } F ( \\mathbf { x } ^ { 0 } ) ,", + "type": "interline_equation", + "image_path": "5b59805a4af053ee4fb95c5b42b27008a7e356ff92846c332e497cfc3ff31f58.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 242, + 397, + 367, + 415 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 418, + 240, + 431 + ], + "lines": [ + { + "bbox": [ + 105, + 416, + 239, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 164, + 433 + ], + "score": 1.0, + "content": "and define for", + "type": "text" + }, + { + "bbox": [ + 164, + 419, + 189, + 430 + ], + "score": 0.9, + "content": "k \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 416, + 207, + 433 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 208, + 419, + 239, + 431 + ], + "score": 0.87, + "content": "i \\in [ s _ { 2 } ]", + "type": "inline_equation" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 176, + 436, + 434, + 453 + ], + "lines": [ + { + "bbox": [ + 176, + 436, + 434, + 453 + ], + "spans": [ + { + "bbox": [ + 176, + 436, + 434, + 453 + ], + "score": 0.9, + "content": "\\mathbf { B } _ { i } ^ { k } \\stackrel { \\mathrm { d e f } } { = } \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) - \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k - 1 } ) - \\big ( \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) - \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k - 1 } ) \\big ) .", + "type": "interline_equation", + "image_path": "1289057aa766bac4d1c804c34e177cedc7345fe754faf9636136f226ac77e204.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 176, + 436, + 434, + 453 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 374, + 469 + ], + "lines": [ + { + "bbox": [ + 107, + 454, + 370, + 471 + ], + "spans": [ + { + "bbox": [ + 107, + 456, + 131, + 469 + ], + "score": 0.92, + "content": "\\{ \\mathbf { B } _ { i } ^ { k } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 454, + 341, + 471 + ], + "score": 1.0, + "content": "is a martingale difference sequence. We have for all", + "type": "text" + }, + { + "bbox": [ + 341, + 457, + 348, + 467 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 454, + 366, + 471 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 366, + 458, + 370, + 467 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 274, + 473, + 336, + 488 + ], + "lines": [ + { + "bbox": [ + 274, + 473, + 336, + 488 + ], + "spans": [ + { + "bbox": [ + 274, + 473, + 336, + 488 + ], + "score": 0.91, + "content": "\\mathbb { E } [ \\mathbf { B } _ { i } ^ { k } | \\mathbf { x } ^ { k } ] = 0 .", + "type": "interline_equation", + "image_path": "1ab1248d3ac8b56df88bfe2bdb5750638f45686b37cd3daf9ffb75238c215762.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 274, + 473, + 336, + 488 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 492, + 316, + 504 + ], + "lines": [ + { + "bbox": [ + 106, + 492, + 316, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 252, + 505 + ], + "score": 1.0, + "content": "Besides, we use Assumption 2.2 for", + "type": "text" + }, + { + "bbox": [ + 252, + 493, + 277, + 502 + ], + "score": 0.9, + "content": "k = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 492, + 316, + 505 + ], + "score": 1.0, + "content": "to bound", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 215, + 507, + 395, + 523 + ], + "lines": [ + { + "bbox": [ + 215, + 507, + 395, + 523 + ], + "spans": [ + { + "bbox": [ + 215, + 507, + 395, + 523 + ], + "score": 0.92, + "content": "\\| \\mathbf { B } _ { i } ^ { 0 } \\| \\leq \\| \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { 0 } ) \\| + \\| \\nabla ^ { 2 } F ( \\mathbf { x } ^ { 0 } ) \\| = 2 L _ { 1 } ,", + "type": "interline_equation", + "image_path": "a96ab3c9c32ec3a722ba047b878da8bddd46e3b9d866c8b308c6d60bf12ac7a8.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 215, + 507, + 395, + 523 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 526, + 283, + 538 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 283, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 219, + 539 + ], + "score": 1.0, + "content": "and use Assumption 2.3 for", + "type": "text" + }, + { + "bbox": [ + 219, + 527, + 244, + 538 + ], + "score": 0.92, + "content": "k \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 525, + 283, + 539 + ], + "score": 1.0, + "content": "to bound", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 142, + 543, + 458, + 559 + ], + "lines": [ + { + "bbox": [ + 142, + 543, + 458, + 559 + ], + "spans": [ + { + "bbox": [ + 142, + 543, + 458, + 559 + ], + "score": 0.91, + "content": "\\| \\mathbf { B } _ { i } ^ { k } \\| \\leq \\| \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) - \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k - 1 } ) \\| + \\| \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) - \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k - 1 } ) \\| \\leq 2 \\sqrt { \\epsilon L _ { 2 } } .", + "type": "interline_equation", + "image_path": "b76551264722383e915a5aec6eda7153275e6f93c5b9292ca46b4842c46f24cc.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 142, + 543, + 458, + 559 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 564, + 261, + 576 + ], + "lines": [ + { + "bbox": [ + 106, + 563, + 262, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 208, + 577 + ], + "score": 1.0, + "content": "From the construction of", + "type": "text" + }, + { + "bbox": [ + 208, + 564, + 222, + 575 + ], + "score": 0.89, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + }, + { + 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393, + 618.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 621, + 493, + 634 + ], + "lines": [ + { + "bbox": [ + 106, + 620, + 495, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 426, + 636 + ], + "score": 1.0, + "content": "Thus using the matrix Azuma’s Inequality in Theorem 7.1 of (Tropp, 2012) and", + "type": "text" + }, + { + "bbox": [ + 426, + 622, + 455, + 633 + ], + "score": 0.92, + "content": "k \\leq p _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 620, + 495, + 636 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "interline_equation", + "bbox": [ + 138, + 637, + 472, + 700 + ], + "lines": [ + { + "bbox": [ + 138, + 637, + 472, + 700 + ], + "spans": [ + { + "bbox": [ + 138, + 637, + 472, + 700 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { P r \\{ \\| \\mathbf H ^ { k } - \\nabla ^ { 2 } F ( { \\mathbf x } ^ { k } ) \\| \\ge t \\} \\le d \\cdot \\exp \\{ - \\frac { t ^ { 2 } / 8 } { \\sum _ { i = 1 } ^ { s _ { 2 } ^ { \\prime } } 4 L _ { 1 } ^ { 2 } / s _ { 2 } ^ { \\prime 2 } + \\sum _ { j = 1 } ^ { k } \\sum _ { i = 1 } ^ { s _ { 2 } } 4 \\epsilon L _ { 2 } / s _ { 2 } ^ { 2 } } \\} } \\\\ & { \\qquad \\le d \\cdot \\exp \\{ - \\frac { t ^ { 2 } / 8 } { 4 L _ { 1 } ^ { 2 } / s _ { 2 } ^ { \\prime } + 4 p _ { 2 } \\epsilon L _ { 2 } / s _ { 2 } } \\} . } \\end{array}", + "type": "interline_equation", + "image_path": "f24e6e87a452eaf0ec8450b5212fb193ef6015d4900c4dabd7c2aca6fe52901e.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 138, + 637, + 472, + 658.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 138, + 658.0, + 472, + 679.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 138, + 679.0, + 472, + 700.0 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 702, + 201, + 713 + ], + "lines": [ + { + "bbox": [ 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+ 152 + ], + "score": 0.94, + "content": "\\| \\mathbf { g } ^ { k } + \\mathbf { H } ^ { k } \\mathbf { h } ^ { k } \\| = \\frac { L _ { 2 } \\lambda ^ { k } } { 2 } \\cdot \\| \\mathbf { h } ^ { k } \\| \\leq 2 \\epsilon .", + "type": "interline_equation", + "image_path": "fe5176f9e4fc73913bf17cea59355d763b3609c4969c2274a129d80ca54486ee.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 230, + 126, + 381, + 152 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 156, + 506, + 182 + ], + "lines": [ + { + "bbox": [ + 105, + 155, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 178, + 172 + ], + "score": 1.0, + "content": "The assumptions", + "type": "text" + }, + { + "bbox": [ + 178, + 157, + 231, + 170 + ], + "score": 0.92, + "content": "\\lVert \\nabla _ { k } \\rVert \\leq \\epsilon / 6", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 155, + 251, + 172 + ], + "score": 1.0, + "content": "and", + 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+ 324, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 246, + 265 + ], + "score": 1.0, + "content": "Combining these two results gives", + "type": "text" + }, + { + "bbox": [ + 246, + 251, + 324, + 264 + ], + "score": 0.91, + "content": "\\| \\nabla F ( \\mathbf { x } ^ { k + 1 } ) \\| \\leq 3 \\epsilon", + "type": "inline_equation" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 106, + 250, + 324, + 265 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 263, + 489, + 276 + ], + "lines": [ + { + "bbox": [ + 106, + 261, + 490, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 236, + 277 + ], + "score": 1.0, + "content": "Besides, using Assumption 2.3,", + "type": "text" + }, + { + "bbox": [ + 236, + 263, + 306, + 276 + ], + "score": 0.9, + "content": "\\| \\nabla _ { k } ^ { 2 } \\| \\le \\sqrt { \\epsilon L _ { 2 } } / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 261, + 490, + 277 + ], + "score": 1.0, + "content": ", and (14), we derive the Hessian lower bound", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 106, + 261, + 490, + 277 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 129, + 280, + 471, + 297 + ], + "lines": [ + { + "bbox": [ + 129, + 280, + 471, + 297 + ], + "spans": [ + { + "bbox": [ + 129, + 280, + 471, + 297 + ], + "score": 0.89, + "content": "\\nabla ^ { 2 } F ( \\mathbf { x } ^ { k + 1 } ) \\approx \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) - L _ { 2 } \\cdot \\Vert \\mathbf { h } ^ { k } \\Vert \\mathbf { I } \\approx \\mathbf { H } ^ { k } - \\sqrt { \\epsilon L _ { 2 } } / 3 \\mathbf { I } - L _ { 2 } \\Vert \\mathbf { h } ^ { k } \\Vert \\mathbf { I } \\approx - \\sqrt { 1 2 \\epsilon L _ { 2 } } \\mathbf { I } .", + "type": "interline_equation", + "image_path": "e36cc1c7ae4617bfd606174d203031bbdf0f504a87031e63a7bc9653d9b34b71.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 129, + 280, + 471, + 297 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 302, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 300, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 134, + 316 + ], + "score": 1.0, + "content": "Hence", + "type": "text" + }, + { + "bbox": [ + 134, + 301, + 156, + 313 + ], + "score": 0.9, + "content": "\\mathbf { x } ^ { k + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 300, + 173, + 316 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 174, + 303, + 189, + 313 + ], + "score": 0.78, + "content": "1 2 \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 300, + 347, + 316 + ], + "score": 1.0, + "content": "-stationary point. Additionally, we have", + "type": "text" + }, + { + "bbox": [ + 347, + 302, + 388, + 315 + ], + "score": 0.93, + "content": "\\| \\mathbf { h } ^ { k } \\| = r", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 300, + 506, + 316 + ], + "score": 1.0, + "content": "according to the complemen-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 313, + 504, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 294, + 326 + ], + "score": 1.0, + "content": "tary condition (15) for all but the last iteration.", + "type": "text" + }, + { + "bbox": [ + 495, + 316, + 504, + 322 + ], + "score": 0.989, + "content": "□", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 300, + 506, + 326 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 338, + 228, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 336, + 229, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 229, + 351 + ], + "score": 1.0, + "content": "B.2 PROOF OF LEMMA 4.1", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 105, + 358, + 505, + 381 + ], + "lines": [ + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 332, + 371 + ], + "score": 1.0, + "content": "Proof. Without loss of generality, we analyze the case", + "type": "text" + }, + { + "bbox": [ + 332, + 359, + 382, + 370 + ], + "score": 0.92, + "content": "0 \\leq k < q _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "for ease of notation. We first", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 369, + 396, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 396, + 383 + ], + "score": 1.0, + "content": "focus on Option II. The proof for Option I follows the similar argument.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 358, + 505, + 383 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 381, + 271, + 392 + ], + "lines": [ + { + "bbox": [ + 106, + 378, + 270, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 195, + 395 + ], + "score": 1.0, + "content": "Option II: Define for", + "type": "text" + }, + { + "bbox": [ + 196, + 381, + 221, + 390 + ], + "score": 0.9, + "content": "k = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 378, + 239, + 395 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 239, + 380, + 270, + 393 + ], + "score": 0.91, + "content": "i \\in [ s _ { 2 } ^ { \\prime } ]", + "type": "inline_equation" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 106, + 378, + 270, + 395 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 242, + 397, + 367, + 415 + ], + "lines": [ + { + "bbox": [ + 242, + 397, + 367, + 415 + ], + "spans": [ + { + "bbox": [ + 242, + 397, + 367, + 415 + ], + "score": 0.91, + "content": "\\mathbf { B } _ { i } ^ { 0 } \\overset { \\mathrm { d e f } } { = } \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { 0 } ) - \\nabla ^ { 2 } F ( \\mathbf { x } ^ { 0 } ) ,", + "type": "interline_equation", + "image_path": "5b59805a4af053ee4fb95c5b42b27008a7e356ff92846c332e497cfc3ff31f58.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 242, + 397, + 367, + 415 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 418, + 240, + 431 + ], + "lines": [ + { + "bbox": [ + 105, + 416, + 239, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 164, + 433 + ], + "score": 1.0, + "content": "and define for", + "type": "text" + }, + { + "bbox": [ + 164, + 419, + 189, + 430 + ], + "score": 0.9, + "content": "k \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 416, + 207, + 433 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 208, + 419, + 239, + 431 + ], + "score": 0.87, + "content": "i \\in [ s _ { 2 } ]", + "type": "inline_equation" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 416, + 239, + 433 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 176, + 436, + 434, + 453 + ], + "lines": [ + { + "bbox": [ + 176, + 436, + 434, + 453 + ], + "spans": [ + { + "bbox": [ + 176, + 436, + 434, + 453 + ], + "score": 0.9, + "content": "\\mathbf { B } _ { i } ^ { k } \\stackrel { \\mathrm { d e f } } { = } \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) - \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k - 1 } ) - \\big ( \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) - \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k - 1 } ) \\big ) .", + "type": "interline_equation", + "image_path": "1289057aa766bac4d1c804c34e177cedc7345fe754faf9636136f226ac77e204.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 176, + 436, + 434, + 453 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 374, + 469 + ], + "lines": [ + { + "bbox": [ + 107, + 454, + 370, + 471 + ], + "spans": [ + { + "bbox": [ + 107, + 456, + 131, + 469 + ], + "score": 0.92, + "content": "\\{ \\mathbf { B } _ { i } ^ { k } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 454, + 341, + 471 + ], + "score": 1.0, + "content": "is a martingale difference sequence. We have for all", + "type": "text" + }, + { + "bbox": [ + 341, + 457, + 348, + 467 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 454, + 366, + 471 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 366, + 458, + 370, + 467 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 107, + 454, + 370, + 471 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 274, + 473, + 336, + 488 + ], + "lines": [ + { + "bbox": [ + 274, + 473, + 336, + 488 + ], + "spans": [ + { + "bbox": [ + 274, + 473, + 336, + 488 + ], + "score": 0.91, + "content": "\\mathbb { E } [ \\mathbf { B } _ { i } ^ { k } | \\mathbf { x } ^ { k } ] = 0 .", + "type": "interline_equation", + "image_path": "1ab1248d3ac8b56df88bfe2bdb5750638f45686b37cd3daf9ffb75238c215762.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 274, + 473, + 336, + 488 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 492, + 316, + 504 + ], + "lines": [ + { + "bbox": [ + 106, + 492, + 316, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 252, + 505 + ], + "score": 1.0, + "content": "Besides, we use Assumption 2.2 for", + "type": "text" + }, + { + "bbox": [ + 252, + 493, + 277, + 502 + ], + "score": 0.9, + "content": "k = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 492, + 316, + 505 + ], + "score": 1.0, + "content": "to bound", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 106, + 492, + 316, + 505 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 215, + 507, + 395, + 523 + ], + "lines": [ + { + "bbox": [ + 215, + 507, + 395, + 523 + ], + "spans": [ + { + "bbox": [ + 215, + 507, + 395, + 523 + ], + "score": 0.92, + "content": "\\| \\mathbf { B } _ { i } ^ { 0 } \\| \\leq \\| \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { 0 } ) \\| + \\| \\nabla ^ { 2 } F ( \\mathbf { x } ^ { 0 } ) \\| = 2 L _ { 1 } ,", + "type": "interline_equation", + "image_path": "a96ab3c9c32ec3a722ba047b878da8bddd46e3b9d866c8b308c6d60bf12ac7a8.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 215, + 507, + 395, + 523 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 526, + 283, + 538 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 283, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 219, + 539 + ], + "score": 1.0, + "content": "and use Assumption 2.3 for", + "type": "text" + }, + { + "bbox": [ + 219, + 527, + 244, + 538 + ], + "score": 0.92, + "content": "k \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 525, + 283, + 539 + ], + "score": 1.0, + "content": "to bound", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 525, + 283, + 539 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 142, + 543, + 458, + 559 + ], + "lines": [ + { + "bbox": [ + 142, + 543, + 458, + 559 + ], + "spans": [ + { + "bbox": [ + 142, + 543, + 458, + 559 + ], + "score": 0.91, + "content": "\\| \\mathbf { B } _ { i } ^ { k } \\| \\leq \\| \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) - \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k - 1 } ) \\| + \\| \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) - \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k - 1 } ) \\| \\leq 2 \\sqrt { \\epsilon L _ { 2 } } .", + "type": "interline_equation", + "image_path": "b76551264722383e915a5aec6eda7153275e6f93c5b9292ca46b4842c46f24cc.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 142, + 543, + 458, + 559 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 564, + 261, + 576 + ], + "lines": [ + { + "bbox": [ + 106, + 563, + 262, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 208, + 577 + ], + "score": 1.0, + "content": "From the construction of", + "type": "text" + }, + { + "bbox": [ + 208, + 564, + 222, + 575 + ], + "score": 0.89, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 563, + 262, + 577 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 106, + 563, + 262, + 577 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 581, + 393, + 618 + ], + "lines": [ + { + "bbox": [ + 218, + 581, + 393, + 618 + ], + "spans": [ + { + "bbox": [ + 218, + 581, + 393, + 618 + ], + "score": 0.94, + "content": "\\mathbf { H } ^ { k } - \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) = \\sum _ { i = 1 } ^ { s _ { 2 } ^ { \\prime } } { \\frac { \\mathbf { B } _ { i } ^ { 0 } } { s _ { 2 } ^ { \\prime } } } + \\sum _ { j = 1 } ^ { k } \\sum _ { i = 1 } ^ { s _ { 2 } } { \\frac { \\mathbf { B } _ { i } ^ { j } } { s _ { 2 } } } .", + "type": "interline_equation", + "image_path": "31737fe57bead9ab4a5a9f1ab377ff3604127e5e0167ea5d6d85fd8f50827453.jpg" + } + ] + } + ], + "index": 28.5, + "virtual_lines": [ + { + "bbox": [ + 218, + 581, + 393, + 599.5 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 218, + 599.5, + 393, + 618.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 621, + 493, + 634 + ], + "lines": [ + { + "bbox": [ + 106, + 620, + 495, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 426, + 636 + ], + "score": 1.0, + "content": "Thus using the matrix Azuma’s Inequality in Theorem 7.1 of (Tropp, 2012) and", + "type": "text" + }, + { + "bbox": [ + 426, + 622, + 455, + 633 + ], + "score": 0.92, + "content": "k \\leq p _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 620, + 495, + 636 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 106, + 620, + 495, + 636 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 138, + 637, + 472, + 700 + ], + "lines": [ + { + "bbox": [ + 138, + 637, + 472, + 700 + ], + "spans": [ + { + "bbox": [ + 138, + 637, + 472, + 700 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { P r \\{ \\| \\mathbf H ^ { k } - \\nabla ^ { 2 } F ( { \\mathbf x } ^ { k } ) \\| \\ge t \\} \\le d \\cdot \\exp \\{ - \\frac { t ^ { 2 } / 8 } { \\sum _ { i = 1 } ^ { s _ { 2 } ^ { \\prime } } 4 L _ { 1 } ^ { 2 } / s _ { 2 } ^ { \\prime 2 } + \\sum _ { j = 1 } ^ { k } \\sum _ { i = 1 } ^ { s _ { 2 } } 4 \\epsilon L _ { 2 } / s _ { 2 } ^ { 2 } } \\} } \\\\ & { \\qquad \\le d \\cdot \\exp \\{ - \\frac { t ^ { 2 } / 8 } { 4 L _ { 1 } ^ { 2 } / s _ { 2 } ^ { \\prime } + 4 p _ { 2 } \\epsilon L _ { 2 } / s _ { 2 } } \\} . } \\end{array}", + "type": "interline_equation", + "image_path": "f24e6e87a452eaf0ec8450b5212fb193ef6015d4900c4dabd7c2aca6fe52901e.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 138, + 637, + 472, + 658.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 138, + 658.0, + 472, + 679.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 138, + 679.0, + 472, + 700.0 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 702, + 201, + 713 + ], + "lines": [ + { + "bbox": [ + 106, + 701, + 201, + 716 + ], + "spans": [ + { + "bbox": [ + 106, + 701, + 201, + 716 + ], + "score": 1.0, + "content": "Consequently, we have", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 106, + 701, + 201, + 716 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 718, + 397, + 734 + ], + "lines": [ + { + "bbox": [ + 203, + 718, + 397, + 734 + ], + "spans": [ + { + "bbox": [ + 203, + 718, + 397, + 734 + ], + "score": 0.9, + "content": "P r \\{ \\| \\mathbf { H } ^ { k } - \\nabla ^ { 2 } F ( \\mathbf { x } ^ { k } ) \\| \\leq \\sqrt { \\epsilon L _ { 2 } } \\} \\geq 1 - \\delta / K _ { 0 } .", + "type": "interline_equation", + "image_path": "0aadfd3a3d67afea7174dde94c4e59b87792f85f47f57d09eb4924bbfb771bad.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 203, + 718, + 397, + 734 + ], + "spans": [], + "index": 35 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 103, + 81, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 148, + 97 + ], + "score": 1.0, + "content": "by taking √", + "type": "text" + }, + { + "bbox": [ + 148, + 82, + 192, + 94 + ], + "score": 0.89, + "content": "t = \\sqrt { \\epsilon L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 79, + 196, + 97 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 196, + 82, + 322, + 95 + ], + "score": 0.52, + "content": "s _ { 2 } ^ { \\prime } = 1 6 L _ { 1 } ^ { 2 } / ( \\epsilon L _ { 2 } ) \\log ( d K _ { 0 } / \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 79, + 326, + 97 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 326, + 81, + 459, + 95 + ], + "score": 0.8, + "content": "s _ { 2 } = 3 2 L _ { 1 } / ( \\sqrt { \\epsilon L _ { 2 } } ) \\log ( d K _ { 0 } / \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 79, + 481, + 97 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 482, + 83, + 505, + 95 + ], + "score": 0.83, + "content": "p _ { 2 } =", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 92, + 165, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 160, + 107 + ], + "score": 0.9, + "content": "L _ { 1 } / ( 2 \\sqrt { \\epsilon L _ { 2 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 92, + 165, + 108 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 105, + 110, + 505, + 135 + ], + "lines": [ + { + "bbox": [ + 106, + 111, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 310, + 124 + ], + "score": 1.0, + "content": "Option I: The proof is similar to the one of Option", + "type": "text" + }, + { + "bbox": [ + 310, + 111, + 318, + 121 + ], + "score": 0.42, + "content": "\\mathrm { I I }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 111, + 408, + 124 + ], + "score": 1.0, + "content": "except that we replace", + "type": "text" + }, + { + "bbox": [ + 409, + 111, + 423, + 124 + ], + "score": 0.91, + "content": "\\mathbf { B } _ { i } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 111, + 505, + 124 + ], + "score": 1.0, + "content": "with zero matrix. In", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 122, + 306, + 135 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 306, + 135 + ], + "score": 1.0, + "content": "such case, the matrix Azuma’s Inequality implies", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "interline_equation", + "bbox": [ + 118, + 138, + 492, + 171 + ], + "lines": [ + { + "bbox": [ + 118, + 138, + 492, + 171 + ], + "spans": [ + { + "bbox": [ + 118, + 138, + 492, + 171 + ], + "score": 0.94, + "content": "P r \\{ \\| \\mathbf H ^ { k } - \\nabla ^ { 2 } F ( \\mathbf x ^ { k } ) \\| \\geq t \\} \\leq d \\cdot \\exp \\{ - \\frac { t ^ { 2 } / 8 } { \\sum _ { j = 1 } ^ { k } \\sum _ { i = 1 } ^ { s _ { 2 } } 4 \\epsilon L _ { 2 } / s _ { 2 } ^ { 2 } } \\} \\leq d \\cdot \\exp \\{ - \\frac { t ^ { 2 } / 8 } { 4 p _ { 2 } \\epsilon L _ { 2 } / s _ { 2 } } \\} .", + "type": "interline_equation", + "image_path": "5105acae8fe1efbc1894a40154a10678befebbbcf1d7c774d44a663b7a40f1b8.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 118, + 138, + 492, + 149.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 118, + 149.0, + 492, + 160.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 118, + 160.0, + 492, + 171.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 176, + 451, + 190 + ], + "lines": [ + { + "bbox": [ + 105, + 175, + 452, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 168, + 191 + ], + "score": 1.0, + "content": "Thus by taking", + "type": "text" + }, + { + "bbox": [ + 168, + 176, + 210, + 189 + ], + "score": 0.89, + "content": "t = \\sqrt { \\epsilon L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 175, + 214, + 191 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 214, + 176, + 312, + 190 + ], + "score": 0.88, + "content": "s _ { 2 } = 3 2 \\sqrt { n } \\log ( d K _ { 0 } / \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 175, + 332, + 191 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 333, + 176, + 371, + 189 + ], + "score": 0.93, + "content": "p _ { 2 } = { \\sqrt { n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 175, + 452, + 191 + ], + "score": 1.0, + "content": ", we have the result.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 194, + 506, + 227 + ], + "lines": [ + { + "bbox": [ + 105, + 192, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 419, + 208 + ], + "score": 1.0, + "content": "Amortized Complexity: In option I, the choice of parameters ensures that:", + "type": "text" + }, + { + "bbox": [ + 419, + 194, + 475, + 206 + ], + "score": 0.92, + "content": "s _ { 2 } ^ { \\prime } \\leq p _ { 2 } \\times s _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 192, + 506, + 208 + ], + "score": 1.0, + "content": "and in", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 204, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 147, + 218 + ], + "score": 1.0, + "content": "option II:", + "type": "text" + }, + { + "bbox": [ + 147, + 206, + 199, + 216 + ], + "score": 0.9, + "content": "n \\leq p _ { 2 } \\times s _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 204, + 506, + 218 + ], + "score": 1.0, + "content": ". 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\\nabla f _ { 1 } ( \\mathbf { x } ^ { k - 1 } ) - \\nabla ^ { 2 } f _ { i } ( \\widetilde { \\mathbf { x } } ) ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) \\| } \\\\ & { \\qquad + \\| \\nabla F ( \\mathbf { x } ^ { k } ) - \\nabla F ( \\mathbf { x } ^ { k - 1 } ) - \\nabla ^ { 2 } F ( \\widetilde { \\mathbf { x } } ) ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) \\| . } \\\\ & { \\quad + \\| \\nabla f _ { i } ( \\mathbf { x } ^ { k } ) - \\nabla f _ { \\widetilde { \\mathbf { x } } } ( \\mathbf { x } ^ { k - 1 } ) - \\nabla ^ { 2 } f _ { i } ( \\widetilde { \\mathbf { x } } ) ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) \\| \\arcsin ^ { k } , } \\\\ & { \\| \\nabla f _ { i } ( \\mathbf { x } ^ { k } ) - \\nabla f _ { i } ( \\mathbf { x } ^ { k - 1 } ) - \\nabla ^ { 2 } f _ { j } ( \\widetilde { \\mathbf { x } } ) ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) \\| \\arcsin ^ { k } } \\\\ & { \\quad = \\| \\int _ { 0 } ^ { 1 } \\left[ \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k - 1 } + t ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) ) - \\nabla ^ { 2 } f _ { i } ( \\widetilde { \\mathbf { x } } ) \\right] ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) d t \\| } \\\\ & { \\quad \\leq \\int _ { 0 } ^ { 1 } L _ { 2 } \\bigg \\| \\mathbf { x } ^ { k } + ( 1 - t ) \\mathbf { x } ^ { k - 1 } - \\widetilde { \\mathbf { x } } \\bigg \\| d t \\cdot \\| \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } \\| } \\\\ & { \\quad \\leq \\int _ { 0 } ^ { 1 } \\big ( L \\| \\mathbf { x } ^ { k } - \\widetilde { \\mathbf { x } } \\| + ( 1 - t ) \\| \\mathbf { x } ^ { k - 1 } - \\widetilde { \\mathbf { x } } \\big \\| d t \\cdot \\| \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } \\| } \\\\ & \\quad \\leq \\int _ { 0 } ^ { 1 } \\big ( L \\| \\mathbf { x } ^ { k } - \\widetilde { \\mathbf { x } } \\end{array}", + "type": "interline_equation", + "image_path": "07dcf33b984b0e393b017fb1bdbd957b6bc029cdd60d7705e31f39932eefb42d.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 107, + 202, + 448, + 257.6666666666667 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 107, + 257.6666666666667, + 448, + 313.33333333333337 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 107, + 313.33333333333337, + 448, + 369.00000000000006 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 369, + 505, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 477, + 381 + ], + "score": 1.0, + "content": "where the first inequality follows from Assumption 2.3 and the last inequality holds because", + "type": "text" + }, + { + "bbox": [ + 477, + 369, + 505, + 381 + ], + "score": 0.85, + "content": "\\| \\mathbf { x } ^ { k } - ", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 378, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 142, + 392 + ], + "score": 0.89, + "content": "\\tilde { \\mathbf { x } } \\| \\leq k r", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 378, + 161, + 394 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 161, + 380, + 235, + 392 + ], + "score": 0.89, + "content": "\\| \\mathbf { x } ^ { k - 1 } - \\bar { \\tilde { \\mathbf { x } } } \\| \\leq k r", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 378, + 265, + 394 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 266, + 383, + 272, + 390 + ], + "score": 0.76, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 378, + 453, + 394 + ], + "score": 1.0, + "content": "is the trust region radius. Similarly, we have", + "type": "text" + }, + { + "bbox": [ + 453, + 381, + 505, + 393 + ], + "score": 0.88, + "content": "\\| \\nabla F ( \\mathbf { x } ^ { k } ) -", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 390, + 367, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 295, + 404 + ], + "score": 0.85, + "content": "\\begin{array} { r } { \\nabla F ( { \\mathbf x } ^ { k - 1 } ) - \\nabla ^ { 2 } F ( \\tilde { { \\mathbf x } } ) ( \\dot { \\mathbf x } ^ { k } - { \\mathbf x } ^ { k - 1 } ) \\| \\leq L _ { 2 } k r ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 390, + 367, + 405 + ], + "score": 1.0, + "content": ". Thus, we bound", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 255, + 405, + 356, + 420 + ], + "lines": [ + { + "bbox": [ + 255, + 405, + 356, + 420 + ], + "spans": [ + { + "bbox": [ + 255, + 405, + 356, + 420 + ], + "score": 0.9, + "content": "\\| \\mathbf { b } _ { i } ^ { k } \\| \\leq 2 L _ { 2 } k r ^ { 2 } \\leq 2 p _ { 1 } \\epsilon", + "type": "interline_equation", + "image_path": "181c05d64df01cda8dc29b06a074116dec1fc8866c54aaa09648aedfa82ae906.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 255, + 405, + 356, + 420 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 421, + 258, + 433 + ], + "lines": [ + { + "bbox": [ + 106, + 420, + 259, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 208, + 434 + ], + "score": 1.0, + "content": "From the construction of", + "type": "text" + }, + { + "bbox": [ + 208, + 421, + 219, + 433 + ], + "score": 0.89, + "content": "\\mathbf { g } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 420, + 259, + 434 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 434, + 366, + 469 + ], + "lines": [ + { + "bbox": [ + 244, + 434, + 366, + 469 + ], + "spans": [ + { + "bbox": [ + 244, + 434, + 366, + 469 + ], + "score": 0.94, + "content": "\\mathbf { g } ^ { k } - \\nabla F ( \\mathbf { x } ^ { k } ) = \\sum _ { j = 1 } ^ { k } \\sum _ { i = 1 } ^ { s _ { 1 } } \\frac { \\mathbf { b } _ { i } ^ { j } } { s _ { 1 } } .", + "type": "interline_equation", + "image_path": "0d859d459e644d295ac6fee6bd28d3a95a5f459bca8bfbe286fb71b07ab7df7f.jpg" + } + ] + } + ], + "index": 16.5, + "virtual_lines": [ + { + "bbox": [ + 244, + 434, + 366, + 451.5 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 244, + 451.5, + 366, + 469.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 470, + 318, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 318, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 137, + 484 + ], + "score": 1.0, + "content": "We use", + "type": "text" + }, + { + "bbox": [ + 137, + 471, + 167, + 482 + ], + "score": 0.92, + "content": "k \\leq p _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 469, + 318, + 484 + ], + "score": 1.0, + "content": "and the Azuma’s inequality to bound", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 483, + 410, + 533 + ], + "lines": [ + { + "bbox": [ + 200, + 483, + 410, + 533 + ], + "spans": [ + { + "bbox": [ + 200, + 483, + 410, + 533 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\quad P r \\{ | | \\mathbf { g } ^ { k } - \\nabla F ( { \\mathbf { x } } ^ { k } ) | | \\geq t \\} } \\\\ & { \\leq \\exp \\{ - \\frac { t ^ { 2 } / 8 } { \\sum _ { j = 1 } ^ { k } \\sum _ { i = 1 } ^ { s _ { 1 } } \\frac { 4 p _ { 1 } ^ { 2 } \\epsilon ^ { 2 } } { s _ { 1 } ^ { 2 } } } \\} \\leq \\exp \\{ - \\frac { t ^ { 2 } / 8 } { 4 \\epsilon ^ { 2 } p _ { 1 } ^ { 3 } / s _ { 1 } } \\} . } \\end{array}", + "type": "interline_equation", + "image_path": "b5064eab451bb9795910961730cc6cfd5f285422035967e6fc64ebe9dce20a50.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 200, + 483, + 410, + 499.6666666666667 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 200, + 499.6666666666667, + 410, + 516.3333333333334 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 200, + 516.3333333333334, + 410, + 533.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 536, + 507, + 576 + ], + "lines": [ + { + "bbox": [ + 106, + 532, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 174, + 550 + ], + "score": 1.0, + "content": "Thus, by taking", + "type": "text" + }, + { + "bbox": [ + 174, + 536, + 208, + 549 + ], + "score": 0.92, + "content": "t = \\epsilon / 6", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 532, + 227, + 550 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 228, + 537, + 269, + 547 + ], + "score": 0.86, + "content": "c = 1 1 5 2", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 532, + 311, + 550 + ], + "score": 1.0, + "content": ", we need", + "type": "text" + }, + { + "bbox": [ + 311, + 535, + 370, + 551 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { s _ { 1 } } { p _ { 1 } ^ { 3 } } \\geq c \\log \\frac { K _ { 0 } } { \\delta } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 532, + 448, + 550 + ], + "score": 1.0, + "content": ". Further we want", + "type": "text" + }, + { + "bbox": [ + 448, + 536, + 505, + 549 + ], + "score": 0.92, + "content": "s _ { 1 } p _ { 1 } \\simeq \\mathcal { O } ( n )", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 546, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 104, + 546, + 183, + 568 + ], + "score": 1.0, + "content": "and hence we take", + "type": "text" + }, + { + "bbox": [ + 184, + 551, + 228, + 564 + ], + "score": 0.9, + "content": "p _ { 1 } = n ^ { 0 . 2 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 546, + 247, + 568 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 248, + 550, + 325, + 565 + ], + "score": 0.91, + "content": "\\begin{array} { r } { s _ { 1 } = n ^ { 0 . 7 5 } c \\log { \\frac { \\dot { K _ { 0 } } } { \\delta } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 546, + 506, + 568 + ], + "score": 1.0, + "content": ". The amortized stochastic first-order oracle", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 563, + 231, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 212, + 576 + ], + "score": 1.0, + "content": "complexity is bounded by", + "type": "text" + }, + { + "bbox": [ + 212, + 564, + 227, + 574 + ], + "score": 0.87, + "content": "2 s _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 563, + 231, + 576 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "title", + "bbox": [ + 106, + 590, + 305, + 603 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 306, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 306, + 604 + ], + "score": 1.0, + "content": "C ANALYSIS OF METAALGORITHM 7", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 613, + 505, + 637 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 232, + 627 + ], + "score": 1.0, + "content": "We first show that INEXACTTR", + "type": "text" + }, + { + "bbox": [ + 233, + 615, + 252, + 626 + ], + "score": 0.65, + "content": "\\mathbf { W E A K }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 613, + 287, + 627 + ], + "score": 1.0, + "content": "finds an", + "type": "text" + }, + { + "bbox": [ + 287, + 614, + 308, + 626 + ], + "score": 0.88, + "content": "\\mathcal { O } ( \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 613, + 347, + 627 + ], + "score": 1.0, + "content": "-SOSP in", + "type": "text" + }, + { + "bbox": [ + 347, + 614, + 388, + 626 + ], + "score": 0.92, + "content": "\\mathcal { O } ( 1 / \\epsilon ^ { 1 . 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 613, + 506, + 627 + ], + "score": 1.0, + "content": "iterations with probability at", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 625, + 278, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 625, + 127, + 638 + ], + "score": 1.0, + "content": "least", + "type": "text" + }, + { + "bbox": [ + 127, + 625, + 144, + 637 + ], + "score": 0.52, + "content": "2 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 625, + 278, + 638 + ], + "score": 1.0, + "content": "as stated in the following lemma.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 106, + 638, + 506, + 678 + ], + "lines": [ + { + "bbox": [ + 105, + 637, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 237, + 652 + ], + "score": 1.0, + "content": "Lemma C.1. Consider problem", + "type": "text" + }, + { + "bbox": [ + 237, + 639, + 250, + 650 + ], + "score": 0.33, + "content": "( ~ )", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 637, + 506, + 652 + ], + "score": 1.0, + "content": "under Assumptions 2.1-2.3. Suppose that the differential estima-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 648, + 507, + 667 + ], + "spans": [ + { + "bbox": [ + 104, + 648, + 124, + 667 + ], + "score": 1.0, + "content": "tors", + "type": "text" + }, + { + "bbox": [ + 125, + 650, + 137, + 663 + ], + "score": 0.88, + "content": "\\mathbf { g } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 648, + 155, + 667 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 156, + 650, + 171, + 661 + ], + "score": 0.88, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 648, + 336, + 667 + ], + "score": 1.0, + "content": "satisfy Eqn. (9) with probability at least", + "type": "text" + }, + { + "bbox": [ + 336, + 650, + 375, + 664 + ], + "score": 0.92, + "content": "\\begin{array} { r } { ( 1 - \\frac { \\zeta } { 4 K } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 648, + 469, + 667 + ], + "score": 1.0, + "content": ". 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By setting", + "type": "text" + }, + { + "bbox": [ + 307, + 709, + 344, + 721 + ], + "score": 0.8, + "content": "\\zeta = 1 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 707, + 348, + 723 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 349, + 708, + 401, + 722 + ], + "score": 0.87, + "content": "r = \\sqrt { \\epsilon / L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 707, + 424, + 723 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 424, + 708, + 502, + 722 + ], + "score": 0.91, + "content": "K = 4 \\sqrt { L _ { 2 } } \\Delta / \\epsilon ^ { 1 . 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 707, + 506, + 723 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 719, + 338, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 159, + 733 + ], + "score": 1.0, + "content": "INEXACTTR", + "type": "text" + }, + { + "bbox": [ + 159, + 721, + 179, + 732 + ], + "score": 0.3, + "content": "\\mathbf { \\partial } \\cdot \\mathbf { W } \\mathbf { E } \\mathbf { A } \\mathbf { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 719, + 317, + 733 + ], + "score": 1.0, + "content": "outputs a 500\u000f-SOSP w.p. at least", + "type": "text" + }, + { + "bbox": [ + 318, + 721, + 333, + 732 + ], + "score": 0.72, + "content": "2 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 719, + 338, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + } + ], + "page_idx": 15, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 309, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "16", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 228, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 229, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 229, + 95 + ], + "score": 1.0, + "content": "B.4 PROOF OF LEMMA 5.1", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 102, + 505, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 101, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 101, + 300, + 116 + ], + "score": 1.0, + "content": "Without loss of generality, we analyze the case", + "type": "text" + }, + { + "bbox": [ + 300, + 103, + 348, + 114 + ], + "score": 0.92, + "content": "0 \\leq k < q _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 101, + 479, + 116 + ], + "score": 1.0, + "content": "for ease of notation. 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\\nabla ^ { 2 } f _ { i } ( \\tilde { \\mathbf { x } } ) ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) } & { } \\\\ { - \\left[ \\nabla F ( \\mathbf { x } ^ { k } ) - \\nabla F ( \\mathbf { x } ^ { k - 1 } ) - \\nabla ^ { 2 } F ( \\tilde { \\mathbf { x } } ) ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) \\right] . } & { } \\end{array}", + "type": "interline_equation", + "image_path": "e52ebd66685c9729a0c3e478a34d1f0d5d56f5fedb103736c94d30b06dbe61d0.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 185, + 124, + 425, + 141.5 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 185, + 141.5, + 425, + 159.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 159, + 333, + 171 + ], + "lines": [ + { + "bbox": [ + 107, + 158, + 333, + 173 + ], + "spans": [ + { + "bbox": [ + 107, + 159, + 129, + 172 + ], + "score": 0.91, + "content": "\\{ \\mathbf { b } _ { i } ^ { k } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 158, + 303, + 173 + ], + "score": 1.0, + "content": "is a martingale difference sequence: for all", + "type": "text" + }, + { + "bbox": [ + 303, + 160, + 310, + 169 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 158, + 327, + 173 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 327, + 160, + 333, + 169 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 107, + 158, + 333, + 173 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 275, + 173, + 335, + 187 + ], + "lines": [ + { + "bbox": [ + 275, + 173, + 335, + 187 + ], + "spans": [ + { + "bbox": [ + 275, + 173, + 335, + 187 + ], + "score": 0.91, + "content": "\\mathbb { E } [ \\mathbf { b } _ { i } ^ { k } | \\mathbf { x } ^ { k } ] = 0 .", + "type": "interline_equation", + "image_path": "f660f0e6f977b918184ec2c064d86ac6b34604eab53e949ab1a4d4fe716f9881.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 275, + 173, + 335, + 187 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 188, + 234, + 201 + ], + "lines": [ + { + "bbox": [ + 105, + 187, + 235, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 142, + 202 + ], + "score": 1.0, + "content": "Besides,", + "type": "text" + }, + { + "bbox": [ + 142, + 188, + 155, + 201 + ], + "score": 0.9, + "content": "\\mathbf { b } _ { i } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 187, + 235, + 202 + ], + "score": 1.0, + "content": "has bounded norm:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 187, + 235, + 202 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 107, + 202, + 448, + 369 + ], + "lines": [ + { + "bbox": [ + 159, + 203, + 448, + 369 + ], + "spans": [ + { + "bbox": [ + 159, + 203, + 448, + 369 + ], + "score": 0.67, + "content": "\\begin{array} { r l } & { \\quad \\| \\mathbf { b } _ { \\star } ^ { k } \\| \\leq \\| \\nabla f _ { i } ( \\mathbf { x } ^ { k } ) - \\nabla f _ { 1 } ( \\mathbf { x } ^ { k - 1 } ) - \\nabla ^ { 2 } f _ { i } ( \\widetilde { \\mathbf { x } } ) ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) \\| } \\\\ & { \\qquad + \\| \\nabla F ( \\mathbf { x } ^ { k } ) - \\nabla F ( \\mathbf { x } ^ { k - 1 } ) - \\nabla ^ { 2 } F ( \\widetilde { \\mathbf { x } } ) ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) \\| . } \\\\ & { \\quad + \\| \\nabla f _ { i } ( \\mathbf { x } ^ { k } ) - \\nabla f _ { \\widetilde { \\mathbf { x } } } ( \\mathbf { x } ^ { k - 1 } ) - \\nabla ^ { 2 } f _ { i } ( \\widetilde { \\mathbf { x } } ) ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) \\| \\arcsin ^ { k } , } \\\\ & { \\| \\nabla f _ { i } ( \\mathbf { x } ^ { k } ) - \\nabla f _ { i } ( \\mathbf { x } ^ { k - 1 } ) - \\nabla ^ { 2 } f _ { j } ( \\widetilde { \\mathbf { x } } ) ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) \\| \\arcsin ^ { k } } \\\\ & { \\quad = \\| \\int _ { 0 } ^ { 1 } \\left[ \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k - 1 } + t ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) ) - \\nabla ^ { 2 } f _ { i } ( \\widetilde { \\mathbf { x } } ) \\right] ( \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } ) d t \\| } \\\\ & { \\quad \\leq \\int _ { 0 } ^ { 1 } L _ { 2 } \\bigg \\| \\mathbf { x } ^ { k } + ( 1 - t ) \\mathbf { x } ^ { k - 1 } - \\widetilde { \\mathbf { x } } \\bigg \\| d t \\cdot \\| \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } \\| } \\\\ & { \\quad \\leq \\int _ { 0 } ^ { 1 } \\big ( L \\| \\mathbf { x } ^ { k } - \\widetilde { \\mathbf { x } } \\| + ( 1 - t ) \\| \\mathbf { x } ^ { k - 1 } - \\widetilde { \\mathbf { x } } \\big \\| d t \\cdot \\| \\mathbf { x } ^ { k } - \\mathbf { x } ^ { k - 1 } \\| } \\\\ & \\quad \\leq \\int _ { 0 } ^ { 1 } \\big ( L \\| \\mathbf { x } ^ { k } - \\widetilde { \\mathbf { x } } \\end{array}", + "type": "interline_equation", + "image_path": "07dcf33b984b0e393b017fb1bdbd957b6bc029cdd60d7705e31f39932eefb42d.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 107, + 202, + 448, + 257.6666666666667 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 107, + 257.6666666666667, + 448, + 313.33333333333337 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 107, + 313.33333333333337, + 448, + 369.00000000000006 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 369, + 505, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 477, + 381 + ], + "score": 1.0, + "content": "where the first inequality follows from Assumption 2.3 and the last inequality holds because", + "type": "text" + }, + { + "bbox": [ + 477, + 369, + 505, + 381 + ], + "score": 0.85, + "content": "\\| \\mathbf { x } ^ { k } - ", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 378, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 142, + 392 + ], + "score": 0.89, + "content": "\\tilde { \\mathbf { x } } \\| \\leq k r", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 378, + 161, + 394 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 161, + 380, + 235, + 392 + ], + "score": 0.89, + "content": "\\| \\mathbf { x } ^ { k - 1 } - \\bar { \\tilde { \\mathbf { x } } } \\| \\leq k r", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 378, + 265, + 394 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 266, + 383, + 272, + 390 + ], + "score": 0.76, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 378, + 453, + 394 + ], + "score": 1.0, + "content": "is the trust region radius. Similarly, we have", + "type": "text" + }, + { + "bbox": [ + 453, + 381, + 505, + 393 + ], + "score": 0.88, + "content": "\\| \\nabla F ( \\mathbf { x } ^ { k } ) -", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 390, + 367, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 295, + 404 + ], + "score": 0.85, + "content": "\\begin{array} { r } { \\nabla F ( { \\mathbf x } ^ { k - 1 } ) - \\nabla ^ { 2 } F ( \\tilde { { \\mathbf x } } ) ( \\dot { \\mathbf x } ^ { k } - { \\mathbf x } ^ { k - 1 } ) \\| \\leq L _ { 2 } k r ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 390, + 367, + 405 + ], + "score": 1.0, + "content": ". Thus, we bound", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 369, + 505, + 405 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 255, + 405, + 356, + 420 + ], + "lines": [ + { + "bbox": [ + 255, + 405, + 356, + 420 + ], + "spans": [ + { + "bbox": [ + 255, + 405, + 356, + 420 + ], + "score": 0.9, + "content": "\\| \\mathbf { b } _ { i } ^ { k } \\| \\leq 2 L _ { 2 } k r ^ { 2 } \\leq 2 p _ { 1 } \\epsilon", + "type": "interline_equation", + "image_path": "181c05d64df01cda8dc29b06a074116dec1fc8866c54aaa09648aedfa82ae906.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 255, + 405, + 356, + 420 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 421, + 258, + 433 + ], + "lines": [ + { + "bbox": [ + 106, + 420, + 259, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 208, + 434 + ], + "score": 1.0, + "content": "From the construction of", + "type": "text" + }, + { + "bbox": [ + 208, + 421, + 219, + 433 + ], + "score": 0.89, + "content": "\\mathbf { g } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 420, + 259, + 434 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 106, + 420, + 259, + 434 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 434, + 366, + 469 + ], + "lines": [ + { + "bbox": [ + 244, + 434, + 366, + 469 + ], + "spans": [ + { + "bbox": [ + 244, + 434, + 366, + 469 + ], + "score": 0.94, + "content": "\\mathbf { g } ^ { k } - \\nabla F ( \\mathbf { x } ^ { k } ) = \\sum _ { j = 1 } ^ { k } \\sum _ { i = 1 } ^ { s _ { 1 } } \\frac { \\mathbf { b } _ { i } ^ { j } } { s _ { 1 } } .", + "type": "interline_equation", + "image_path": "0d859d459e644d295ac6fee6bd28d3a95a5f459bca8bfbe286fb71b07ab7df7f.jpg" + } + ] + } + ], + "index": 16.5, + "virtual_lines": [ + { + "bbox": [ + 244, + 434, + 366, + 451.5 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 244, + 451.5, + 366, + 469.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 470, + 318, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 318, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 137, + 484 + ], + "score": 1.0, + "content": "We use", + "type": "text" + }, + { + "bbox": [ + 137, + 471, + 167, + 482 + ], + "score": 0.92, + "content": "k \\leq p _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 469, + 318, + 484 + ], + "score": 1.0, + "content": "and the Azuma’s inequality to bound", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 469, + 318, + 484 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 483, + 410, + 533 + ], + "lines": [ + { + "bbox": [ + 200, + 483, + 410, + 533 + ], + "spans": [ + { + "bbox": [ + 200, + 483, + 410, + 533 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\quad P r \\{ | | \\mathbf { g } ^ { k } - \\nabla F ( { \\mathbf { x } } ^ { k } ) | | \\geq t \\} } \\\\ & { \\leq \\exp \\{ - \\frac { t ^ { 2 } / 8 } { \\sum _ { j = 1 } ^ { k } \\sum _ { i = 1 } ^ { s _ { 1 } } \\frac { 4 p _ { 1 } ^ { 2 } \\epsilon ^ { 2 } } { s _ { 1 } ^ { 2 } } } \\} \\leq \\exp \\{ - \\frac { t ^ { 2 } / 8 } { 4 \\epsilon ^ { 2 } p _ { 1 } ^ { 3 } / s _ { 1 } } \\} . } \\end{array}", + "type": "interline_equation", + "image_path": "b5064eab451bb9795910961730cc6cfd5f285422035967e6fc64ebe9dce20a50.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 200, + 483, + 410, + 499.6666666666667 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 200, + 499.6666666666667, + 410, + 516.3333333333334 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 200, + 516.3333333333334, + 410, + 533.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 536, + 507, + 576 + ], + "lines": [ + { + "bbox": [ + 106, + 532, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 174, + 550 + ], + "score": 1.0, + "content": "Thus, by taking", + "type": "text" + }, + { + "bbox": [ + 174, + 536, + 208, + 549 + ], + "score": 0.92, + "content": "t = \\epsilon / 6", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 532, + 227, + 550 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 228, + 537, + 269, + 547 + ], + "score": 0.86, + "content": "c = 1 1 5 2", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 532, + 311, + 550 + ], + "score": 1.0, + "content": ", we need", + "type": "text" + }, + { + "bbox": [ + 311, + 535, + 370, + 551 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { s _ { 1 } } { p _ { 1 } ^ { 3 } } \\geq c \\log \\frac { K _ { 0 } } { \\delta } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 532, + 448, + 550 + ], + "score": 1.0, + "content": ". Further we want", + "type": "text" + }, + { + "bbox": [ + 448, + 536, + 505, + 549 + ], + "score": 0.92, + "content": "s _ { 1 } p _ { 1 } \\simeq \\mathcal { O } ( n )", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 546, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 104, + 546, + 183, + 568 + ], + "score": 1.0, + "content": "and hence we take", + "type": "text" + }, + { + "bbox": [ + 184, + 551, + 228, + 564 + ], + "score": 0.9, + "content": "p _ { 1 } = n ^ { 0 . 2 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 546, + 247, + 568 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 248, + 550, + 325, + 565 + ], + "score": 0.91, + "content": "\\begin{array} { r } { s _ { 1 } = n ^ { 0 . 7 5 } c \\log { \\frac { \\dot { K _ { 0 } } } { \\delta } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 546, + 506, + 568 + ], + "score": 1.0, + "content": ". The amortized stochastic first-order oracle", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 563, + 231, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 212, + 576 + ], + "score": 1.0, + "content": "complexity is bounded by", + "type": "text" + }, + { + "bbox": [ + 212, + 564, + 227, + 574 + ], + "score": 0.87, + "content": "2 s _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 563, + 231, + 576 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23, + "bbox_fs": [ + 104, + 532, + 506, + 576 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 590, + 305, + 603 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 306, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 306, + 604 + ], + "score": 1.0, + "content": "C ANALYSIS OF METAALGORITHM 7", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 613, + 505, + 637 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 232, + 627 + ], + "score": 1.0, + "content": "We first show that INEXACTTR", + "type": "text" + }, + { + "bbox": [ + 233, + 615, + 252, + 626 + ], + "score": 0.65, + "content": "\\mathbf { W E A K }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 613, + 287, + 627 + ], + "score": 1.0, + "content": "finds an", + "type": "text" + }, + { + "bbox": [ + 287, + 614, + 308, + 626 + ], + "score": 0.88, + "content": "\\mathcal { O } ( \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 613, + 347, + 627 + ], + "score": 1.0, + "content": "-SOSP in", + "type": "text" + }, + { + "bbox": [ + 347, + 614, + 388, + 626 + ], + "score": 0.92, + "content": "\\mathcal { O } ( 1 / \\epsilon ^ { 1 . 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 613, + 506, + 627 + ], + "score": 1.0, + "content": "iterations with probability at", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 625, + 278, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 625, + 127, + 638 + ], + "score": 1.0, + "content": "least", + "type": "text" + }, + { + "bbox": [ + 127, + 625, + 144, + 637 + ], + "score": 0.52, + "content": "2 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 625, + 278, + 638 + ], + "score": 1.0, + "content": "as stated in the following lemma.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 613, + 506, + 638 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 638, + 506, + 678 + ], + "lines": [ + { + "bbox": [ + 105, + 637, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 237, + 652 + ], + "score": 1.0, + "content": "Lemma C.1. Consider problem", + "type": "text" + }, + { + "bbox": [ + 237, + 639, + 250, + 650 + ], + "score": 0.33, + "content": "( ~ )", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 637, + 506, + 652 + ], + "score": 1.0, + "content": "under Assumptions 2.1-2.3. Suppose that the differential estima-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 648, + 507, + 667 + ], + "spans": [ + { + "bbox": [ + 104, + 648, + 124, + 667 + ], + "score": 1.0, + "content": "tors", + "type": "text" + }, + { + "bbox": [ + 125, + 650, + 137, + 663 + ], + "score": 0.88, + "content": "\\mathbf { g } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 648, + 155, + 667 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 156, + 650, + 171, + 661 + ], + "score": 0.88, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 648, + 336, + 667 + ], + "score": 1.0, + "content": "satisfy Eqn. (9) with probability at least", + "type": "text" + }, + { + "bbox": [ + 336, + 650, + 375, + 664 + ], + "score": 0.92, + "content": "\\begin{array} { r } { ( 1 - \\frac { \\zeta } { 4 K } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 648, + 469, + 667 + ], + "score": 1.0, + "content": ". Besides, suppose that", + "type": "text" + }, + { + "bbox": [ + 470, + 650, + 482, + 661 + ], + "score": 0.87, + "content": "\\tilde { \\mathbf { h } } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 648, + 507, + 667 + ], + "score": 1.0, + "content": "is an", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 663, + 318, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 275, + 680 + ], + "score": 1.0, + "content": "approximate solution to (8) such that w.p.", + "type": "text" + }, + { + "bbox": [ + 276, + 663, + 313, + 678 + ], + "score": 0.91, + "content": "\\begin{array} { r } { ( 1 - \\frac { \\zeta } { 4 K } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 663, + 318, + 680 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 104, + 637, + 507, + 680 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 176, + 679, + 434, + 705 + ], + "lines": [ + { + "bbox": [ + 176, + 679, + 434, + 705 + ], + "spans": [ + { + "bbox": [ + 176, + 679, + 434, + 705 + ], + "score": 0.91, + "content": "\\langle \\mathbf { g } ^ { k } , \\tilde { \\mathbf { h } } ^ { k } \\rangle + \\frac { 1 } { 2 } \\langle \\mathbf { H } ^ { k } \\tilde { \\mathbf { h } } ^ { k } , \\tilde { \\mathbf { h } } ^ { k } \\rangle \\leq \\langle \\mathbf { g } ^ { k } , \\mathbf { h } ^ { k } \\rangle + \\frac { 1 } { 2 } \\langle \\mathbf { H } ^ { k } \\mathbf { h } ^ { k } , \\mathbf { h } ^ { k } \\rangle + \\frac { \\epsilon ^ { 1 . 5 } } { \\sqrt { L _ { 2 } } } ,", + "type": "interline_equation", + "image_path": "2c16ec5c53bf827dae529153d298ea592a3a6e2d9f69bce18185bac35265930a.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 176, + 679, + 434, + 705 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 708, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 104, + 707, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 707, + 133, + 723 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 709, + 147, + 720 + ], + "score": 0.86, + "content": "\\mathbf { h } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 707, + 307, + 723 + ], + "score": 1.0, + "content": "is a global solution to (8). By setting", + "type": "text" + }, + { + "bbox": [ + 307, + 709, + 344, + 721 + ], + "score": 0.8, + "content": "\\zeta = 1 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 707, + 348, + 723 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 349, + 708, + 401, + 722 + ], + "score": 0.87, + "content": "r = \\sqrt { \\epsilon / L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 707, + 424, + 723 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 424, + 708, + 502, + 722 + ], + "score": 0.91, + "content": "K = 4 \\sqrt { L _ { 2 } } \\Delta / \\epsilon ^ { 1 . 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 707, + 506, + 723 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 719, + 338, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 159, + 733 + ], + "score": 1.0, + "content": "INEXACTTR", + "type": "text" + }, + { + "bbox": [ + 159, + 721, + 179, + 732 + ], + "score": 0.3, + "content": "\\mathbf { \\partial } \\cdot \\mathbf { W } \\mathbf { E } \\mathbf { A } \\mathbf { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 719, + 317, + 733 + ], + "score": 1.0, + "content": "outputs a 500\u000f-SOSP w.p. at least", + "type": "text" + }, + { + "bbox": [ + 318, + 721, + 333, + 732 + ], + "score": 0.72, + "content": "2 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 719, + 338, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 104, + 707, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 338, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 78, + 334, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 295, + 98 + ], + "score": 1.0, + "content": "Proof. 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We", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 171, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 119, + 185 + ], + "score": 1.0, + "content": "let", + "type": "text" + }, + { + "bbox": [ + 119, + 172, + 131, + 182 + ], + "score": 0.88, + "content": "\\lambda ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 171, + 373, + 185 + ], + "score": 1.0, + "content": "denote the dual variable corresponding to the global solution", + "type": "text" + }, + { + "bbox": [ + 373, + 172, + 385, + 182 + ], + "score": 0.88, + "content": "\\mathbf { h } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 171, + 506, + 185 + ], + "score": 1.0, + "content": "as defined in Lemma 2.1. We", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 181, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 104, + 181, + 144, + 196 + ], + "score": 1.0, + "content": "note that", + "type": "text" + }, + { + "bbox": [ + 145, + 182, + 157, + 193 + ], + "score": 0.88, + "content": "\\mathbf { h } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 181, + 176, + 196 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 176, + 182, + 188, + 193 + ], + "score": 0.89, + "content": "\\lambda ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 181, + 371, + 196 + ], + "score": 1.0, + "content": "are used only in our analysis. The INEXACT", + "type": "text" + }, + { + "bbox": [ + 371, + 185, + 390, + 195 + ], + "score": 0.35, + "content": "\\mathrm { w e A K }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 181, + 506, + 196 + ], + "score": 1.0, + "content": "algorithm only requires the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 194, + 337, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 193, + 206 + ], + "score": 1.0, + "content": "approximate solution", + "type": "text" + }, + { + "bbox": [ + 194, + 194, + 205, + 204 + ], + "score": 0.87, + "content": "\\tilde { \\mathbf { x } } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 194, + 297, + 206 + ], + "score": 1.0, + "content": "without knowledge of", + "type": "text" + }, + { + "bbox": [ + 297, + 194, + 309, + 204 + ], + "score": 0.88, + "content": "\\mathbf { h } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 194, + 321, + 206 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 321, + 194, + 333, + 204 + ], + "score": 0.89, + "content": "\\lambda ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 194, + 337, + 206 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 210, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 105, + 209, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 304, + 228 + ], + "score": 1.0, + "content": "By the assumption that (9) holds with probability", + "type": "text" + }, + { + "bbox": [ + 304, + 210, + 343, + 226 + ], + "score": 0.92, + "content": "\\textstyle { \\left( { 1 - { \\frac { \\zeta } { 4 K } } } \\right) }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 209, + 410, + 228 + ], + "score": 1.0, + "content": "and the fact that", + "type": "text" + }, + { + "bbox": [ + 410, + 210, + 487, + 225 + ], + "score": 0.93, + "content": "\\| \\tilde { \\mathbf { h } } ^ { k } \\| \\le r = \\sqrt { L _ { 2 } \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 209, + 506, + 228 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 222, + 192, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 147, + 243 + ], + "score": 1.0, + "content": "have w.p.", + "type": "text" + }, + { + "bbox": [ + 147, + 224, + 186, + 240 + ], + "score": 0.92, + "content": "\\textstyle { \\left( { 1 - { \\frac { \\zeta } { 4 K } } } \\right) }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 222, + 192, + 243 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 244, + 411, + 273 + ], + "lines": [ + { + "bbox": [ + 200, + 244, + 411, + 273 + ], + "spans": [ + { + "bbox": [ + 200, + 244, + 411, + 273 + ], + "score": 0.92, + "content": "\\frac { L _ { 2 } } { 6 } \\| \\tilde { \\mathbf { h } } ^ { k } \\| ^ { 3 } + \\| \\nabla _ { k } \\| \\| \\tilde { \\mathbf { h } } ^ { k } \\| + \\frac { 1 } { 2 } \\| \\nabla _ { k } ^ { 2 } \\| \\| \\tilde { \\mathbf { h } } ^ { k } \\| ^ { 2 } \\leq \\frac { \\epsilon ^ { 1 . 5 } } { 2 \\sqrt { L _ { 2 } } } .", + "type": "interline_equation", + "image_path": "f4fa9225b6f536e955cf7b583aa298251879902ffc6088129ea61db36e3512bd.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 200, + 244, + 411, + 273 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 285, + 505, + 299 + ], + "lines": [ + { + "bbox": [ + 105, + 284, + 502, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 464, + 302 + ], + "score": 1.0, + "content": "Plugging (16), (17), and (26) into (25) and applying the union bound, we have w.p. at least", + "type": "text" + }, + { + "bbox": [ + 464, + 284, + 502, + 299 + ], + "score": 0.93, + "content": "\\begin{array} { r } { ( 1 - \\frac { \\zeta } { 2 K } ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 306, + 459, + 334 + ], + "lines": [ + { + "bbox": [ + 151, + 306, + 459, + 334 + ], + "spans": [ + { + "bbox": [ + 151, + 306, + 459, + 334 + ], + "score": 0.92, + "content": "F ( { \\mathbf x } ^ { k + 1 } ) \\le F ( { \\mathbf x } ^ { k } ) - \\frac { L _ { 2 } \\lambda ^ { k } \\| { \\mathbf h } ^ { k } \\| ^ { 2 } } { 4 } + \\frac { 3 \\epsilon ^ { 1 . 5 } } { 2 \\sqrt { L _ { 2 } } } = F ( { \\mathbf x } ^ { k } ) - \\frac { L _ { 2 } \\lambda ^ { k } r ^ { 2 } } { 4 } + \\frac { 3 \\epsilon ^ { 1 . 5 } } { 2 \\sqrt { L _ { 2 } } } ,", + "type": "interline_equation", + "image_path": "8a02aa228f0aec3132cde4a2fc73e578b05458ea77195a0b69b1c21caf94d387.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 151, + 306, + 459, + 334 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 336, + 295, + 349 + ], + "lines": [ + { + "bbox": [ + 106, + 336, + 296, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 296, + 350 + ], + "score": 1.0, + "content": "where the second inequality follows from (15):", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 168, + 354, + 443, + 370 + ], + "lines": [ + { + "bbox": [ + 168, + 354, + 443, + 370 + ], + "spans": [ + { + "bbox": [ + 168, + 354, + 443, + 370 + ], + "score": 0.9, + "content": "0 = \\lambda ^ { k } ( \\| \\mathbf h ^ { k } \\| - r ) = \\lambda ^ { k } ( \\| \\mathbf h ^ { k } \\| - r ) ( \\| \\mathbf h ^ { k } \\| + r ) = \\lambda ^ { k } ( \\| \\mathbf h ^ { k } \\| ^ { 2 } - r ^ { 2 } ) .", + "type": "interline_equation", + "image_path": "8cfbf8890382660d3b47c333a2048ce9c50cfd1b23059f2c3065e521e2bb78ad.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 168, + 354, + 443, + 370 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 375, + 506, + 398 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 231, + 388 + ], + "score": 1.0, + "content": "Summing inequality (27) from", + "type": "text" + }, + { + "bbox": [ + 231, + 376, + 256, + 385 + ], + "score": 0.92, + "content": "k = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 374, + 267, + 388 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 268, + 376, + 295, + 386 + ], + "score": 0.89, + "content": "K - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 374, + 505, + 388 + ], + "score": 1.0, + "content": "and applying the union bound, we have w.p. at least", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 384, + 152, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 146, + 398 + ], + "score": 0.89, + "content": "\\left( 1 - \\zeta / 2 \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 384, + 152, + 400 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 176, + 398, + 435, + 432 + ], + "lines": [ + { + "bbox": [ + 176, + 398, + 435, + 432 + ], + "spans": [ + { + "bbox": [ + 176, + 398, + 435, + 432 + ], + "score": 0.94, + "content": "\\frac { 1 } { K } \\sum _ { k = 0 } ^ { K - 1 } \\lambda ^ { k } \\le \\frac { 4 ( F ( \\mathbf { x } ^ { 0 } ) - F ( \\mathbf { x } ^ { K + 1 } ) ) } { L _ { 2 } r ^ { 2 } K } + \\frac { 6 \\epsilon ^ { 1 . 5 } } { L _ { 2 } ^ { 1 . 5 } r ^ { 2 } } \\le \\frac { 4 \\Delta } { \\epsilon K } + \\frac { 6 \\sqrt { \\epsilon } } { \\sqrt { L _ { 2 } } } ,", + "type": "interline_equation", + "image_path": "bf123e3aef16dd9879bec6b5bb1b84059c93b84a438af694c98cb738aac4d7d7.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 176, + 398, + 435, + 409.3333333333333 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 176, + 409.3333333333333, + 435, + 420.66666666666663 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 176, + 420.66666666666663, + 435, + 431.99999999999994 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 504, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 433, + 504, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 504, + 449 + ], + "score": 1.0, + "content": "where the second inequality follows from Assumption 2.1 and our choice of the trust region radius.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 451, + 342, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 159, + 465 + ], + "score": 1.0, + "content": "By sampling", + "type": "text" + }, + { + "bbox": [ + 160, + 451, + 166, + 462 + ], + "score": 0.82, + "content": "\\bar { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 451, + 232, + 465 + ], + "score": 1.0, + "content": "uniformly from", + "type": "text" + }, + { + "bbox": [ + 232, + 452, + 296, + 464 + ], + "score": 0.93, + "content": "\\{ 0 , \\ldots , K - 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 451, + 342, + 465 + ], + "score": 1.0, + "content": ", we obtain", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "interline_equation", + "bbox": [ + 263, + 470, + 347, + 505 + ], + "lines": [ + { + "bbox": [ + 263, + 470, + 347, + 505 + ], + "spans": [ + { + "bbox": [ + 263, + 470, + 347, + 505 + ], + "score": 0.94, + "content": "\\mathbb { E } [ \\lambda ^ { \\bar { k } } ] = \\frac { 1 } { K } \\sum _ { k = 0 } ^ { K - 1 } \\lambda ^ { k } ,", + "type": "interline_equation", + "image_path": "107fcb40c963dcd1c5c33403eea00221a5f4ae50e2ed106e29f62798bef2b787.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 263, + 470, + 347, + 487.5 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 263, + 487.5, + 347, + 505.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 104, + 508, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 104, + 508, + 337, + 526 + ], + "score": 1.0, + "content": "where the expectation is taken over the randomness of √", + "type": "text" + }, + { + "bbox": [ + 338, + 511, + 344, + 522 + ], + "score": 0.83, + "content": "\\bar { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 508, + 506, + 526 + ], + "score": 1.0, + "content": ". Combining (29) and (30) and taking", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 520, + 313, + 536 + ], + "spans": [ + { + "bbox": [ + 107, + 522, + 183, + 534 + ], + "score": 0.93, + "content": "K = 4 \\Delta \\sqrt { L _ { 2 } } / \\epsilon ^ { 1 . 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 520, + 272, + 536 + ], + "score": 1.0, + "content": ", we have w.p. at least", + "type": "text" + }, + { + "bbox": [ + 272, + 522, + 312, + 535 + ], + "score": 0.91, + "content": "\\left( 1 - \\zeta / 2 \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 520, + 313, + 536 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 273, + 540, + 337, + 568 + ], + "lines": [ + { + "bbox": [ + 273, + 540, + 337, + 568 + ], + "spans": [ + { + "bbox": [ + 273, + 540, + 337, + 568 + ], + "score": 0.95, + "content": "\\mathbb { E } [ \\lambda ^ { \\bar { k } } ] \\leq \\frac { 7 \\sqrt \\epsilon } { \\sqrt { L _ { 2 } } } .", + "type": "interline_equation", + "image_path": "2ef0ee6dcaed518fd4bee5fe578fc7e61dd7447a85f9346fbe07a3431e91c938.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 273, + 540, + 337, + 568 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 574, + 506, + 596 + ], + "lines": [ + { + "bbox": [ + 105, + 573, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 130, + 588 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 574, + 142, + 585 + ], + "score": 0.88, + "content": "\\lambda ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 573, + 506, + 588 + ], + "score": 1.0, + "content": "is always no-negative, by Markov’s inequality and the union bound, with probability at least", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 584, + 169, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 129, + 597 + ], + "score": 0.88, + "content": "1 - \\zeta", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 584, + 169, + 599 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "interline_equation", + "bbox": [ + 277, + 596, + 334, + 624 + ], + "lines": [ + { + "bbox": [ + 277, + 596, + 334, + 624 + ], + "spans": [ + { + "bbox": [ + 277, + 596, + 334, + 624 + ], + "score": 0.94, + "content": "\\lambda ^ { \\bar { k } } \\leq \\frac { 1 4 \\sqrt { \\epsilon } } { \\zeta \\sqrt { L _ { 2 } } } .", + "type": "interline_equation", + "image_path": "08ed327e45b2a8ba5c293d38c0b17968eabac4292c4c3c7f0fed7a71e005cc76.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 277, + 596, + 334, + 624 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 628, + 506, + 652 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 149, + 643 + ], + "score": 1.0, + "content": "By taking", + "type": "text" + }, + { + "bbox": [ + 149, + 630, + 185, + 641 + ], + "score": 0.91, + "content": "\\zeta = 1 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 627, + 276, + 643 + ], + "score": 1.0, + "content": ", we have w.p. at least", + "type": "text" + }, + { + "bbox": [ + 276, + 627, + 362, + 642 + ], + "score": 0.9, + "content": "2 / 3 , \\lambda ^ { \\bar { k } } \\leq 4 2 \\sqrt { \\epsilon / L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 627, + 506, + 643 + ], + "score": 1.0, + "content": ". The rest of the proof is similar to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 640, + 253, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 253, + 652 + ], + "score": 1.0, + "content": "Theorem 3.1 and we have the result.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 666, + 505, + 690 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 350, + 681 + ], + "score": 1.0, + "content": "The following theorem shows that MetaAlgorithm 7 finds an", + "type": "text" + }, + { + "bbox": [ + 351, + 667, + 371, + 679 + ], + "score": 0.91, + "content": "\\mathcal { O } ( \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 664, + 419, + 681 + ], + "score": 1.0, + "content": "-SOSP w.p.", + "type": "text" + }, + { + "bbox": [ + 419, + 667, + 449, + 679 + ], + "score": 0.89, + "content": "( 1 - \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 664, + 505, + 681 + ], + "score": 1.0, + "content": "after running", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 677, + 274, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 159, + 690 + ], + "score": 1.0, + "content": "INEXACTTR", + "type": "text" + }, + { + "bbox": [ + 159, + 678, + 179, + 690 + ], + "score": 0.26, + "content": "\\mathbf { \\partial } \\cdot \\mathbf { W } \\mathbf { E } \\mathbf { A } \\mathbf { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 677, + 194, + 690 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 195, + 678, + 246, + 690 + ], + "score": 0.92, + "content": "\\Theta ( \\log ( 1 / \\delta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 677, + 274, + 690 + ], + "score": 1.0, + "content": "times.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 693, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 106, + 692, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 692, + 505, + 705 + ], + "score": 1.0, + "content": "Theorem C.1 (Iteration Complexity of MetaAlgorithm 7). In the same setting as Lemma C.1, let", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 700, + 509, + 727 + ], + "spans": [ + { + "bbox": [ + 106, + 707, + 173, + 721 + ], + "score": 0.89, + "content": "\\begin{array} { r } { T = \\frac { 3 } { 2 } \\mathrm { l o g } ( 2 / \\delta ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 705, + 177, + 721 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 177, + 708, + 219, + 719 + ], + "score": 0.82, + "content": "c _ { 1 } = 6 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 705, + 223, + 721 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 223, + 708, + 265, + 719 + ], + "score": 0.82, + "content": "c _ { 2 } = 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 705, + 288, + 721 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 304, + 700, + 509, + 727 + ], + "score": 1.0, + "content": "32L21\u000fL2 log(4d/δ). Then MetaAlgorithm 7 finds a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 720, + 286, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 252, + 733 + ], + "score": 1.0, + "content": "600\u000f-SOSP with probability at least", + "type": "text" + }, + { + "bbox": [ + 253, + 720, + 282, + 732 + ], + "score": 0.93, + "content": "( 1 - \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 720, + 286, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 309, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "17", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 338, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 78, + 334, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 295, + 98 + ], + "score": 1.0, + "content": "Proof. Combining (11) and (24), we have w.p.", + "type": "text" + }, + { + "bbox": [ + 296, + 81, + 334, + 96 + ], + "score": 0.91, + "content": "\\begin{array} { r } { ( 1 - \\frac { \\zeta } { 4 K } ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 78, + 334, + 98 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 101, + 420, + 155 + ], + "lines": [ + { + "bbox": [ + 190, + 101, + 420, + 155 + ], + "spans": [ + { + "bbox": [ + 190, + 101, + 420, + 155 + ], + "score": 0.94, + "content": "\\begin{array} { r } { F ( { \\mathbf x } ^ { k + 1 } ) \\le F ( { \\mathbf x } ^ { k } ) + \\langle { \\mathbf g } ^ { k } , { \\mathbf h } ^ { k } \\rangle + \\displaystyle \\frac { 1 } { 2 } \\langle { \\mathbf H } ^ { k } { \\mathbf h } ^ { k } , { \\mathbf h } ^ { k } \\rangle + \\displaystyle \\frac { \\epsilon ^ { 1 . 5 } } { \\sqrt { L _ { 2 } } } } \\\\ { + \\displaystyle \\frac { L _ { 2 } } { 6 } \\| \\tilde { \\mathbf h } ^ { k } \\| ^ { 3 } + \\| \\nabla _ { k } \\| \\| \\tilde { \\mathbf h } ^ { k } \\| + \\displaystyle \\frac { 1 } { 2 } \\| \\nabla _ { k } ^ { 2 } \\| \\| \\tilde { \\mathbf h } ^ { k } \\| ^ { 2 } , } \\end{array}", + "type": "interline_equation", + "image_path": "2f16a65cfa817451421902b5bee412c3789951b3c8f68cdb9bbf2e16f4347e7f.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 190, + 101, + 420, + 119.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 190, + 119.0, + 420, + 137.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 190, + 137.0, + 420, + 155.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 159, + 506, + 206 + ], + "lines": [ + { + "bbox": [ + 104, + 159, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 104, + 159, + 132, + 174 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 160, + 145, + 171 + ], + "score": 0.88, + "content": "\\mathbf { h } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 159, + 307, + 174 + ], + "score": 1.0, + "content": "is a global solution to the QCQP (8) and", + "type": "text" + }, + { + "bbox": [ + 307, + 159, + 320, + 172 + ], + "score": 0.88, + "content": "\\tilde { \\mathbf { h } } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 159, + 506, + 174 + ], + "score": 1.0, + "content": "is an approximate solution satisfying (24). We", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 171, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 119, + 185 + ], + "score": 1.0, + "content": "let", + "type": "text" + }, + { + "bbox": [ + 119, + 172, + 131, + 182 + ], + "score": 0.88, + "content": "\\lambda ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 171, + 373, + 185 + ], + "score": 1.0, + "content": "denote the dual variable corresponding to the global solution", + "type": "text" + }, + { + "bbox": [ + 373, + 172, + 385, + 182 + ], + "score": 0.88, + "content": "\\mathbf { h } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 171, + 506, + 185 + ], + "score": 1.0, + "content": "as defined in Lemma 2.1. We", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 181, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 104, + 181, + 144, + 196 + ], + "score": 1.0, + "content": "note that", + "type": "text" + }, + { + "bbox": [ + 145, + 182, + 157, + 193 + ], + "score": 0.88, + "content": "\\mathbf { h } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 181, + 176, + 196 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 176, + 182, + 188, + 193 + ], + "score": 0.89, + "content": "\\lambda ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 181, + 371, + 196 + ], + "score": 1.0, + "content": "are used only in our analysis. The INEXACT", + "type": "text" + }, + { + "bbox": [ + 371, + 185, + 390, + 195 + ], + "score": 0.35, + "content": "\\mathrm { w e A K }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 181, + 506, + 196 + ], + "score": 1.0, + "content": "algorithm only requires the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 194, + 337, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 193, + 206 + ], + "score": 1.0, + "content": "approximate solution", + "type": "text" + }, + { + "bbox": [ + 194, + 194, + 205, + 204 + ], + "score": 0.87, + "content": "\\tilde { \\mathbf { x } } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 194, + 297, + 206 + ], + "score": 1.0, + "content": "without knowledge of", + "type": "text" + }, + { + "bbox": [ + 297, + 194, + 309, + 204 + ], + "score": 0.88, + "content": "\\mathbf { h } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 194, + 321, + 206 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 321, + 194, + 333, + 204 + ], + "score": 0.89, + "content": "\\lambda ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 194, + 337, + 206 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5, + "bbox_fs": [ + 104, + 159, + 506, + 206 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 210, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 105, + 209, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 304, + 228 + ], + "score": 1.0, + "content": "By the assumption that (9) holds with probability", + "type": "text" + }, + { + "bbox": [ + 304, + 210, + 343, + 226 + ], + "score": 0.92, + "content": "\\textstyle { \\left( { 1 - { \\frac { \\zeta } { 4 K } } } \\right) }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 209, + 410, + 228 + ], + "score": 1.0, + "content": "and the fact that", + "type": "text" + }, + { + "bbox": [ + 410, + 210, + 487, + 225 + ], + "score": 0.93, + "content": "\\| \\tilde { \\mathbf { h } } ^ { k } \\| \\le r = \\sqrt { L _ { 2 } \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 209, + 506, + 228 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 222, + 192, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 147, + 243 + ], + "score": 1.0, + "content": "have w.p.", + "type": "text" + }, + { + "bbox": [ + 147, + 224, + 186, + 240 + ], + "score": 0.92, + "content": "\\textstyle { \\left( { 1 - { \\frac { \\zeta } { 4 K } } } \\right) }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 222, + 192, + 243 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 209, + 506, + 243 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 244, + 411, + 273 + ], + "lines": [ + { + "bbox": [ + 200, + 244, + 411, + 273 + ], + "spans": [ + { + "bbox": [ + 200, + 244, + 411, + 273 + ], + "score": 0.92, + "content": "\\frac { L _ { 2 } } { 6 } \\| \\tilde { \\mathbf { h } } ^ { k } \\| ^ { 3 } + \\| \\nabla _ { k } \\| \\| \\tilde { \\mathbf { h } } ^ { k } \\| + \\frac { 1 } { 2 } \\| \\nabla _ { k } ^ { 2 } \\| \\| \\tilde { \\mathbf { h } } ^ { k } \\| ^ { 2 } \\leq \\frac { \\epsilon ^ { 1 . 5 } } { 2 \\sqrt { L _ { 2 } } } .", + "type": "interline_equation", + "image_path": "f4fa9225b6f536e955cf7b583aa298251879902ffc6088129ea61db36e3512bd.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 200, + 244, + 411, + 273 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 285, + 505, + 299 + ], + "lines": [ + { + "bbox": [ + 105, + 284, + 502, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 464, + 302 + ], + "score": 1.0, + "content": "Plugging (16), (17), and (26) into (25) and applying the union bound, we have w.p. at least", + "type": "text" + }, + { + "bbox": [ + 464, + 284, + 502, + 299 + ], + "score": 0.93, + "content": "\\begin{array} { r } { ( 1 - \\frac { \\zeta } { 2 K } ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 284, + 502, + 302 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 306, + 459, + 334 + ], + "lines": [ + { + "bbox": [ + 151, + 306, + 459, + 334 + ], + "spans": [ + { + "bbox": [ + 151, + 306, + 459, + 334 + ], + "score": 0.92, + "content": "F ( { \\mathbf x } ^ { k + 1 } ) \\le F ( { \\mathbf x } ^ { k } ) - \\frac { L _ { 2 } \\lambda ^ { k } \\| { \\mathbf h } ^ { k } \\| ^ { 2 } } { 4 } + \\frac { 3 \\epsilon ^ { 1 . 5 } } { 2 \\sqrt { L _ { 2 } } } = F ( { \\mathbf x } ^ { k } ) - \\frac { L _ { 2 } \\lambda ^ { k } r ^ { 2 } } { 4 } + \\frac { 3 \\epsilon ^ { 1 . 5 } } { 2 \\sqrt { L _ { 2 } } } ,", + "type": "interline_equation", + "image_path": "8a02aa228f0aec3132cde4a2fc73e578b05458ea77195a0b69b1c21caf94d387.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 151, + 306, + 459, + 334 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 336, + 295, + 349 + ], + "lines": [ + { + "bbox": [ + 106, + 336, + 296, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 296, + 350 + ], + "score": 1.0, + "content": "where the second inequality follows from (15):", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 106, + 336, + 296, + 350 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 168, + 354, + 443, + 370 + ], + "lines": [ + { + "bbox": [ + 168, + 354, + 443, + 370 + ], + "spans": [ + { + "bbox": [ + 168, + 354, + 443, + 370 + ], + "score": 0.9, + "content": "0 = \\lambda ^ { k } ( \\| \\mathbf h ^ { k } \\| - r ) = \\lambda ^ { k } ( \\| \\mathbf h ^ { k } \\| - r ) ( \\| \\mathbf h ^ { k } \\| + r ) = \\lambda ^ { k } ( \\| \\mathbf h ^ { k } \\| ^ { 2 } - r ^ { 2 } ) .", + "type": "interline_equation", + "image_path": "8cfbf8890382660d3b47c333a2048ce9c50cfd1b23059f2c3065e521e2bb78ad.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 168, + 354, + 443, + 370 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 375, + 506, + 398 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 231, + 388 + ], + "score": 1.0, + "content": "Summing inequality (27) from", + "type": "text" + }, + { + "bbox": [ + 231, + 376, + 256, + 385 + ], + "score": 0.92, + "content": "k = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 374, + 267, + 388 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 268, + 376, + 295, + 386 + ], + "score": 0.89, + "content": "K - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 374, + 505, + 388 + ], + "score": 1.0, + "content": "and applying the union bound, we have w.p. at least", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 384, + 152, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 146, + 398 + ], + "score": 0.89, + "content": "\\left( 1 - \\zeta / 2 \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 384, + 152, + 400 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 374, + 505, + 400 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 176, + 398, + 435, + 432 + ], + "lines": [ + { + "bbox": [ + 176, + 398, + 435, + 432 + ], + "spans": [ + { + "bbox": [ + 176, + 398, + 435, + 432 + ], + "score": 0.94, + "content": "\\frac { 1 } { K } \\sum _ { k = 0 } ^ { K - 1 } \\lambda ^ { k } \\le \\frac { 4 ( F ( \\mathbf { x } ^ { 0 } ) - F ( \\mathbf { x } ^ { K + 1 } ) ) } { L _ { 2 } r ^ { 2 } K } + \\frac { 6 \\epsilon ^ { 1 . 5 } } { L _ { 2 } ^ { 1 . 5 } r ^ { 2 } } \\le \\frac { 4 \\Delta } { \\epsilon K } + \\frac { 6 \\sqrt { \\epsilon } } { \\sqrt { L _ { 2 } } } ,", + "type": "interline_equation", + "image_path": "bf123e3aef16dd9879bec6b5bb1b84059c93b84a438af694c98cb738aac4d7d7.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 176, + 398, + 435, + 409.3333333333333 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 176, + 409.3333333333333, + 435, + 420.66666666666663 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 176, + 420.66666666666663, + 435, + 431.99999999999994 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 504, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 433, + 504, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 504, + 449 + ], + "score": 1.0, + "content": "where the second inequality follows from Assumption 2.1 and our choice of the trust region radius.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 451, + 342, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 159, + 465 + ], + "score": 1.0, + "content": "By sampling", + "type": "text" + }, + { + "bbox": [ + 160, + 451, + 166, + 462 + ], + "score": 0.82, + "content": "\\bar { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 451, + 232, + 465 + ], + "score": 1.0, + "content": "uniformly from", + "type": "text" + }, + { + "bbox": [ + 232, + 452, + 296, + 464 + ], + "score": 0.93, + "content": "\\{ 0 , \\ldots , K - 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 451, + 342, + 465 + ], + "score": 1.0, + "content": ", we obtain", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 433, + 504, + 465 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 263, + 470, + 347, + 505 + ], + "lines": [ + { + "bbox": [ + 263, + 470, + 347, + 505 + ], + "spans": [ + { + "bbox": [ + 263, + 470, + 347, + 505 + ], + "score": 0.94, + "content": "\\mathbb { E } [ \\lambda ^ { \\bar { k } } ] = \\frac { 1 } { K } \\sum _ { k = 0 } ^ { K - 1 } \\lambda ^ { k } ,", + "type": "interline_equation", + "image_path": "107fcb40c963dcd1c5c33403eea00221a5f4ae50e2ed106e29f62798bef2b787.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 263, + 470, + 347, + 487.5 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 263, + 487.5, + 347, + 505.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 104, + 508, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 104, + 508, + 337, + 526 + ], + "score": 1.0, + "content": "where the expectation is taken over the randomness of √", + "type": "text" + }, + { + "bbox": [ + 338, + 511, + 344, + 522 + ], + "score": 0.83, + "content": "\\bar { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 508, + 506, + 526 + ], + "score": 1.0, + "content": ". Combining (29) and (30) and taking", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 520, + 313, + 536 + ], + "spans": [ + { + "bbox": [ + 107, + 522, + 183, + 534 + ], + "score": 0.93, + "content": "K = 4 \\Delta \\sqrt { L _ { 2 } } / \\epsilon ^ { 1 . 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 520, + 272, + 536 + ], + "score": 1.0, + "content": ", we have w.p. at least", + "type": "text" + }, + { + "bbox": [ + 272, + 522, + 312, + 535 + ], + "score": 0.91, + "content": "\\left( 1 - \\zeta / 2 \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 520, + 313, + 536 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 508, + 506, + 536 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 273, + 540, + 337, + 568 + ], + "lines": [ + { + "bbox": [ + 273, + 540, + 337, + 568 + ], + "spans": [ + { + "bbox": [ + 273, + 540, + 337, + 568 + ], + "score": 0.95, + "content": "\\mathbb { E } [ \\lambda ^ { \\bar { k } } ] \\leq \\frac { 7 \\sqrt \\epsilon } { \\sqrt { L _ { 2 } } } .", + "type": "interline_equation", + "image_path": "2ef0ee6dcaed518fd4bee5fe578fc7e61dd7447a85f9346fbe07a3431e91c938.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 273, + 540, + 337, + 568 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 574, + 506, + 596 + ], + "lines": [ + { + "bbox": [ + 105, + 573, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 130, + 588 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 574, + 142, + 585 + ], + "score": 0.88, + "content": "\\lambda ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 573, + 506, + 588 + ], + "score": 1.0, + "content": "is always no-negative, by Markov’s inequality and the union bound, with probability at least", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 584, + 169, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 129, + 597 + ], + "score": 0.88, + "content": "1 - \\zeta", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 584, + 169, + 599 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 573, + 506, + 599 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 277, + 596, + 334, + 624 + ], + "lines": [ + { + "bbox": [ + 277, + 596, + 334, + 624 + ], + "spans": [ + { + "bbox": [ + 277, + 596, + 334, + 624 + ], + "score": 0.94, + "content": "\\lambda ^ { \\bar { k } } \\leq \\frac { 1 4 \\sqrt { \\epsilon } } { \\zeta \\sqrt { L _ { 2 } } } .", + "type": "interline_equation", + "image_path": "08ed327e45b2a8ba5c293d38c0b17968eabac4292c4c3c7f0fed7a71e005cc76.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 277, + 596, + 334, + 624 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 628, + 506, + 652 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 149, + 643 + ], + "score": 1.0, + "content": "By taking", + "type": "text" + }, + { + "bbox": [ + 149, + 630, + 185, + 641 + ], + "score": 0.91, + "content": "\\zeta = 1 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 627, + 276, + 643 + ], + "score": 1.0, + "content": ", we have w.p. at least", + "type": "text" + }, + { + "bbox": [ + 276, + 627, + 362, + 642 + ], + "score": 0.9, + "content": "2 / 3 , \\lambda ^ { \\bar { k } } \\leq 4 2 \\sqrt { \\epsilon / L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 627, + 506, + 643 + ], + "score": 1.0, + "content": ". The rest of the proof is similar to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 640, + 253, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 253, + 652 + ], + "score": 1.0, + "content": "Theorem 3.1 and we have the result.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 627, + 506, + 652 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 666, + 505, + 690 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 350, + 681 + ], + "score": 1.0, + "content": "The following theorem shows that MetaAlgorithm 7 finds an", + "type": "text" + }, + { + "bbox": [ + 351, + 667, + 371, + 679 + ], + "score": 0.91, + "content": "\\mathcal { O } ( \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 664, + 419, + 681 + ], + "score": 1.0, + "content": "-SOSP w.p.", + "type": "text" + }, + { + "bbox": [ + 419, + 667, + 449, + 679 + ], + "score": 0.89, + "content": "( 1 - \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 664, + 505, + 681 + ], + "score": 1.0, + "content": "after running", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 677, + 274, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 159, + 690 + ], + "score": 1.0, + "content": "INEXACTTR", + "type": "text" + }, + { + "bbox": [ + 159, + 678, + 179, + 690 + ], + "score": 0.26, + "content": "\\mathbf { \\partial } \\cdot \\mathbf { W } \\mathbf { E } \\mathbf { A } \\mathbf { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 677, + 194, + 690 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 195, + 678, + 246, + 690 + ], + "score": 0.92, + "content": "\\Theta ( \\log ( 1 / \\delta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 677, + 274, + 690 + ], + "score": 1.0, + "content": "times.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 664, + 505, + 690 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 693, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 106, + 692, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 692, + 505, + 705 + ], + "score": 1.0, + "content": "Theorem C.1 (Iteration Complexity of MetaAlgorithm 7). In the same setting as Lemma C.1, let", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 700, + 509, + 727 + ], + "spans": [ + { + "bbox": [ + 106, + 707, + 173, + 721 + ], + "score": 0.89, + "content": "\\begin{array} { r } { T = \\frac { 3 } { 2 } \\mathrm { l o g } ( 2 / \\delta ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 705, + 177, + 721 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 177, + 708, + 219, + 719 + ], + "score": 0.82, + "content": "c _ { 1 } = 6 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 705, + 223, + 721 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 223, + 708, + 265, + 719 + ], + "score": 0.82, + "content": "c _ { 2 } = 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 705, + 288, + 721 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 304, + 700, + 509, + 727 + ], + "score": 1.0, + "content": "32L21\u000fL2 log(4d/δ). Then MetaAlgorithm 7 finds a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 720, + 286, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 252, + 733 + ], + "score": 1.0, + "content": "600\u000f-SOSP with probability at least", + "type": "text" + }, + { + "bbox": [ + 253, + 720, + 282, + 732 + ], + "score": 0.93, + "content": "( 1 - \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 720, + 286, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 106, + 692, + 509, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 506, + 118 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 284, + 95 + ], + "score": 1.0, + "content": "Proof. By Lemma C.1 and our choice of", + "type": "text" + }, + { + "bbox": [ + 284, + 83, + 293, + 93 + ], + "score": 0.81, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 81, + 369, + 95 + ], + "score": 1.0, + "content": ", with probability", + "type": "text" + }, + { + "bbox": [ + 370, + 82, + 411, + 95 + ], + "score": 0.91, + "content": "\\left( 1 - \\delta / 2 \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 81, + 482, + 95 + ], + "score": 1.0, + "content": ", at least one of", + "type": "text" + }, + { + "bbox": [ + 482, + 82, + 493, + 93 + ], + "score": 0.87, + "content": "\\mathbf { x } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 92, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 92, + 114, + 108 + ], + "score": 1.0, + "content": "a", + "type": "text" + }, + { + "bbox": [ + 114, + 95, + 135, + 105 + ], + "score": 0.56, + "content": "5 0 0 \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 92, + 273, + 108 + ], + "score": 1.0, + "content": "-SOSP. On the other hand, since √", + "type": "text" + }, + { + "bbox": [ + 273, + 94, + 384, + 106 + ], + "score": 0.92, + "content": "\\psi _ { t } ( \\tilde { \\mathbf { v } } ^ { t } ) \\leq \\psi _ { t } ( \\mathbf { v } ^ { t } ) + \\sqrt { \\epsilon L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 92, + 456, + 108 + ], + "score": 1.0, + "content": "with probability", + "type": "text" + }, + { + "bbox": [ + 456, + 94, + 491, + 106 + ], + "score": 0.91, + "content": "1 - \\delta / 4", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 92, + 506, + 108 + ], + "score": 1.0, + "content": ", if", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 106, + 271, + 120 + ], + "spans": [ + { + "bbox": [ + 107, + 106, + 189, + 119 + ], + "score": 0.93, + "content": "\\psi _ { t } ( \\tilde { \\mathbf { v } } ^ { t } ) \\geq - \\sqrt { c _ { 2 } \\epsilon L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 106, + 233, + 120 + ], + "score": 1.0, + "content": ", then, w.p.", + "type": "text" + }, + { + "bbox": [ + 234, + 106, + 266, + 119 + ], + "score": 0.91, + "content": "1 - \\delta / 4", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 106, + 271, + 120 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 122, + 437, + 137 + ], + "lines": [ + { + "bbox": [ + 173, + 122, + 437, + 137 + ], + "spans": [ + { + "bbox": [ + 173, + 122, + 437, + 137 + ], + "score": 0.87, + "content": "\\psi _ { t } ( \\mathbf { v } ^ { t } ) \\geq \\psi _ { t } ( \\tilde { \\mathbf { v } } ^ { t } ) - \\sqrt { \\epsilon L _ { 2 } } \\geq - \\sqrt { c _ { 2 } \\epsilon L _ { 2 } } - \\sqrt { \\epsilon L _ { 2 } } \\geq - \\sqrt { 5 5 0 \\epsilon L _ { 2 } } ,", + "type": "interline_equation", + "image_path": "5431faf34f9f7e6007dd3f9fa5c9b878277fae0c6e37d2c7365e0e69be97bd8f.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 173, + 122, + 437, + 137 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 140, + 329, + 152 + ], + "lines": [ + { + "bbox": [ + 105, + 137, + 330, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 315, + 155 + ], + "score": 1.0, + "content": "where the last inequality follows from our choice of", + "type": "text" + }, + { + "bbox": [ + 316, + 143, + 325, + 151 + ], + "score": 0.82, + "content": "c _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 137, + 330, + 155 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 156, + 505, + 180 + ], + "lines": [ + { + "bbox": [ + 105, + 156, + 506, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 173, + 170 + ], + "score": 1.0, + "content": "Option I: Since", + "type": "text" + }, + { + "bbox": [ + 174, + 156, + 238, + 169 + ], + "score": 0.93, + "content": "\\mathbf { H } ^ { t } = \\nabla ^ { 2 } F ( \\mathbf { x } ^ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 156, + 317, + 170 + ], + "score": 1.0, + "content": "is the full Hessian,", + "type": "text" + }, + { + "bbox": [ + 317, + 157, + 346, + 169 + ], + "score": 0.92, + "content": "\\psi _ { t } ( \\mathbf { v } ^ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 156, + 464, + 170 + ], + "score": 1.0, + "content": "is the smallest eigenvalue of", + "type": "text" + }, + { + "bbox": [ + 464, + 156, + 503, + 169 + ], + "score": 0.92, + "content": "\\nabla ^ { 2 } F ( \\mathbf { x } ^ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 156, + 506, + 170 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 167, + 501, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 396, + 182 + ], + "score": 1.0, + "content": "Applying the union bound, we conclude that MetaAlgorithm 7 outputs a", + "type": "text" + }, + { + "bbox": [ + 397, + 169, + 417, + 179 + ], + "score": 0.5, + "content": "6 0 0 \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 167, + 465, + 182 + ], + "score": 1.0, + "content": "-SOSP w.p.", + "type": "text" + }, + { + "bbox": [ + 465, + 169, + 495, + 181 + ], + "score": 0.91, + "content": "( 1 - \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 167, + 501, + 182 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 106, + 184, + 350, + 198 + ], + "lines": [ + { + "bbox": [ + 105, + 183, + 350, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 168, + 200 + ], + "score": 1.0, + "content": "Option II: Let", + "type": "text" + }, + { + "bbox": [ + 168, + 185, + 285, + 198 + ], + "score": 0.93, + "content": "\\mathbf { B } _ { i } : = \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { t } ) - \\nabla ^ { 2 } F ( \\mathbf { x } ^ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 183, + 300, + 200 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 301, + 186, + 326, + 196 + ], + "score": 0.89, + "content": "i \\in \\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 183, + 350, + 200 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 200, + 365, + 233 + ], + "lines": [ + { + "bbox": [ + 246, + 200, + 365, + 233 + ], + "spans": [ + { + "bbox": [ + 246, + 200, + 365, + 233 + ], + "score": 0.95, + "content": "\\mathbf { H } ^ { t } - \\boldsymbol { \\nabla } ^ { 2 } F ( \\mathbf { x } ^ { t } ) = \\frac { 1 } { s } \\sum _ { i = 1 } ^ { s } \\mathbf { B } _ { i } .", + "type": "interline_equation", + "image_path": "82330d1f335568caf45cf8e4281942893a73d002f570e62e7e857cb02656db0c.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 246, + 200, + 365, + 216.5 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 246, + 216.5, + 365, + 233.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 235, + 223, + 247 + ], + "lines": [ + { + "bbox": [ + 106, + 234, + 224, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 224, + 249 + ], + "score": 1.0, + "content": "By Assumption 2.2, we have", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 249, + 394, + 264 + ], + "lines": [ + { + "bbox": [ + 216, + 249, + 394, + 264 + ], + "spans": [ + { + "bbox": [ + 216, + 249, + 394, + 264 + ], + "score": 0.92, + "content": "\\| \\mathbf { B } _ { i } \\| \\leq \\| \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { t } ) \\| + \\| \\nabla ^ { 2 } F ( \\mathbf { x } ^ { t } ) \\| \\leq 2 L _ { 1 } .", + "type": "interline_equation", + "image_path": "a66b657f6d5e008d7e819d605f1b1407257caed0799a469954f3e07b2a273a77.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 216, + 249, + 394, + 264 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 267, + 434, + 279 + ], + "lines": [ + { + "bbox": [ + 105, + 266, + 434, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 434, + 281 + ], + "score": 1.0, + "content": "Applying the matrix Azuma’s Inequality in Theorem 7.1 of Tropp (2012) leads to", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 281, + 414, + 308 + ], + "lines": [ + { + "bbox": [ + 196, + 281, + 414, + 308 + ], + "spans": [ + { + "bbox": [ + 196, + 281, + 414, + 308 + ], + "score": 0.91, + "content": "P r \\{ \\| \\mathbf { H } ^ { t } - \\nabla ^ { 2 } F ( \\mathbf { x } ^ { t } ) \\| \\ge \\sqrt { \\epsilon L _ { 2 } } \\} \\le d \\cdot \\exp ( \\frac { - \\epsilon L _ { 2 } s } { 3 2 L _ { 1 } ^ { 2 } } ) .", + "type": "interline_equation", + "image_path": "93b547fd9a178797b861f3f6b660d4c1e3c7334f23d4c7a6543ffda48ce10664.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 196, + 281, + 414, + 308 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 312, + 480, + 328 + ], + "lines": [ + { + "bbox": [ + 106, + 308, + 477, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 147, + 330 + ], + "score": 1.0, + "content": "By taking", + "type": "text" + }, + { + "bbox": [ + 148, + 311, + 227, + 329 + ], + "score": 0.94, + "content": "\\begin{array} { r } { s = \\frac { 3 2 L _ { 1 } ^ { 2 } } { \\epsilon L _ { 2 } } \\mathrm { l o g } ( 4 d / \\delta ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 308, + 454, + 330 + ], + "score": 1.0, + "content": "and applying the union bound, we have with probability", + "type": "text" + }, + { + "bbox": [ + 455, + 316, + 477, + 326 + ], + "score": 0.86, + "content": "1 - \\delta", + "type": "inline_equation" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 332, + 438, + 348 + ], + "lines": [ + { + "bbox": [ + 172, + 332, + 438, + 348 + ], + "spans": [ + { + "bbox": [ + 172, + 332, + 438, + 348 + ], + "score": 0.88, + "content": "\\nabla ^ { 2 } F ( \\mathbf { x } ^ { t } ) \\approx \\mathbf { H } ^ { t } - \\sqrt { \\epsilon L _ { 2 } } \\mathbf { I } \\approx ( \\psi _ { t } ( \\mathbf { v } ^ { t } ) - \\sqrt { \\epsilon L _ { 2 } } ) \\mathbf { I } \\approx - \\sqrt { 6 0 0 \\epsilon L _ { 2 } } \\mathbf { I } ,", + "type": "interline_equation", + "image_path": "82fc59a55d1163015ccb3dcb872124de25d906f486abf3f8a4e8261b20f94afb.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 172, + 332, + 438, + 348 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 386, + 363 + ], + "lines": [ + { + "bbox": [ + 105, + 350, + 387, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 387, + 365 + ], + "score": 1.0, + "content": "where the last inequality follows from (33). This completes the proof.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 108, + 378, + 253, + 391 + ], + "lines": [ + { + "bbox": [ + 105, + 377, + 255, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 255, + 393 + ], + "score": 1.0, + "content": "D PROOF OF THEOREM 6.1", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 402, + 506, + 501 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 506, + 416 + ], + "score": 1.0, + "content": "Proof. We first analyze the computational cost of Lanczos method. By Corollary 2 in (Carmon &", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 102, + 411, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 102, + 411, + 446, + 436 + ], + "score": 1.0, + "content": "Duchi, 2018), for any desired accuracy \u000f˜, Lanczos method achieves this accuracy in", + "type": "text" + }, + { + "bbox": [ + 446, + 414, + 505, + 432 + ], + "score": 0.95, + "content": "\\begin{array} { r } { \\mathcal { O } ( \\frac { r } { \\sqrt { \\tilde { \\epsilon } } } \\mathrm { l o g } \\frac { r \\sqrt { d } } { \\tilde { \\epsilon } p } ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 234, + 443 + ], + "score": 1.0, + "content": "Lanczos iterations w.p. at least", + "type": "text" + }, + { + "bbox": [ + 235, + 430, + 265, + 443 + ], + "score": 0.93, + "content": "( 1 - p )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 430, + 506, + 443 + ], + "score": 1.0, + "content": ". Without loss of generality, we assume that the number of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 441, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 339, + 455 + ], + "score": 1.0, + "content": "Lanczos iterations is strictly smaller than the dimension", + "type": "text" + }, + { + "bbox": [ + 339, + 442, + 345, + 452 + ], + "score": 0.76, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 441, + 506, + 455 + ], + "score": 1.0, + "content": ", otherwise the QCQP subproblem can", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 452, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 465 + ], + "score": 1.0, + "content": "be solved exactly. We note that each Lanczos iteration involves computation of one matrix-vector", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 463, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 104, + 463, + 440, + 478 + ], + "score": 1.0, + "content": "product. Therefore, to satisfy the condition (24) in Lemma C.1, one needs to evaluate", + "type": "text" + }, + { + "bbox": [ + 441, + 463, + 505, + 477 + ], + "score": 0.91, + "content": "\\tilde { \\mathcal { O } } ( 1 / ( L _ { 2 } \\epsilon ) ^ { 0 . 2 5 } )", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 475, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 252, + 491 + ], + "score": 1.0, + "content": "Hessian-vector products of the form", + "type": "text" + }, + { + "bbox": [ + 253, + 476, + 274, + 487 + ], + "score": 0.89, + "content": "\\mathbf { H } ^ { k } \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 475, + 433, + 491 + ], + "score": 1.0, + "content": ". Similarly, to solve (10) up to accuracy", + "type": "text" + }, + { + "bbox": [ + 433, + 477, + 458, + 488 + ], + "score": 0.91, + "content": "\\sqrt { \\epsilon L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 475, + 506, + 491 + ], + "score": 1.0, + "content": "w.h.p., one", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 488, + 414, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 177, + 502 + ], + "score": 1.0, + "content": "needs to evaluate", + "type": "text" + }, + { + "bbox": [ + 178, + 488, + 241, + 502 + ], + "score": 0.93, + "content": "\\tilde { \\mathcal { O } } ( 1 / ( L _ { 2 } \\epsilon ) ^ { 0 . 2 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 488, + 389, + 502 + ], + "score": 1.0, + "content": "Hessian-vector products of the form", + "type": "text" + }, + { + "bbox": [ + 390, + 489, + 410, + 500 + ], + "score": 0.88, + "content": "\\mathbf { H } ^ { t } \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 488, + 414, + 502 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 505, + 506, + 634 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 345, + 518 + ], + "score": 1.0, + "content": "In MetaAlgorithm 7, to verify whether the candidate solution", + "type": "text" + }, + { + "bbox": [ + 345, + 506, + 356, + 516 + ], + "score": 0.86, + "content": "\\mathbf { v } _ { } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 505, + 405, + 518 + ], + "score": 1.0, + "content": "is indeed an", + "type": "text" + }, + { + "bbox": [ + 405, + 506, + 426, + 518 + ], + "score": 0.85, + "content": "\\mathcal { O } ( \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 505, + 506, + 518 + ], + "score": 1.0, + "content": "-SOSP, one needs at", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 515, + 508, + 531 + ], + "spans": [ + { + "bbox": [ + 104, + 515, + 128, + 531 + ], + "score": 1.0, + "content": "most", + "type": "text" + }, + { + "bbox": [ + 128, + 517, + 151, + 529 + ], + "score": 0.91, + "content": "{ \\mathcal { O } } ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 515, + 291, + 531 + ], + "score": 1.0, + "content": "stochastic gradient evaluations and", + "type": "text" + }, + { + "bbox": [ + 291, + 516, + 469, + 529 + ], + "score": 0.92, + "content": "\\mathcal { O } ( \\operatorname* { m i n } \\{ n , \\log ( 4 d / \\delta ) L _ { 1 } ^ { 2 } / ( L _ { 2 } \\epsilon ) \\} / ( L _ { 2 } \\epsilon ) ^ { 0 . 2 5 } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 515, + 508, + 531 + ], + "score": 1.0, + "content": "stochas-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 528, + 507, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 507, + 541 + ], + "score": 1.0, + "content": "tic Hessian-vector product evaluations, where the latter one follows from the proof of Theorem 7.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "score": 1.0, + "content": "We proceed to analyze the computational complexity of the INEXACTTRWEAK procedure. Recall that", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 548, + 507, + 563 + ], + "spans": [ + { + "bbox": [ + 104, + 548, + 295, + 563 + ], + "score": 1.0, + "content": "the iteration complexity of MetaAlgorithm 7 is", + "type": "text" + }, + { + "bbox": [ + 295, + 550, + 366, + 562 + ], + "score": 0.91, + "content": "\\mathcal { O } ( \\log ( 1 / \\delta ) / \\epsilon ^ { 1 . 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 548, + 507, + 563 + ], + "score": 1.0, + "content": ". Following Lemma 4.2 and Corol-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 560, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 104, + 560, + 321, + 576 + ], + "score": 1.0, + "content": "lary 4.1, the stochastic first-order oracle complexity is", + "type": "text" + }, + { + "bbox": [ + 321, + 562, + 457, + 575 + ], + "score": 0.87, + "content": "\\tilde { \\mathcal { O } } ( \\operatorname* { m i n } \\{ n / \\epsilon ^ { 1 . 5 } , \\sqrt { n } / \\epsilon ^ { 2 } \\} \\mathrm { l o g } ( 1 / \\delta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 560, + 506, + 576 + ], + "score": 1.0, + "content": ". Following", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 573, + 507, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 301, + 587 + ], + "score": 1.0, + "content": "the proof of Lemma 4.1 and Corollary 4.1, when", + "type": "text" + }, + { + "bbox": [ + 301, + 574, + 330, + 585 + ], + "score": 0.91, + "content": "p _ { 2 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 573, + 507, + 587 + ], + "score": 1.0, + "content": ", the overall stochastic Hessian sample com-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 583, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 104, + 583, + 145, + 599 + ], + "score": 1.0, + "content": "plexity is", + "type": "text" + }, + { + "bbox": [ + 145, + 584, + 279, + 598 + ], + "score": 0.9, + "content": "\\tilde { \\mathcal { O } } ( \\operatorname* { m i n } \\{ n / \\epsilon ^ { 1 . 5 } , 1 / \\epsilon ^ { 2 . 5 } \\} \\log ( 1 / \\delta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 583, + 339, + 599 + ], + "score": 1.0, + "content": ". 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By Lemma C.1 and our choice of", + "type": "text" + }, + { + "bbox": [ + 284, + 83, + 293, + 93 + ], + "score": 0.81, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 81, + 369, + 95 + ], + "score": 1.0, + "content": ", with probability", + "type": "text" + }, + { + "bbox": [ + 370, + 82, + 411, + 95 + ], + "score": 0.91, + "content": "\\left( 1 - \\delta / 2 \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 81, + 482, + 95 + ], + "score": 1.0, + "content": ", at least one of", + "type": "text" + }, + { + "bbox": [ + 482, + 82, + 493, + 93 + ], + "score": 0.87, + "content": "\\mathbf { x } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 92, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 92, + 114, + 108 + ], + "score": 1.0, + "content": "a", + "type": "text" + }, + { + "bbox": [ + 114, + 95, + 135, + 105 + ], + "score": 0.56, + "content": "5 0 0 \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 92, + 273, + 108 + ], + "score": 1.0, + "content": "-SOSP. 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This completes the proof.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 350, + 387, + 365 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 378, + 253, + 391 + ], + "lines": [ + { + "bbox": [ + 105, + 377, + 255, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 255, + 393 + ], + "score": 1.0, + "content": "D PROOF OF THEOREM 6.1", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 402, + 506, + 501 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 506, + 416 + ], + "score": 1.0, + "content": "Proof. We first analyze the computational cost of Lanczos method. By Corollary 2 in (Carmon &", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 102, + 411, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 102, + 411, + 446, + 436 + ], + "score": 1.0, + "content": "Duchi, 2018), for any desired accuracy \u000f˜, Lanczos method achieves this accuracy in", + "type": "text" + }, + { + "bbox": [ + 446, + 414, + 505, + 432 + ], + "score": 0.95, + "content": "\\begin{array} { r } { \\mathcal { O } ( \\frac { r } { \\sqrt { \\tilde { \\epsilon } } } \\mathrm { l o g } \\frac { r \\sqrt { d } } { \\tilde { \\epsilon } p } ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 234, + 443 + ], + "score": 1.0, + "content": "Lanczos iterations w.p. at least", + "type": "text" + }, + { + "bbox": [ + 235, + 430, + 265, + 443 + ], + "score": 0.93, + "content": "( 1 - p )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 430, + 506, + 443 + ], + "score": 1.0, + "content": ". Without loss of generality, we assume that the number of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 441, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 339, + 455 + ], + "score": 1.0, + "content": "Lanczos iterations is strictly smaller than the dimension", + "type": "text" + }, + { + "bbox": [ + 339, + 442, + 345, + 452 + ], + "score": 0.76, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 441, + 506, + 455 + ], + "score": 1.0, + "content": ", otherwise the QCQP subproblem can", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 452, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 465 + ], + "score": 1.0, + "content": "be solved exactly. We note that each Lanczos iteration involves computation of one matrix-vector", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 463, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 104, + 463, + 440, + 478 + ], + "score": 1.0, + "content": "product. Therefore, to satisfy the condition (24) in Lemma C.1, one needs to evaluate", + "type": "text" + }, + { + "bbox": [ + 441, + 463, + 505, + 477 + ], + "score": 0.91, + "content": "\\tilde { \\mathcal { O } } ( 1 / ( L _ { 2 } \\epsilon ) ^ { 0 . 2 5 } )", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 475, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 252, + 491 + ], + "score": 1.0, + "content": "Hessian-vector products of the form", + "type": "text" + }, + { + "bbox": [ + 253, + 476, + 274, + 487 + ], + "score": 0.89, + "content": "\\mathbf { H } ^ { k } \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 475, + 433, + 491 + ], + "score": 1.0, + "content": ". Similarly, to solve (10) up to accuracy", + "type": "text" + }, + { + "bbox": [ + 433, + 477, + 458, + 488 + ], + "score": 0.91, + "content": "\\sqrt { \\epsilon L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 475, + 506, + 491 + ], + "score": 1.0, + "content": "w.h.p., one", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 488, + 414, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 177, + 502 + ], + "score": 1.0, + "content": "needs to evaluate", + "type": "text" + }, + { + "bbox": [ + 178, + 488, + 241, + 502 + ], + "score": 0.93, + "content": "\\tilde { \\mathcal { O } } ( 1 / ( L _ { 2 } \\epsilon ) ^ { 0 . 2 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 488, + 389, + 502 + ], + "score": 1.0, + "content": "Hessian-vector products of the form", + "type": "text" + }, + { + "bbox": [ + 390, + 489, + 410, + 500 + ], + "score": 0.88, + "content": "\\mathbf { H } ^ { t } \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 488, + 414, + 502 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5, + "bbox_fs": [ + 102, + 403, + 506, + 502 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 505, + 506, + 634 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 345, + 518 + ], + "score": 1.0, + "content": "In MetaAlgorithm 7, to verify whether the candidate solution", + "type": "text" + }, + { + "bbox": [ + 345, + 506, + 356, + 516 + ], + "score": 0.86, + "content": "\\mathbf { v } _ { } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 505, + 405, + 518 + ], + "score": 1.0, + "content": "is indeed an", + "type": "text" + }, + { + "bbox": [ + 405, + 506, + 426, + 518 + ], + "score": 0.85, + "content": "\\mathcal { O } ( \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 505, + 506, + 518 + ], + "score": 1.0, + "content": "-SOSP, one needs at", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 515, + 508, + 531 + ], + "spans": [ + { + "bbox": [ + 104, + 515, + 128, + 531 + ], + "score": 1.0, + "content": "most", + "type": "text" + }, + { + "bbox": [ + 128, + 517, + 151, + 529 + ], + "score": 0.91, + "content": "{ \\mathcal { O } } ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 515, + 291, + 531 + ], + "score": 1.0, + "content": "stochastic gradient evaluations and", + "type": "text" + }, + { + "bbox": [ + 291, + 516, + 469, + 529 + ], + "score": 0.92, + "content": "\\mathcal { O } ( \\operatorname* { m i n } \\{ n , \\log ( 4 d / \\delta ) L _ { 1 } ^ { 2 } / ( L _ { 2 } \\epsilon ) \\} / ( L _ { 2 } \\epsilon ) ^ { 0 . 2 5 } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 515, + 508, + 531 + ], + "score": 1.0, + "content": "stochas-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 528, + 507, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 507, + 541 + ], + "score": 1.0, + "content": "tic Hessian-vector product evaluations, where the latter one follows from the proof of Theorem 7.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "score": 1.0, + "content": "We proceed to analyze the computational complexity of the INEXACTTRWEAK procedure. Recall that", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 548, + 507, + 563 + ], + "spans": [ + { + "bbox": [ + 104, + 548, + 295, + 563 + ], + "score": 1.0, + "content": "the iteration complexity of MetaAlgorithm 7 is", + "type": "text" + }, + { + "bbox": [ + 295, + 550, + 366, + 562 + ], + "score": 0.91, + "content": "\\mathcal { O } ( \\log ( 1 / \\delta ) / \\epsilon ^ { 1 . 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 548, + 507, + 563 + ], + "score": 1.0, + "content": ". Following Lemma 4.2 and Corol-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 560, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 104, + 560, + 321, + 576 + ], + "score": 1.0, + "content": "lary 4.1, the stochastic first-order oracle complexity is", + "type": "text" + }, + { + "bbox": [ + 321, + 562, + 457, + 575 + ], + "score": 0.87, + "content": "\\tilde { \\mathcal { O } } ( \\operatorname* { m i n } \\{ n / \\epsilon ^ { 1 . 5 } , \\sqrt { n } / \\epsilon ^ { 2 } \\} \\mathrm { l o g } ( 1 / \\delta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 560, + 506, + 576 + ], + "score": 1.0, + "content": ". Following", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 573, + 507, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 301, + 587 + ], + "score": 1.0, + "content": "the proof of Lemma 4.1 and Corollary 4.1, when", + "type": "text" + }, + { + "bbox": [ + 301, + 574, + 330, + 585 + ], + "score": 0.91, + "content": "p _ { 2 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 573, + 507, + 587 + ], + "score": 1.0, + "content": ", the overall stochastic Hessian sample com-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 583, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 104, + 583, + 145, + 599 + ], + "score": 1.0, + "content": "plexity is", + "type": "text" + }, + { + "bbox": [ + 145, + 584, + 279, + 598 + ], + "score": 0.9, + "content": "\\tilde { \\mathcal { O } } ( \\operatorname* { m i n } \\{ n / \\epsilon ^ { 1 . 5 } , 1 / \\epsilon ^ { 2 . 5 } \\} \\log ( 1 / \\delta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 583, + 339, + 599 + ], + "score": 1.0, + "content": ". Since it takes", + "type": "text" + }, + { + "bbox": [ + 339, + 584, + 384, + 598 + ], + "score": 0.94, + "content": "\\tilde { \\mathcal { O } } ( 1 / \\epsilon ^ { 0 . 2 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 583, + 506, + 599 + ], + "score": 1.0, + "content": "Lanczos iterations to meet the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 596, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 506, + 610 + ], + "score": 1.0, + "content": "condition (24) as stated above, the overall stochastic Hessian-vector product oracle complexity is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 606, + 507, + 623 + ], + "spans": [ + { + "bbox": [ + 107, + 608, + 249, + 621 + ], + "score": 0.91, + "content": "\\tilde { \\mathcal { O } } ( \\operatorname* { m i n } \\{ n / \\epsilon ^ { 1 . 7 5 } , 1 / \\epsilon ^ { 2 . 7 5 } \\} \\log ( 1 / \\delta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 606, + 507, + 623 + ], + "score": 1.0, + "content": ". Combining the stochastic first-order and Hessian-vector product", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 619, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 249, + 636 + ], + "score": 1.0, + "content": "complexities, the overall runtime is", + "type": "text" + }, + { + "bbox": [ + 250, + 620, + 438, + 634 + ], + "score": 0.93, + "content": "\\tilde { \\mathcal { O } } ( d \\operatorname* { m i n } \\{ n / \\epsilon ^ { 1 . 7 5 } , 1 / \\epsilon ^ { 2 . 7 5 } + \\sqrt { n } / \\epsilon ^ { 2 } \\} \\log ( 1 / \\delta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 619, + 443, + 636 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 622, + 506, + 634 + ], + "score": 0.993, + "content": "□", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31, + "bbox_fs": [ + 104, + 505, + 508, + 636 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 648, + 397, + 662 + ], + "lines": [ + { + "bbox": [ + 104, + 648, + 398, + 665 + ], + "spans": [ + { + "bbox": [ + 104, + 648, + 398, + 665 + ], + "score": 1.0, + "content": "E A FASTER HESSIAN-VECTOR BASED QCQP SOLVER", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 673, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 672, + 506, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 471, + 687 + ], + "score": 1.0, + "content": "We recall from the previous section that, to approximately solve a quadratic subproblem in", + "type": "text" + }, + { + "bbox": [ + 472, + 674, + 502, + 685 + ], + "score": 0.86, + "content": "{ \\mathrm { { S T R } } } _ { \\mathrm { { f r e e } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 672, + 506, + 687 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 683, + 507, + 700 + ], + "spans": [ + { + "bbox": [ + 104, + 683, + 210, + 700 + ], + "score": 1.0, + "content": "Lanczos method requires", + "type": "text" + }, + { + "bbox": [ + 210, + 684, + 316, + 698 + ], + "score": 0.9, + "content": "\\tilde { \\mathcal { O } } ( \\operatorname* { m i n } \\{ n / \\epsilon ^ { 0 . 2 5 } , 1 / \\bar { \\epsilon } ^ { 1 . 2 5 } \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 683, + 507, + 700 + ], + "score": 1.0, + "content": "stochastic Hessian-vector product evaluations.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 695, + 508, + 713 + ], + "spans": [ + { + "bbox": [ + 104, + 695, + 333, + 713 + ], + "score": 1.0, + "content": "In this section, we propose a faster QCQP solver with an", + "type": "text" + }, + { + "bbox": [ + 333, + 697, + 456, + 711 + ], + "score": 0.91, + "content": "\\tilde { \\mathcal { O } } ( \\operatorname* { m i n } \\{ n + n ^ { 0 . 7 5 } / \\epsilon ^ { 0 . 2 5 } , 1 / \\epsilon \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 695, + 508, + 713 + ], + "score": 1.0, + "content": "complexity.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 318, + 722 + ], + "score": 1.0, + "content": "Replacing Lanczos method with this QCQP solver in", + "type": "text" + }, + { + "bbox": [ + 318, + 710, + 349, + 721 + ], + "score": 0.78, + "content": "\\operatorname { S T R } _ { \\mathrm { f r e e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "results in a faster Hessian-free method,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 719, + 230, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 189, + 734 + ], + "score": 1.0, + "content": "which we refer to as", + "type": "text" + }, + { + "bbox": [ + 190, + 721, + 226, + 732 + ], + "score": 0.9, + "content": "{ \\mathrm { S T R } } _ { \\mathrm { f r e e } } +", + "type": "inline_equation" + }, + { + "bbox": [ + 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], + "score": 1.0, + "content": "We summarize this result in Lemma E.1 which shows that the trust region subproblem is equivalent", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 126, + 184, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 184, + 137 + ], + "score": 1.0, + "content": "to a convex QCQP.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 138, + 506, + 173 + ], + "lines": [ + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "Lemma E.1. (Convex Reformulation of QCQP (Flippo & Jansen, 1996; Wang & Xia, 2017)) Denote", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 149, + 506, + 164 + ], + "spans": [ + { + "bbox": [ + 107, + 150, + 127, + 161 + ], + "score": 0.87, + "content": "\\lambda _ { \\mathrm { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 149, + 242, + 164 + ], + "score": 1.0, + "content": "as the smallest eigenvalue of", + "type": "text" + }, + { + "bbox": [ + 243, + 149, + 257, + 160 + ], + "score": 0.88, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 149, + 276, + 164 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 277, + 151, + 298, + 161 + ], + "score": 0.81, + "content": "\\mathbf { u } _ { \\mathrm { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 149, + 506, + 164 + ], + "score": 1.0, + "content": "be a corresponding eigenvector. W.l.o.g., we assume", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 159, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 125, + 174 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 125, + 160, + 187, + 173 + ], + "score": 0.91, + "content": "\\langle \\mathbf { g } ^ { k } , \\mathbf { u } _ { \\mathrm { m i n } } \\rangle \\leq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 159, + 206, + 174 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 206, + 161, + 281, + 173 + ], + "score": 0.92, + "content": "\\mu = \\operatorname* { m i n } \\{ \\lambda _ { \\operatorname* { m i n } } , 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 159, + 506, + 174 + ], + "score": 1.0, + "content": ". Then the QCQP (8) is equivalent to the convex problem", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 184, + 179, + 425, + 205 + ], + "lines": [ + { + "bbox": [ + 184, + 179, + 425, + 205 + ], + "spans": [ + { + "bbox": [ + 184, + 179, + 425, + 205 + ], + "score": 0.93, + "content": "\\operatorname* { m i n } _ { \\substack { \\mathbf { h } \\in \\mathbb { R } ^ { d } , \\| \\mathbf { h } \\| \\leq r } } q ^ { k } ( \\mathbf { h } ) = \\langle \\mathbf { g } ^ { k } , \\mathbf { h } \\rangle + \\frac { 1 } { 2 } \\langle ( \\mathbf { H } ^ { k } - \\mu I ) \\mathbf { h } , \\mathbf { h } \\rangle + \\frac { 1 } { 2 } \\mu r ^ { 2 }", + "type": "interline_equation", + "image_path": "3c259b5f80ea3f9312154d89b17ab811f991e2da8e57c54e7a7d957cbd040c5d.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 184, + 179, + 425, + 205 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 206, + 506, + 229 + ], + "lines": [ + { + "bbox": [ + 105, + 204, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 448, + 219 + ], + "score": 1.0, + "content": "in the sense that (8) and (38) have the same minimum function value. Moreover, when", + "type": "text" + }, + { + "bbox": [ + 449, + 206, + 488, + 217 + ], + "score": 0.89, + "content": "\\lambda _ { \\operatorname* { m i n } } < 0 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 204, + 506, + 219 + ], + "score": 1.0, + "content": ", for", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 216, + 286, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 270, + 230 + ], + "score": 1.0, + "content": "any optimal solution of (38), denoted by", + "type": "text" + }, + { + "bbox": [ + 270, + 216, + 282, + 229 + ], + "score": 0.89, + "content": "\\mathbf { h } _ { c } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 216, + 286, + 230 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 231, + 438, + 260 + ], + "lines": [ + { + "bbox": [ + 173, + 231, + 438, + 260 + ], + "spans": [ + { + "bbox": [ + 173, + 231, + 438, + 260 + ], + "score": 0.93, + "content": "\\mathbf { h } _ { c } ^ { k } + \\frac { \\sqrt { \\langle \\mathbf { h } _ { c } ^ { k } , \\mathbf { u } _ { \\mathrm { m i n } } \\rangle ^ { 2 } - \\| \\mathbf { u } _ { \\mathrm { m i n } } \\| ^ { 2 } \\big ( \\| \\mathbf { h } _ { c } ^ { k } \\| ^ { 2 } - r ^ { 2 } \\big ) } - \\langle \\mathbf { h } _ { c } ^ { k } , \\mathbf { u } _ { \\mathrm { m i n } } \\rangle } { \\| \\mathbf { u } _ { \\mathrm { m i n } } \\| ^ { 2 } } \\mathbf { u } _ { \\mathrm { m i n } }", + "type": "interline_equation", + "image_path": "4a84f0a66a70d1fd6e8264a455b9841b44c7da5a8599d395dd6b5468a6e797b6.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 173, + 231, + 438, + 260 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 262, + 297, + 274 + ], + "lines": [ + { + "bbox": [ + 105, + 261, + 299, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 299, + 276 + ], + "score": 1.0, + "content": "is a global minimizer of the original QCQP (8).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 281, + 504, + 306 + ], + "lines": [ + { + "bbox": [ + 105, + 281, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 428, + 295 + ], + "score": 1.0, + "content": "To perform the above reformulation, one needs to compute the exact eigenpair", + "type": "text" + }, + { + "bbox": [ + 428, + 282, + 479, + 293 + ], + "score": 0.85, + "content": "\\left( \\lambda _ { \\operatorname* { m i n } } , \\mathbf { u } _ { \\operatorname* { m i n } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 281, + 505, + 295 + ], + "score": 1.0, + "content": ". Nev-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 293, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 437, + 308 + ], + "score": 1.0, + "content": "ertheless, as we shall see, it is sufficient to compute an approximate eigenpair", + "type": "text" + }, + { + "bbox": [ + 438, + 294, + 462, + 307 + ], + "score": 0.92, + "content": "( \\tilde { \\lambda } , \\tilde { \\mathbf { u } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 293, + 506, + 308 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 313, + 374, + 328 + ], + "lines": [ + { + "bbox": [ + 236, + 313, + 374, + 328 + ], + "spans": [ + { + "bbox": [ + 236, + 313, + 374, + 328 + ], + "score": 0.92, + "content": "\\lambda _ { \\operatorname* { m i n } } \\leq \\tilde { \\lambda } = \\tilde { \\mathbf { u } } ^ { T } \\mathbf { H } ^ { k } \\tilde { \\mathbf { u } } \\leq \\lambda _ { \\operatorname* { m i n } } + \\tilde { \\epsilon } ,", + "type": "interline_equation", + "image_path": "a85cc13309ad658a543ec13c6a3f9f3866bf41c82b4eb5baea8eb426643121dc.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 236, + 313, + 374, + 328 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 330, + 505, + 352 + ], + "lines": [ + { + "bbox": [ + 105, + 328, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 362, + 343 + ], + "score": 1.0, + "content": "where \u000f˜ is a target accuracy to be determined later. We note that", + "type": "text" + }, + { + "bbox": [ + 363, + 330, + 397, + 341 + ], + "score": 0.91, + "content": "\\tilde { \\epsilon } \\leq 2 L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 328, + 453, + 343 + ], + "score": 1.0, + "content": "w.l.o.g. since", + "type": "text" + }, + { + "bbox": [ + 454, + 329, + 503, + 342 + ], + "score": 0.93, + "content": "\\| \\mathbf { H } ^ { k } \\| \\leq L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 328, + 506, + 343 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 341, + 437, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 437, + 352 + ], + "score": 1.0, + "content": "With this approximate eigenpair, it remains to solve the following convex problem", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 184, + 354, + 425, + 380 + ], + "lines": [ + { + "bbox": [ + 184, + 354, + 425, + 380 + ], + "spans": [ + { + "bbox": [ + 184, + 354, + 425, + 380 + ], + "score": 0.93, + "content": "\\operatorname* { m i n } _ { \\mathbf { h } \\in \\mathbb { R } ^ { d } , \\| \\mathbf { h } \\| \\leq r } \\tilde { q } ^ { k } ( \\mathbf { h } ) = \\langle \\mathbf { g } ^ { k } , \\mathbf { h } \\rangle + \\frac { 1 } { 2 } \\langle ( \\mathbf { H } ^ { k } - \\tilde { \\mu } I ) \\mathbf { h } , \\mathbf { h } \\rangle + \\frac { 1 } { 2 } \\tilde { \\mu } r ^ { 2 }", + "type": "interline_equation", + "image_path": "92370e579ccde63752f9debd6649e54b60e3b2dda0048a056aaecb071c1a9b48.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 184, + 354, + 425, + 380 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 462, + 397 + ], + "lines": [ + { + "bbox": [ + 105, + 383, + 462, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 133, + 399 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 383, + 212, + 397 + ], + "score": 0.93, + "content": "\\tilde { \\mu } = \\mathrm { m i n } \\{ 0 , \\tilde { \\lambda } - \\tilde { \\epsilon } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 384, + 462, + 399 + ], + "score": 1.0, + "content": ". One can check that the problem (41) well approximates (38).", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 399, + 503, + 423 + ], + "lines": [ + { + "bbox": [ + 106, + 397, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 187, + 413 + ], + "score": 1.0, + "content": "Corollary E.1. Let", + "type": "text" + }, + { + "bbox": [ + 187, + 398, + 198, + 411 + ], + "score": 0.87, + "content": "q _ { * } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 397, + 216, + 413 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 216, + 398, + 227, + 411 + ], + "score": 0.86, + "content": "\\tilde { q } _ { * } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 397, + 505, + 413 + ], + "score": 1.0, + "content": "be the minimum function value of (38) and (41), respectively. Assume", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 410, + 269, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 243, + 424 + ], + "score": 0.9, + "content": "\\lambda _ { \\operatorname* { m i n } } \\leq \\tilde { \\lambda } = \\tilde { \\mathbf { u } } ^ { T } \\mathbf { H } ^ { k } \\tilde { \\mathbf { u } } \\leq \\lambda _ { \\operatorname* { m i n } } + \\tilde { \\epsilon } .", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 411, + 269, + 425 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "interline_equation", + "bbox": [ + 271, + 425, + 340, + 440 + ], + "lines": [ + { + "bbox": [ + 271, + 425, + 340, + 440 + ], + "spans": [ + { + "bbox": [ + 271, + 425, + 340, + 440 + ], + "score": 0.92, + "content": "| q _ { * } ^ { k } - \\tilde { q } _ { * } ^ { k } | \\leq \\tilde { \\epsilon } r ^ { 2 } .", + "type": "interline_equation", + "image_path": "144b1097a8e6cfa77799b3ee5d9151f9f89fa95e3828e733548c43633f5362a3.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 271, + 425, + 340, + 440 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 447, + 506, + 494 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "We note that the above convex reformulation approach divides an indefinite QCQP into two subprob-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 459, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 312, + 474 + ], + "score": 1.0, + "content": "lems: (i) computation of an approximate eigenpair", + "type": "text" + }, + { + "bbox": [ + 312, + 459, + 336, + 472 + ], + "score": 0.91, + "content": "( \\tilde { \\lambda } , \\tilde { \\mathbf { u } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 460, + 506, + 474 + ], + "score": 1.0, + "content": "; (ii) solving the convex problem (41). As", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 365, + 484 + ], + "score": 1.0, + "content": "we shall see, by exploiting the finite-sum structure of the Hessian", + "type": "text" + }, + { + "bbox": [ + 366, + 471, + 380, + 482 + ], + "score": 0.88, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 471, + 506, + 484 + ], + "score": 1.0, + "content": ", these two subproblems can be", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 482, + 496, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 496, + 495 + ], + "score": 1.0, + "content": "efficiently solved. We treat these two subproblems in the following two subsections, respectively.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "title", + "bbox": [ + 108, + 506, + 303, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 304, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 304, + 519 + ], + "score": 1.0, + "content": "E.2 FINDING THE SMALLEST EIGENVECTOR", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 527, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 526, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 505, + 540 + ], + "score": 1.0, + "content": "To find a unit vector that satisfies requirement (40), we resort to the AppxPCA method (Allen-Zhu", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 537, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 104, + 537, + 338, + 552 + ], + "score": 1.0, + "content": "& Li, 2016), which first finds an approximate eigenvalue", + "type": "text" + }, + { + "bbox": [ + 339, + 539, + 395, + 549 + ], + "score": 0.9, + "content": "\\lambda = \\lambda _ { \\operatorname* { m i n } } - \\tilde { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 537, + 506, + 552 + ], + "score": 1.0, + "content": "via binary search and then", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 548, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 104, + 548, + 327, + 562 + ], + "score": 1.0, + "content": "applies Power method to the positive definite matrix", + "type": "text" + }, + { + "bbox": [ + 327, + 549, + 384, + 561 + ], + "score": 0.91, + "content": "( \\mathbf { H } ^ { k } - \\lambda I ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 548, + 506, + 562 + ], + "score": 1.0, + "content": "for a logarithmic number of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 558, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 104, + 558, + 195, + 574 + ], + "score": 1.0, + "content": "iterations. Computing", + "type": "text" + }, + { + "bbox": [ + 196, + 560, + 258, + 573 + ], + "score": 0.92, + "content": "( \\mathbf { H } ^ { k } - \\lambda I ) ^ { - 1 } \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 558, + 316, + 574 + ], + "score": 1.0, + "content": "for any vector", + "type": "text" + }, + { + "bbox": [ + 317, + 563, + 324, + 570 + ], + "score": 0.71, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 558, + 433, + 574 + ], + "score": 1.0, + "content": "is equivalent to solving the", + "type": "text" + }, + { + "bbox": [ + 433, + 561, + 439, + 570 + ], + "score": 0.41, + "content": "\\tilde { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 558, + 506, + 574 + ], + "score": 1.0, + "content": "-strongly convex", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 570, + 240, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 240, + 584 + ], + "score": 1.0, + "content": "problem (Allen-Zhu & Li, 2018)", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 585, + 390, + 609 + ], + "lines": [ + { + "bbox": [ + 219, + 585, + 390, + 609 + ], + "spans": [ + { + "bbox": [ + 219, + 585, + 390, + 609 + ], + "score": 0.94, + "content": "\\operatorname* { m i n } _ { \\mathbf { u } } \\phi ^ { k } ( \\mathbf { u } ) : = \\frac { 1 } { 2 } \\mathbf { u } ^ { T } ( \\mathbf { H } ^ { k } - \\lambda I ) \\mathbf { u } - \\langle \\mathbf { v } , \\mathbf { u } \\rangle", + "type": "interline_equation", + "image_path": "dac887be88ba8de15c2391acc4b84c9a52c388b533f2345e8611cffc16c8fef5.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 219, + 585, + 390, + 609 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 611, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 161, + 627 + ], + "score": 1.0, + "content": "We note that", + "type": "text" + }, + { + "bbox": [ + 162, + 611, + 272, + 627 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathbf { H } ^ { k } = \\frac { 1 } { | \\cal { S } | } \\sum _ { i \\in \\cal { S } } \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 610, + 342, + 627 + ], + "score": 1.0, + "content": ". Specifically, in", + "type": "text" + }, + { + "bbox": [ + 343, + 612, + 374, + 624 + ], + "score": 0.87, + "content": "\\mathrm { { S T R } _ { \\mathrm { { f r e e } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 610, + 405, + 627 + ], + "score": 1.0, + "content": ", either", + "type": "text" + }, + { + "bbox": [ + 405, + 612, + 440, + 624 + ], + "score": 0.92, + "content": "| S | = n", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 610, + 463, + 627 + ], + "score": 1.0, + "content": "(i.e.,", + "type": "text" + }, + { + "bbox": [ + 463, + 612, + 478, + 623 + ], + "score": 0.88, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 610, + 506, + 627 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 173, + 641 + ], + "score": 1.0, + "content": "full Hessian) or", + "type": "text" + }, + { + "bbox": [ + 173, + 627, + 255, + 640 + ], + "score": 0.92, + "content": "| S | = \\tilde { \\mathcal { O } } ( L _ { 1 } ^ { 2 } / ( L _ { 2 } \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 626, + 367, + 641 + ], + "score": 1.0, + "content": "by Lemma 4.1. Therefore,", + "type": "text" + }, + { + "bbox": [ + 368, + 627, + 390, + 640 + ], + "score": 0.91, + "content": "\\phi ^ { k } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 626, + 506, + 641 + ], + "score": 1.0, + "content": "can be expressed as sum of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 639, + 195, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 195, + 650 + ], + "score": 1.0, + "content": "non-convex functions", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34 + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 650, + 452, + 680 + ], + "lines": [ + { + "bbox": [ + 159, + 650, + 452, + 680 + ], + "spans": [ + { + "bbox": [ + 159, + 650, + 452, + 680 + ], + "score": 0.94, + "content": "\\phi ^ { k } ( \\mathbf { u } ) = \\frac { 1 } { | S | } \\sum _ { i \\in S } \\phi _ { i } ^ { k } ( \\mathbf { u } ) = \\frac { 1 } { | S | } \\sum _ { i \\in S } \\left( \\frac { 1 } { 2 } \\mathbf { u } ^ { T } ( \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) - \\lambda I ) \\mathbf { u } - \\langle \\mathbf { v } , \\mathbf { u } \\rangle \\right) .", + "type": "interline_equation", + "image_path": "eec1192412c143e85db823e72aa5225f155ff2f2d760377afc4b2cc2ceb28aab.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 159, + 650, + 452, + 660.0 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 159, + 660.0, + 452, + 670.0 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 159, + 670.0, + 452, + 680.0 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 683, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 682, + 506, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 215, + 697 + ], + "score": 1.0, + "content": "By observing that each", + "type": "text" + }, + { + "bbox": [ + 216, + 683, + 228, + 696 + ], + "score": 0.9, + "content": "\\phi _ { i } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 682, + 338, + 697 + ], + "score": 1.0, + "content": "is non-convex and has", + "type": "text" + }, + { + "bbox": [ + 338, + 684, + 363, + 696 + ], + "score": 0.88, + "content": "\\left( 4 L _ { 2 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 682, + 506, + 697 + ], + "score": 1.0, + "content": "-Lipschitz gradient, we can use", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 695, + 507, + 710 + ], + "spans": [ + { + "bbox": [ + 106, + 695, + 332, + 709 + ], + "score": 1.0, + "content": "KatyushaXS (Allen-Zhu, 2018a) to solve problem (43) in", + "type": "text" + }, + { + "bbox": [ + 333, + 696, + 428, + 710 + ], + "score": 0.92, + "content": "\\tilde { \\mathcal { O } } ( | S | + | S | ^ { 3 / 4 } \\sqrt { L _ { 2 } / \\tilde { \\epsilon } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 695, + 507, + 709 + ], + "score": 1.0, + "content": "stochastic Hessian-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 190, + 722 + ], + "score": 1.0, + "content": "vector product (i.e.,", + "type": "text" + }, + { + "bbox": [ + 190, + 709, + 241, + 722 + ], + "score": 0.92, + "content": "\\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) \\mathbf { u } )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "evaluations. The following result is taken from (Agarwal et al.,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 720, + 438, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 438, + 733 + ], + "score": 1.0, + "content": "2017, Section G.3), which gives the overall computation complexity of AppxPCA.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40.5 + } + ], + "page_idx": 18, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "19", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 292, + 94 + ], + "lines": [ + { + "bbox": [ + 104, + 80, + 294, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 80, + 294, + 96 + ], + "score": 1.0, + "content": "E.1 CONVEX REFORMULATION OF QCQP", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 102, + 504, + 136 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "score": 1.0, + "content": "To begin with, we present a known result that is key to achieve faster algorithm than Lanczos method.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 126 + ], + "score": 1.0, + "content": "We summarize this result in Lemma E.1 which shows that the trust region subproblem is equivalent", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 126, + 184, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 184, + 137 + ], + "score": 1.0, + "content": "to a convex QCQP.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 102, + 506, + 137 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 138, + 506, + 173 + ], + "lines": [ + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "Lemma E.1. (Convex Reformulation of QCQP (Flippo & Jansen, 1996; Wang & Xia, 2017)) Denote", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 149, + 506, + 164 + ], + "spans": [ + { + "bbox": [ + 107, + 150, + 127, + 161 + ], + "score": 0.87, + "content": "\\lambda _ { \\mathrm { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 149, + 242, + 164 + ], + "score": 1.0, + "content": "as the smallest eigenvalue of", + "type": "text" + }, + { + "bbox": [ + 243, + 149, + 257, + 160 + ], + "score": 0.88, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 149, + 276, + 164 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 277, + 151, + 298, + 161 + ], + "score": 0.81, + "content": "\\mathbf { u } _ { \\mathrm { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 149, + 506, + 164 + ], + "score": 1.0, + "content": "be a corresponding eigenvector. W.l.o.g., we assume", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 159, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 125, + 174 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 125, + 160, + 187, + 173 + ], + "score": 0.91, + "content": "\\langle \\mathbf { g } ^ { k } , \\mathbf { u } _ { \\mathrm { m i n } } \\rangle \\leq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 159, + 206, + 174 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 206, + 161, + 281, + 173 + ], + "score": 0.92, + "content": "\\mu = \\operatorname* { m i n } \\{ \\lambda _ { \\operatorname* { m i n } } , 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 159, + 506, + 174 + ], + "score": 1.0, + "content": ". Then the QCQP (8) is equivalent to the convex problem", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 137, + 506, + 174 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 184, + 179, + 425, + 205 + ], + "lines": [ + { + "bbox": [ + 184, + 179, + 425, + 205 + ], + "spans": [ + { + "bbox": [ + 184, + 179, + 425, + 205 + ], + "score": 0.93, + "content": "\\operatorname* { m i n } _ { \\substack { \\mathbf { h } \\in \\mathbb { R } ^ { d } , \\| \\mathbf { h } \\| \\leq r } } q ^ { k } ( \\mathbf { h } ) = \\langle \\mathbf { g } ^ { k } , \\mathbf { h } \\rangle + \\frac { 1 } { 2 } \\langle ( \\mathbf { H } ^ { k } - \\mu I ) \\mathbf { h } , \\mathbf { h } \\rangle + \\frac { 1 } { 2 } \\mu r ^ { 2 }", + "type": "interline_equation", + "image_path": "3c259b5f80ea3f9312154d89b17ab811f991e2da8e57c54e7a7d957cbd040c5d.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 184, + 179, + 425, + 205 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 206, + 506, + 229 + ], + "lines": [ + { + "bbox": [ + 105, + 204, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 448, + 219 + ], + "score": 1.0, + "content": "in the sense that (8) and (38) have the same minimum function value. Moreover, when", + "type": "text" + }, + { + "bbox": [ + 449, + 206, + 488, + 217 + ], + "score": 0.89, + "content": "\\lambda _ { \\operatorname* { m i n } } < 0 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 204, + 506, + 219 + ], + "score": 1.0, + "content": ", for", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 216, + 286, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 270, + 230 + ], + "score": 1.0, + "content": "any optimal solution of (38), denoted by", + "type": "text" + }, + { + "bbox": [ + 270, + 216, + 282, + 229 + ], + "score": 0.89, + "content": "\\mathbf { h } _ { c } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 216, + 286, + 230 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 204, + 506, + 230 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 231, + 438, + 260 + ], + "lines": [ + { + "bbox": [ + 173, + 231, + 438, + 260 + ], + "spans": [ + { + "bbox": [ + 173, + 231, + 438, + 260 + ], + "score": 0.93, + "content": "\\mathbf { h } _ { c } ^ { k } + \\frac { \\sqrt { \\langle \\mathbf { h } _ { c } ^ { k } , \\mathbf { u } _ { \\mathrm { m i n } } \\rangle ^ { 2 } - \\| \\mathbf { u } _ { \\mathrm { m i n } } \\| ^ { 2 } \\big ( \\| \\mathbf { h } _ { c } ^ { k } \\| ^ { 2 } - r ^ { 2 } \\big ) } - \\langle \\mathbf { h } _ { c } ^ { k } , \\mathbf { u } _ { \\mathrm { m i n } } \\rangle } { \\| \\mathbf { u } _ { \\mathrm { m i n } } \\| ^ { 2 } } \\mathbf { u } _ { \\mathrm { m i n } }", + "type": "interline_equation", + "image_path": "4a84f0a66a70d1fd6e8264a455b9841b44c7da5a8599d395dd6b5468a6e797b6.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 173, + 231, + 438, + 260 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 262, + 297, + 274 + ], + "lines": [ + { + "bbox": [ + 105, + 261, + 299, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 299, + 276 + ], + "score": 1.0, + "content": "is a global minimizer of the original QCQP (8).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 261, + 299, + 276 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 281, + 504, + 306 + ], + "lines": [ + { + "bbox": [ + 105, + 281, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 428, + 295 + ], + "score": 1.0, + "content": "To perform the above reformulation, one needs to compute the exact eigenpair", + "type": "text" + }, + { + "bbox": [ + 428, + 282, + 479, + 293 + ], + "score": 0.85, + "content": "\\left( \\lambda _ { \\operatorname* { m i n } } , \\mathbf { u } _ { \\operatorname* { m i n } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 281, + 505, + 295 + ], + "score": 1.0, + "content": ". Nev-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 293, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 437, + 308 + ], + "score": 1.0, + "content": "ertheless, as we shall see, it is sufficient to compute an approximate eigenpair", + "type": "text" + }, + { + "bbox": [ + 438, + 294, + 462, + 307 + ], + "score": 0.92, + "content": "( \\tilde { \\lambda } , \\tilde { \\mathbf { u } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 293, + 506, + 308 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 281, + 506, + 308 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 313, + 374, + 328 + ], + "lines": [ + { + "bbox": [ + 236, + 313, + 374, + 328 + ], + "spans": [ + { + "bbox": [ + 236, + 313, + 374, + 328 + ], + "score": 0.92, + "content": "\\lambda _ { \\operatorname* { m i n } } \\leq \\tilde { \\lambda } = \\tilde { \\mathbf { u } } ^ { T } \\mathbf { H } ^ { k } \\tilde { \\mathbf { u } } \\leq \\lambda _ { \\operatorname* { m i n } } + \\tilde { \\epsilon } ,", + "type": "interline_equation", + "image_path": "a85cc13309ad658a543ec13c6a3f9f3866bf41c82b4eb5baea8eb426643121dc.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 236, + 313, + 374, + 328 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 330, + 505, + 352 + ], + "lines": [ + { + "bbox": [ + 105, + 328, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 362, + 343 + ], + "score": 1.0, + "content": "where \u000f˜ is a target accuracy to be determined later. We note that", + "type": "text" + }, + { + "bbox": [ + 363, + 330, + 397, + 341 + ], + "score": 0.91, + "content": "\\tilde { \\epsilon } \\leq 2 L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 328, + 453, + 343 + ], + "score": 1.0, + "content": "w.l.o.g. since", + "type": "text" + }, + { + "bbox": [ + 454, + 329, + 503, + 342 + ], + "score": 0.93, + "content": "\\| \\mathbf { H } ^ { k } \\| \\leq L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 328, + 506, + 343 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 341, + 437, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 437, + 352 + ], + "score": 1.0, + "content": "With this approximate eigenpair, it remains to solve the following convex problem", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 328, + 506, + 352 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 184, + 354, + 425, + 380 + ], + "lines": [ + { + "bbox": [ + 184, + 354, + 425, + 380 + ], + "spans": [ + { + "bbox": [ + 184, + 354, + 425, + 380 + ], + "score": 0.93, + "content": "\\operatorname* { m i n } _ { \\mathbf { h } \\in \\mathbb { R } ^ { d } , \\| \\mathbf { h } \\| \\leq r } \\tilde { q } ^ { k } ( \\mathbf { h } ) = \\langle \\mathbf { g } ^ { k } , \\mathbf { h } \\rangle + \\frac { 1 } { 2 } \\langle ( \\mathbf { H } ^ { k } - \\tilde { \\mu } I ) \\mathbf { h } , \\mathbf { h } \\rangle + \\frac { 1 } { 2 } \\tilde { \\mu } r ^ { 2 }", + "type": "interline_equation", + "image_path": "92370e579ccde63752f9debd6649e54b60e3b2dda0048a056aaecb071c1a9b48.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 184, + 354, + 425, + 380 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 462, + 397 + ], + "lines": [ + { + "bbox": [ + 105, + 383, + 462, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 133, + 399 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 383, + 212, + 397 + ], + "score": 0.93, + "content": "\\tilde { \\mu } = \\mathrm { m i n } \\{ 0 , \\tilde { \\lambda } - \\tilde { \\epsilon } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 384, + 462, + 399 + ], + "score": 1.0, + "content": ". One can check that the problem (41) well approximates (38).", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 383, + 462, + 399 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 399, + 503, + 423 + ], + "lines": [ + { + "bbox": [ + 106, + 397, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 187, + 413 + ], + "score": 1.0, + "content": "Corollary E.1. Let", + "type": "text" + }, + { + "bbox": [ + 187, + 398, + 198, + 411 + ], + "score": 0.87, + "content": "q _ { * } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 397, + 216, + 413 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 216, + 398, + 227, + 411 + ], + "score": 0.86, + "content": "\\tilde { q } _ { * } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 397, + 505, + 413 + ], + "score": 1.0, + "content": "be the minimum function value of (38) and (41), respectively. Assume", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 410, + 269, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 243, + 424 + ], + "score": 0.9, + "content": "\\lambda _ { \\operatorname* { m i n } } \\leq \\tilde { \\lambda } = \\tilde { \\mathbf { u } } ^ { T } \\mathbf { H } ^ { k } \\tilde { \\mathbf { u } } \\leq \\lambda _ { \\operatorname* { m i n } } + \\tilde { \\epsilon } .", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 411, + 269, + 425 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 106, + 397, + 505, + 425 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 271, + 425, + 340, + 440 + ], + "lines": [ + { + "bbox": [ + 271, + 425, + 340, + 440 + ], + "spans": [ + { + "bbox": [ + 271, + 425, + 340, + 440 + ], + "score": 0.92, + "content": "| q _ { * } ^ { k } - \\tilde { q } _ { * } ^ { k } | \\leq \\tilde { \\epsilon } r ^ { 2 } .", + "type": "interline_equation", + "image_path": "144b1097a8e6cfa77799b3ee5d9151f9f89fa95e3828e733548c43633f5362a3.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 271, + 425, + 340, + 440 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 447, + 506, + 494 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "We note that the above convex reformulation approach divides an indefinite QCQP into two subprob-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 459, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 312, + 474 + ], + "score": 1.0, + "content": "lems: (i) computation of an approximate eigenpair", + "type": "text" + }, + { + "bbox": [ + 312, + 459, + 336, + 472 + ], + "score": 0.91, + "content": "( \\tilde { \\lambda } , \\tilde { \\mathbf { u } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 460, + 506, + 474 + ], + "score": 1.0, + "content": "; (ii) solving the convex problem (41). As", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 365, + 484 + ], + "score": 1.0, + "content": "we shall see, by exploiting the finite-sum structure of the Hessian", + "type": "text" + }, + { + "bbox": [ + 366, + 471, + 380, + 482 + ], + "score": 0.88, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 471, + 506, + 484 + ], + "score": 1.0, + "content": ", these two subproblems can be", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 482, + 496, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 496, + 495 + ], + "score": 1.0, + "content": "efficiently solved. We treat these two subproblems in the following two subsections, respectively.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 447, + 506, + 495 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 506, + 303, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 304, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 304, + 519 + ], + "score": 1.0, + "content": "E.2 FINDING THE SMALLEST EIGENVECTOR", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 527, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 526, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 505, + 540 + ], + "score": 1.0, + "content": "To find a unit vector that satisfies requirement (40), we resort to the AppxPCA method (Allen-Zhu", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 537, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 104, + 537, + 338, + 552 + ], + "score": 1.0, + "content": "& Li, 2016), which first finds an approximate eigenvalue", + "type": "text" + }, + { + "bbox": [ + 339, + 539, + 395, + 549 + ], + "score": 0.9, + "content": "\\lambda = \\lambda _ { \\operatorname* { m i n } } - \\tilde { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 537, + 506, + 552 + ], + "score": 1.0, + "content": "via binary search and then", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 548, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 104, + 548, + 327, + 562 + ], + "score": 1.0, + "content": "applies Power method to the positive definite matrix", + "type": "text" + }, + { + "bbox": [ + 327, + 549, + 384, + 561 + ], + "score": 0.91, + "content": "( \\mathbf { H } ^ { k } - \\lambda I ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 548, + 506, + 562 + ], + "score": 1.0, + "content": "for a logarithmic number of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 558, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 104, + 558, + 195, + 574 + ], + "score": 1.0, + "content": "iterations. Computing", + "type": "text" + }, + { + "bbox": [ + 196, + 560, + 258, + 573 + ], + "score": 0.92, + "content": "( \\mathbf { H } ^ { k } - \\lambda I ) ^ { - 1 } \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 558, + 316, + 574 + ], + "score": 1.0, + "content": "for any vector", + "type": "text" + }, + { + "bbox": [ + 317, + 563, + 324, + 570 + ], + "score": 0.71, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 558, + 433, + 574 + ], + "score": 1.0, + "content": "is equivalent to solving the", + "type": "text" + }, + { + "bbox": [ + 433, + 561, + 439, + 570 + ], + "score": 0.41, + "content": "\\tilde { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 558, + 506, + 574 + ], + "score": 1.0, + "content": "-strongly convex", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 570, + 240, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 240, + 584 + ], + "score": 1.0, + "content": "problem (Allen-Zhu & Li, 2018)", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29, + "bbox_fs": [ + 104, + 526, + 506, + 584 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 585, + 390, + 609 + ], + "lines": [ + { + "bbox": [ + 219, + 585, + 390, + 609 + ], + "spans": [ + { + "bbox": [ + 219, + 585, + 390, + 609 + ], + "score": 0.94, + "content": "\\operatorname* { m i n } _ { \\mathbf { u } } \\phi ^ { k } ( \\mathbf { u } ) : = \\frac { 1 } { 2 } \\mathbf { u } ^ { T } ( \\mathbf { H } ^ { k } - \\lambda I ) \\mathbf { u } - \\langle \\mathbf { v } , \\mathbf { u } \\rangle", + "type": "interline_equation", + "image_path": "dac887be88ba8de15c2391acc4b84c9a52c388b533f2345e8611cffc16c8fef5.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 219, + 585, + 390, + 609 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 611, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 161, + 627 + ], + "score": 1.0, + "content": "We note that", + "type": "text" + }, + { + "bbox": [ + 162, + 611, + 272, + 627 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathbf { H } ^ { k } = \\frac { 1 } { | \\cal { S } | } \\sum _ { i \\in \\cal { S } } \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 610, + 342, + 627 + ], + "score": 1.0, + "content": ". Specifically, in", + "type": "text" + }, + { + "bbox": [ + 343, + 612, + 374, + 624 + ], + "score": 0.87, + "content": "\\mathrm { { S T R } _ { \\mathrm { { f r e e } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 610, + 405, + 627 + ], + "score": 1.0, + "content": ", either", + "type": "text" + }, + { + "bbox": [ + 405, + 612, + 440, + 624 + ], + "score": 0.92, + "content": "| S | = n", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 610, + 463, + 627 + ], + "score": 1.0, + "content": "(i.e.,", + "type": "text" + }, + { + "bbox": [ + 463, + 612, + 478, + 623 + ], + "score": 0.88, + "content": "\\mathbf { H } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 610, + 506, + 627 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 173, + 641 + ], + "score": 1.0, + "content": "full Hessian) or", + "type": "text" + }, + { + "bbox": [ + 173, + 627, + 255, + 640 + ], + "score": 0.92, + "content": "| S | = \\tilde { \\mathcal { O } } ( L _ { 1 } ^ { 2 } / ( L _ { 2 } \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 626, + 367, + 641 + ], + "score": 1.0, + "content": "by Lemma 4.1. Therefore,", + "type": "text" + }, + { + "bbox": [ + 368, + 627, + 390, + 640 + ], + "score": 0.91, + "content": "\\phi ^ { k } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 626, + 506, + 641 + ], + "score": 1.0, + "content": "can be expressed as sum of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 639, + 195, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 195, + 650 + ], + "score": 1.0, + "content": "non-convex functions", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 610, + 506, + 650 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 650, + 452, + 680 + ], + "lines": [ + { + "bbox": [ + 159, + 650, + 452, + 680 + ], + "spans": [ + { + "bbox": [ + 159, + 650, + 452, + 680 + ], + "score": 0.94, + "content": "\\phi ^ { k } ( \\mathbf { u } ) = \\frac { 1 } { | S | } \\sum _ { i \\in S } \\phi _ { i } ^ { k } ( \\mathbf { u } ) = \\frac { 1 } { | S | } \\sum _ { i \\in S } \\left( \\frac { 1 } { 2 } \\mathbf { u } ^ { T } ( \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) - \\lambda I ) \\mathbf { u } - \\langle \\mathbf { v } , \\mathbf { u } \\rangle \\right) .", + "type": "interline_equation", + "image_path": "eec1192412c143e85db823e72aa5225f155ff2f2d760377afc4b2cc2ceb28aab.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 159, + 650, + 452, + 660.0 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 159, + 660.0, + 452, + 670.0 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 159, + 670.0, + 452, + 680.0 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 683, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 682, + 506, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 215, + 697 + ], + "score": 1.0, + "content": "By observing that each", + "type": "text" + }, + { + "bbox": [ + 216, + 683, + 228, + 696 + ], + "score": 0.9, + "content": "\\phi _ { i } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 682, + 338, + 697 + ], + "score": 1.0, + "content": "is non-convex and has", + "type": "text" + }, + { + "bbox": [ + 338, + 684, + 363, + 696 + ], + "score": 0.88, + "content": "\\left( 4 L _ { 2 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 682, + 506, + 697 + ], + "score": 1.0, + "content": "-Lipschitz gradient, we can use", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 695, + 507, + 710 + ], + "spans": [ + { + "bbox": [ + 106, + 695, + 332, + 709 + ], + "score": 1.0, + "content": "KatyushaXS (Allen-Zhu, 2018a) to solve problem (43) in", + "type": "text" + }, + { + "bbox": [ + 333, + 696, + 428, + 710 + ], + "score": 0.92, + "content": "\\tilde { \\mathcal { O } } ( | S | + | S | ^ { 3 / 4 } \\sqrt { L _ { 2 } / \\tilde { \\epsilon } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 695, + 507, + 709 + ], + "score": 1.0, + "content": "stochastic Hessian-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 190, + 722 + ], + "score": 1.0, + "content": "vector product (i.e.,", + "type": "text" + }, + { + "bbox": [ + 190, + 709, + 241, + 722 + ], + "score": 0.92, + "content": "\\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) \\mathbf { u } )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "evaluations. The following result is taken from (Agarwal et al.,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 720, + 438, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 438, + 733 + ], + "score": 1.0, + "content": "2017, Section G.3), which gives the overall computation complexity of AppxPCA.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 682, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 237, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 238, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 238, + 97 + ], + "score": 1.0, + "content": "Algorithm 9 Fast QCQP Solver", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 98, + 201, + 110 + ], + "lines": [ + { + "bbox": [ + 105, + 96, + 202, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 96, + 138, + 113 + ], + "score": 1.0, + "content": 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Let", + "type": "text" + }, + { + "bbox": [ + 178, + 268, + 320, + 285 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathbf { H } ^ { k } = \\frac { 1 } { | \\mathcal { S } | } \\sum _ { i \\in \\cal S } \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) \\in \\mathbb { R } ^ { d \\times d } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 266, + 350, + 285 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 350, + 269, + 425, + 282 + ], + "score": 0.93, + "content": "\\| \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) \\| \\leq L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 266, + 506, + 285 + ], + "score": 1.0, + "content": ". With probability at", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 283, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 127, + 298 + ], + "score": 1.0, + "content": "least", + "type": "text" + }, + { + "bbox": [ + 128, + 285, + 152, + 297 + ], + "score": 0.54, + "content": "1 - p ,", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 283, + 337, + 298 + ], + "score": 1.0, + "content": ", AppxPCA produces a unit vector u satisfying", + "type": "text" + }, + { + "bbox": [ + 337, + 284, + 421, + 296 + ], + "score": 0.9, + "content": "\\mathbf { u } ^ { T } \\mathbf { H } ^ { k } \\mathbf { u } \\leq \\lambda _ { \\operatorname* { m i n } } + \\tilde { \\epsilon } .", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 283, + 505, + 298 + ], + "score": 1.0, + "content": ". The total stochastic", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 295, + 385, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 285, + 310 + ], + "score": 1.0, + "content": "Hessian-vector product oracle complexity is", + "type": "text" + }, + { + "bbox": [ + 285, + 296, + 380, + 310 + ], + "score": 0.92, + "content": "\\tilde { \\mathcal { O } } ( | S | + | S | ^ { 3 / 4 } \\sqrt { L _ { 2 } / \\tilde { \\epsilon } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 295, + 385, + 310 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 108, + 322, + 262, + 335 + ], + "lines": [ + { + "bbox": [ + 104, + 320, + 264, + 338 + ], + "spans": [ + { + "bbox": [ + 104, + 320, + 264, + 338 + ], + "score": 1.0, + "content": "E.3 SOLVING THE CONVEX QCQP", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 344, + 504, + 367 + ], + "lines": [ + { + "bbox": [ + 105, + 343, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 506, + 357 + ], + "score": 1.0, + "content": "In what follows, we show that the convex problem (41) can be solved efficiently. We first observe that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 355, + 503, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 503, + 368 + ], + "score": 1.0, + "content": "problem (41) has a finite-sum structure and can be rewritten as an unconstrained problem of the form", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 110, + 372, + 482, + 403 + ], + "lines": [ + { + "bbox": [ + 110, + 372, + 482, + 403 + ], + "spans": [ + { + "bbox": [ + 110, + 372, + 482, + 403 + ], + "score": 0.93, + "content": "\\operatorname* { m i n } _ { \\mathbf { h \\in \\mathbb { R } } ^ { d } } \\frac { 1 } { | \\mathcal { S } | } \\sum _ { i \\in \\mathcal { S } } \\tilde { q } _ { i } ^ { k } ( \\mathbf { h } ) + \\Psi ( \\mathbf { h } ) = \\frac { 1 } { | \\mathcal { S } | } \\sum _ { i \\in \\mathcal { S } } \\Big ( \\langle \\mathbf { g } ^ { k } , \\mathbf { h } \\rangle + \\frac { 1 } { 2 } \\langle ( \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) - \\tilde { \\mu } I ) \\mathbf { h } , \\mathbf { h } \\rangle + \\frac { 1 } { 2 } \\tilde { \\mu } r ^ { 2 } \\Big ) + \\Psi ( \\mathbf { h } ) ,", + "type": "interline_equation", + "image_path": "3801b173622be69c84e2ae4917f4dd8581c66820bf6fb61fa7f318f2e17b01f4.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 110, + 372, + 482, + 382.3333333333333 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 110, + 382.3333333333333, + 482, + 392.66666666666663 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 110, + 392.66666666666663, + 482, + 402.99999999999994 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 409, + 506, + 468 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 135, + 423 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 410, + 181, + 423 + ], + "score": 0.93, + "content": "\\Psi ( { \\bf h } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 410, + 194, + 423 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 194, + 410, + 234, + 423 + ], + "score": 0.92, + "content": "\\| \\mathbf { h } \\| \\leq r", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 410, + 282, + 423 + ], + "score": 1.0, + "content": ", otherwise", + "type": "text" + }, + { + "bbox": [ + 283, + 410, + 342, + 422 + ], + "score": 0.9, + "content": "\\Psi ( \\mathbf { h } ) \\ = \\ + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 410, + 429, + 423 + ], + "score": 1.0, + "content": ". We note that each", + "type": "text" + }, + { + "bbox": [ + 429, + 410, + 454, + 423 + ], + "score": 0.93, + "content": "\\tilde { q } _ { i } ^ { k } ( \\mathbf { h } )", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 410, + 506, + 423 + ], + "score": 1.0, + "content": "in (45) has", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 422, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 130, + 434 + ], + "score": 0.87, + "content": "\\left( 4 L _ { 2 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 422, + 280, + 435 + ], + "score": 1.0, + "content": "-Lipschitz continuous gradient since", + "type": "text" + }, + { + "bbox": [ + 280, + 422, + 354, + 434 + ], + "score": 0.9, + "content": "\\| \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) \\| \\leq L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 422, + 373, + 435 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 374, + 422, + 408, + 433 + ], + "score": 0.91, + "content": "\\tilde { \\epsilon } \\leq 2 L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 422, + 506, + 435 + ], + "score": 1.0, + "content": ". Therefore, we can use", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 432, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 104, + 432, + 152, + 446 + ], + "score": 1.0, + "content": "KatyushaX", + "type": "text" + }, + { + "bbox": [ + 152, + 433, + 160, + 443 + ], + "score": 0.26, + "content": "\\mathrm { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 432, + 506, + 446 + ], + "score": 1.0, + "content": "(Allen-Zhu, 2018a) to solve (45). By (Allen-Zhu, 2018a, Theorem 4.6), KatyushaXW", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 444, + 507, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 203, + 457 + ], + "score": 1.0, + "content": "finds a point h such that", + "type": "text" + }, + { + "bbox": [ + 204, + 445, + 284, + 458 + ], + "score": 0.93, + "content": "\\mathbb { E } [ \\tilde { q } ^ { k } ( \\mathbf { h } ) - \\tilde { q } _ { * } ^ { k } ] \\leq \\tilde { \\epsilon } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 445, + 309, + 457 + ], + "score": 1.0, + "content": "using", + "type": "text" + }, + { + "bbox": [ + 309, + 444, + 428, + 458 + ], + "score": 0.92, + "content": "\\tilde { \\mathcal { O } } ( | S | + | S | ^ { 3 / 4 } \\sqrt { L _ { 2 } } \\cdot r / \\sqrt { \\tilde { \\epsilon } _ { 1 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 445, + 507, + 457 + ], + "score": 1.0, + "content": "stochastic Hessian-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 457, + 389, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 199, + 468 + ], + "score": 1.0, + "content": "vector products, where", + "type": "text" + }, + { + "bbox": [ + 200, + 457, + 209, + 468 + ], + "score": 0.88, + "content": "\\tilde { \\epsilon } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 457, + 389, + 468 + ], + "score": 1.0, + "content": "is the target accuracy to be determined later.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 108, + 482, + 253, + 494 + ], + "lines": [ + { + "bbox": [ + 106, + 483, + 254, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 254, + 495 + ], + "score": 1.0, + "content": "E.4 PUTTING IT ALL TOGETHER", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 503, + 506, + 574 + ], + "lines": [ + { + "bbox": [ + 106, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "The complete procedure of our fast QCQP solver is summarized in Algorithm 9. Combining all the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 515, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 210, + 529 + ], + "score": 1.0, + "content": "above results and setting", + "type": "text" + }, + { + "bbox": [ + 211, + 515, + 262, + 529 + ], + "score": 0.86, + "content": "r = \\sqrt { \\epsilon / L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 515, + 267, + 529 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 267, + 515, + 321, + 528 + ], + "score": 0.86, + "content": "\\tilde { \\epsilon } = \\sqrt { \\epsilon L _ { 2 } } / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 515, + 343, + 529 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 343, + 515, + 381, + 527 + ], + "score": 0.92, + "content": "\\tilde { \\epsilon } _ { 1 } = \\tilde { \\epsilon } r ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 515, + 506, + 529 + ], + "score": 1.0, + "content": ", one can find an approximate", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 103, + 524, + 508, + 547 + ], + "spans": [ + { + "bbox": [ + 103, + 524, + 315, + 547 + ], + "score": 1.0, + "content": "solution to QCQP (8) satisfying requirement (24) in", + "type": "text" + }, + { + "bbox": [ + 315, + 528, + 426, + 542 + ], + "score": 0.92, + "content": "{ \\tilde { \\mathcal { O } } } ( | S | + | S | ^ { 3 / 4 } L _ { 2 } ^ { 0 . 2 5 } / \\epsilon ^ { 0 . 2 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 524, + 508, + 547 + ], + "score": 1.0, + "content": "stochastic Hessian-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 541, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 406, + 552 + ], + "score": 1.0, + "content": "vector product evaluations. By replacing Lanczos method with this solver in", + "type": "text" + }, + { + "bbox": [ + 406, + 541, + 436, + 551 + ], + "score": 0.87, + "content": "\\operatorname { S T R } _ { \\mathrm { f r e e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 541, + 505, + 552 + ], + "score": 1.0, + "content": ", we derive a new", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 217, + 564 + ], + "score": 1.0, + "content": "Hessian-free method called", + "type": "text" + }, + { + "bbox": [ + 217, + 552, + 253, + 563 + ], + "score": 0.91, + "content": "{ \\mathrm { S T R } } _ { \\mathrm { f r e e } } +", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 551, + 506, + 564 + ], + "score": 1.0, + "content": ", which is summarized in Algorithm 10. The following theorem", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 562, + 417, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 288, + 575 + ], + "score": 1.0, + "content": "establishes the overall runtime complexity of", + "type": "text" + }, + { + "bbox": [ + 289, + 563, + 325, + 574 + ], + "score": 0.89, + "content": "{ \\mathrm { S T R } } _ { \\mathrm { f r e e } ^ { + } }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 562, + 383, + 575 + ], + "score": 1.0, + "content": "for finding an", + "type": "text" + }, + { + "bbox": [ + 383, + 564, + 388, + 572 + ], + "score": 0.66, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 562, + 417, + 575 + ], + "score": 1.0, + "content": "-SOSP.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 578, + 506, + 645 + ], + "lines": [ + { + "bbox": [ + 105, + 577, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 385, + 592 + ], + "score": 1.0, + "content": "Theorem E.1. Consider Algorithm 10 for solving problem (1). Let", + "type": "text" + }, + { + "bbox": [ + 385, + 579, + 422, + 591 + ], + "score": 0.87, + "content": "\\zeta = 1 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 578, + 425, + 592 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 426, + 577, + 477, + 592 + ], + "score": 0.86, + "content": "r = \\sqrt { \\epsilon / L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 578, + 481, + 592 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 482, + 579, + 505, + 591 + ], + "score": 0.77, + "content": "K =", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 588, + 508, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 159, + 607 + ], + "score": 0.91, + "content": "4 \\sqrt { L _ { 2 } } \\Delta / \\epsilon ^ { 1 . 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 592, + 163, + 608 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 164, + 593, + 227, + 608 + ], + "score": 0.89, + "content": "\\begin{array} { r } { T = \\frac { 3 } { 2 } \\mathrm { { l o g } } ( 2 / \\delta ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 592, + 230, + 608 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 231, + 595, + 269, + 606 + ], + "score": 0.82, + "content": "c _ { 1 } = 6 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 592, + 272, + 608 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 273, + 595, + 311, + 606 + ], + "score": 0.85, + "content": "c _ { 2 } = 5 0 0 \\mathrm { \\Omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 592, + 332, + 608 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 346, + 588, + 508, + 612 + ], + "score": 1.0, + "content": "32L21\u000fL2 log(4d/δ). The hyper-parameters√", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 608, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 431, + 620 + ], + "score": 1.0, + "content": "in Estimator 4 are set to the same values as those in Lemma 4.2. Besides, let", + "type": "text" + }, + { + "bbox": [ + 431, + 608, + 486, + 620 + ], + "score": 0.92, + "content": "\\tilde { \\epsilon } = \\sqrt { \\epsilon L _ { 2 } } / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 609, + 505, + 620 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 618, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 146, + 630 + ], + "score": 0.91, + "content": "\\tilde { \\epsilon } _ { 1 } = \\tilde { \\epsilon } r ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 618, + 264, + 632 + ], + "score": 1.0, + "content": "in Algorithm 9. To find an", + "type": "text" + }, + { + "bbox": [ + 264, + 620, + 329, + 631 + ], + "score": 0.65, + "content": "O ( \\epsilon ) { - } S O S P ~ w . p", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 618, + 367, + 632 + ], + "score": 1.0, + "content": ". at least", + "type": "text" + }, + { + "bbox": [ + 368, + 620, + 392, + 630 + ], + "score": 0.83, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 618, + 506, + 632 + ], + "score": 1.0, + "content": ", the runtime complexity is", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 629, + 347, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 343, + 645 + ], + "score": 0.83, + "content": "\\tilde { \\mathcal { O } } ( d \\mathrm { m i n } \\{ n / \\epsilon ^ { 1 . 5 } + n ^ { 0 . 7 5 } / \\epsilon ^ { 1 . 7 5 } , 1 / \\epsilon ^ { 2 . 5 } + \\sqrt { n } / \\epsilon ^ { 2 } \\} \\mathrm { l o g } ( 1 / \\delta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 629, + 347, + 646 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 324, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 658, + 324, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 324, + 672 + ], + "score": 1.0, + "content": "Proof. The proof directly follows from that in Sec. D.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 686, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 686, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 270, + 699 + ], + "score": 1.0, + "content": "We compare the runtime complexity of", + "type": "text" + }, + { + "bbox": [ + 270, + 687, + 301, + 698 + ], + "score": 0.88, + "content": "\\operatorname { S T R } _ { \\mathrm { f r e e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 686, + 320, + 699 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 320, + 687, + 357, + 698 + ], + "score": 0.9, + "content": "{ \\mathrm { S T R } } _ { \\mathrm { f r e e } } +", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 686, + 505, + 699 + ], + "score": 1.0, + "content": "with existing Hessian free methods", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 696, + 504, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 230, + 711 + ], + "score": 1.0, + "content": "in Table 2. One can see that", + "type": "text" + }, + { + "bbox": [ + 230, + 698, + 261, + 709 + ], + "score": 0.85, + "content": "\\mathrm { { S T R } _ { \\mathrm { { f r e e } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 696, + 472, + 711 + ], + "score": 1.0, + "content": "strictly outperforms Hessian-free Cubic. Besides,", + "type": "text" + }, + { + "bbox": [ + 473, + 698, + 504, + 709 + ], + "score": 0.81, + "content": "\\mathrm { { S T R } _ { \\mathrm { { f r e e } } } }", + "type": "inline_equation" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 212, + 723 + ], + "score": 1.0, + "content": "outperforms Fast-Cubic if", + "type": "text" + }, + { + "bbox": [ + 212, + 709, + 273, + 722 + ], + "score": 0.92, + "content": "n \\ge \\Omega ( 1 / \\epsilon ^ { 4 / 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 708, + 505, + 723 + ], + "score": 1.0, + "content": ", which is a mild condition for large-scale problems in the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 206, + 732 + ], + "score": 1.0, + "content": "moderate accuracy case.", + "type": "text" + }, + { + "bbox": [ + 207, + 721, + 243, + 732 + ], + "score": 0.85, + "content": "{ \\mathrm { S T R } } _ { \\mathrm { f r e e } ^ { + } }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "strictly outperforms both Hessian-free Cubic and Fast-Cubic. We", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41.5 + } + ], + "page_idx": 19, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 659, + 504, + 670 + ], + "lines": [ + { + "bbox": [ + 496, + 661, + 504, + 669 + ], + "spans": [ + { + "bbox": [ + 496, + 661, + 504, + 669 + ], + "score": 0.998, + "content": "□", + "type": 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Let", + "type": "text" + }, + { + "bbox": [ + 178, + 268, + 320, + 285 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathbf { H } ^ { k } = \\frac { 1 } { | \\mathcal { S } | } \\sum _ { i \\in \\cal S } \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) \\in \\mathbb { R } ^ { d \\times d } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 266, + 350, + 285 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 350, + 269, + 425, + 282 + ], + "score": 0.93, + "content": "\\| \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) \\| \\leq L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 266, + 506, + 285 + ], + "score": 1.0, + "content": ". With probability at", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 283, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 127, + 298 + ], + "score": 1.0, + "content": "least", + "type": "text" + }, + { + "bbox": [ + 128, + 285, + 152, + 297 + ], + "score": 0.54, + "content": "1 - p ,", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 283, + 337, + 298 + ], + "score": 1.0, + "content": ", AppxPCA produces a unit vector u satisfying", + "type": "text" + }, + { + "bbox": [ + 337, + 284, + 421, + 296 + ], + "score": 0.9, + "content": "\\mathbf { u } ^ { T } \\mathbf { H } ^ { k } \\mathbf { u } \\leq \\lambda _ { \\operatorname* { m i n } } + \\tilde { \\epsilon } .", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 283, + 505, + 298 + ], + "score": 1.0, + "content": ". The total stochastic", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 295, + 385, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 285, + 310 + ], + "score": 1.0, + "content": "Hessian-vector product oracle complexity is", + "type": "text" + }, + { + "bbox": [ + 285, + 296, + 380, + 310 + ], + "score": 0.92, + "content": "\\tilde { \\mathcal { O } } ( | S | + | S | ^ { 3 / 4 } \\sqrt { L _ { 2 } / \\tilde { \\epsilon } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 295, + 385, + 310 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 104, + 266, + 506, + 310 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 322, + 262, + 335 + ], + "lines": [ + { + "bbox": [ + 104, + 320, + 264, + 338 + ], + "spans": [ + { + "bbox": [ + 104, + 320, + 264, + 338 + ], + "score": 1.0, + "content": "E.3 SOLVING THE CONVEX QCQP", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 344, + 504, + 367 + ], + "lines": [ + { + "bbox": [ + 105, + 343, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 506, + 357 + ], + "score": 1.0, + "content": "In what follows, we show that the convex problem (41) can be solved efficiently. We first observe that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 355, + 503, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 503, + 368 + ], + "score": 1.0, + "content": "problem (41) has a finite-sum structure and can be rewritten as an unconstrained problem of the form", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 343, + 506, + 368 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 110, + 372, + 482, + 403 + ], + "lines": [ + { + "bbox": [ + 110, + 372, + 482, + 403 + ], + "spans": [ + { + "bbox": [ + 110, + 372, + 482, + 403 + ], + "score": 0.93, + "content": "\\operatorname* { m i n } _ { \\mathbf { h \\in \\mathbb { R } } ^ { d } } \\frac { 1 } { | \\mathcal { S } | } \\sum _ { i \\in \\mathcal { S } } \\tilde { q } _ { i } ^ { k } ( \\mathbf { h } ) + \\Psi ( \\mathbf { h } ) = \\frac { 1 } { | \\mathcal { S } | } \\sum _ { i \\in \\mathcal { S } } \\Big ( \\langle \\mathbf { g } ^ { k } , \\mathbf { h } \\rangle + \\frac { 1 } { 2 } \\langle ( \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) - \\tilde { \\mu } I ) \\mathbf { h } , \\mathbf { h } \\rangle + \\frac { 1 } { 2 } \\tilde { \\mu } r ^ { 2 } \\Big ) + \\Psi ( \\mathbf { h } ) ,", + "type": "interline_equation", + "image_path": "3801b173622be69c84e2ae4917f4dd8581c66820bf6fb61fa7f318f2e17b01f4.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 110, + 372, + 482, + 382.3333333333333 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 110, + 382.3333333333333, + 482, + 392.66666666666663 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 110, + 392.66666666666663, + 482, + 402.99999999999994 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 409, + 506, + 468 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 135, + 423 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 410, + 181, + 423 + ], + "score": 0.93, + "content": "\\Psi ( { \\bf h } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 410, + 194, + 423 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 194, + 410, + 234, + 423 + ], + "score": 0.92, + "content": "\\| \\mathbf { h } \\| \\leq r", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 410, + 282, + 423 + ], + "score": 1.0, + "content": ", otherwise", + "type": "text" + }, + { + "bbox": [ + 283, + 410, + 342, + 422 + ], + "score": 0.9, + "content": "\\Psi ( \\mathbf { h } ) \\ = \\ + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 410, + 429, + 423 + ], + "score": 1.0, + "content": ". We note that each", + "type": "text" + }, + { + "bbox": [ + 429, + 410, + 454, + 423 + ], + "score": 0.93, + "content": "\\tilde { q } _ { i } ^ { k } ( \\mathbf { h } )", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 410, + 506, + 423 + ], + "score": 1.0, + "content": "in (45) has", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 422, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 130, + 434 + ], + "score": 0.87, + "content": "\\left( 4 L _ { 2 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 422, + 280, + 435 + ], + "score": 1.0, + "content": "-Lipschitz continuous gradient since", + "type": "text" + }, + { + "bbox": [ + 280, + 422, + 354, + 434 + ], + "score": 0.9, + "content": "\\| \\nabla ^ { 2 } f _ { i } ( \\mathbf { x } ^ { k } ) \\| \\leq L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 422, + 373, + 435 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 374, + 422, + 408, + 433 + ], + "score": 0.91, + "content": "\\tilde { \\epsilon } \\leq 2 L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 422, + 506, + 435 + ], + "score": 1.0, + "content": ". Therefore, we can use", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 432, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 104, + 432, + 152, + 446 + ], + "score": 1.0, + "content": "KatyushaX", + "type": "text" + }, + { + "bbox": [ + 152, + 433, + 160, + 443 + ], + "score": 0.26, + "content": "\\mathrm { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 432, + 506, + 446 + ], + "score": 1.0, + "content": "(Allen-Zhu, 2018a) to solve (45). By (Allen-Zhu, 2018a, Theorem 4.6), KatyushaXW", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 444, + 507, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 203, + 457 + ], + "score": 1.0, + "content": "finds a point h such that", + "type": "text" + }, + { + "bbox": [ + 204, + 445, + 284, + 458 + ], + "score": 0.93, + "content": "\\mathbb { E } [ \\tilde { q } ^ { k } ( \\mathbf { h } ) - \\tilde { q } _ { * } ^ { k } ] \\leq \\tilde { \\epsilon } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 445, + 309, + 457 + ], + "score": 1.0, + "content": "using", + "type": "text" + }, + { + "bbox": [ + 309, + 444, + 428, + 458 + ], + "score": 0.92, + "content": "\\tilde { \\mathcal { O } } ( | S | + | S | ^ { 3 / 4 } \\sqrt { L _ { 2 } } \\cdot r / \\sqrt { \\tilde { \\epsilon } _ { 1 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 445, + 507, + 457 + ], + "score": 1.0, + "content": "stochastic Hessian-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 457, + 389, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 199, + 468 + ], + "score": 1.0, + "content": "vector products, where", + "type": "text" + }, + { + "bbox": [ + 200, + 457, + 209, + 468 + ], + "score": 0.88, + "content": "\\tilde { \\epsilon } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 457, + 389, + 468 + ], + "score": 1.0, + "content": "is the target accuracy to be determined later.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 104, + 410, + 507, + 468 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 482, + 253, + 494 + ], + "lines": [ + { + "bbox": [ + 106, + 483, + 254, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 254, + 495 + ], + "score": 1.0, + "content": "E.4 PUTTING IT ALL TOGETHER", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 503, + 506, + 574 + ], + "lines": [ + { + "bbox": [ + 106, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "The complete procedure of our fast QCQP solver is summarized in Algorithm 9. Combining all the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 515, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 210, + 529 + ], + "score": 1.0, + "content": "above results and setting", + "type": "text" + }, + { + "bbox": [ + 211, + 515, + 262, + 529 + ], + "score": 0.86, + "content": "r = \\sqrt { \\epsilon / L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 515, + 267, + 529 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 267, + 515, + 321, + 528 + ], + "score": 0.86, + "content": "\\tilde { \\epsilon } = \\sqrt { \\epsilon L _ { 2 } } / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 515, + 343, + 529 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 343, + 515, + 381, + 527 + ], + "score": 0.92, + "content": "\\tilde { \\epsilon } _ { 1 } = \\tilde { \\epsilon } r ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 515, + 506, + 529 + ], + "score": 1.0, + "content": ", one can find an approximate", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 103, + 524, + 508, + 547 + ], + "spans": [ + { + "bbox": [ + 103, + 524, + 315, + 547 + ], + "score": 1.0, + "content": "solution to QCQP (8) satisfying requirement (24) in", + "type": "text" + }, + { + "bbox": [ + 315, + 528, + 426, + 542 + ], + "score": 0.92, + "content": "{ \\tilde { \\mathcal { O } } } ( | S | + | S | ^ { 3 / 4 } L _ { 2 } ^ { 0 . 2 5 } / \\epsilon ^ { 0 . 2 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 524, + 508, + 547 + ], + "score": 1.0, + "content": "stochastic Hessian-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 541, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 406, + 552 + ], + "score": 1.0, + "content": "vector product evaluations. By replacing Lanczos method with this solver in", + "type": "text" + }, + { + "bbox": [ + 406, + 541, + 436, + 551 + ], + "score": 0.87, + "content": "\\operatorname { S T R } _ { \\mathrm { f r e e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 541, + 505, + 552 + ], + "score": 1.0, + "content": ", we derive a new", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 217, + 564 + ], + "score": 1.0, + "content": "Hessian-free method called", + "type": "text" + }, + { + "bbox": [ + 217, + 552, + 253, + 563 + ], + "score": 0.91, + "content": "{ \\mathrm { S T R } } _ { \\mathrm { f r e e } } +", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 551, + 506, + 564 + ], + "score": 1.0, + "content": ", which is summarized in Algorithm 10. The following theorem", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 562, + 417, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 288, + 575 + ], + "score": 1.0, + "content": "establishes the overall runtime complexity of", + "type": "text" + }, + { + "bbox": [ + 289, + 563, + 325, + 574 + ], + "score": 0.89, + "content": "{ \\mathrm { S T R } } _ { \\mathrm { f r e e } ^ { + } }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 562, + 383, + 575 + ], + "score": 1.0, + "content": "for finding an", + "type": "text" + }, + { + "bbox": [ + 383, + 564, + 388, + 572 + ], + "score": 0.66, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 562, + 417, + 575 + ], + "score": 1.0, + "content": "-SOSP.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 103, + 502, + 508, + 575 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 578, + 506, + 645 + ], + "lines": [ + { + "bbox": [ + 105, + 577, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 385, + 592 + ], + "score": 1.0, + "content": "Theorem E.1. Consider Algorithm 10 for solving problem (1). Let", + "type": "text" + }, + { + "bbox": [ + 385, + 579, + 422, + 591 + ], + "score": 0.87, + "content": "\\zeta = 1 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 578, + 425, + 592 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 426, + 577, + 477, + 592 + ], + "score": 0.86, + "content": "r = \\sqrt { \\epsilon / L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 578, + 481, + 592 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 482, + 579, + 505, + 591 + ], + "score": 0.77, + "content": "K =", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 588, + 508, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 159, + 607 + ], + "score": 0.91, + "content": "4 \\sqrt { L _ { 2 } } \\Delta / \\epsilon ^ { 1 . 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 592, + 163, + 608 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 164, + 593, + 227, + 608 + ], + "score": 0.89, + "content": "\\begin{array} { r } { T = \\frac { 3 } { 2 } \\mathrm { { l o g } } ( 2 / \\delta ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 592, + 230, + 608 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 231, + 595, + 269, + 606 + ], + "score": 0.82, + "content": "c _ { 1 } = 6 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 592, + 272, + 608 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 273, + 595, + 311, + 606 + ], + "score": 0.85, + "content": "c _ { 2 } = 5 0 0 \\mathrm { \\Omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 592, + 332, + 608 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 346, + 588, + 508, + 612 + ], + "score": 1.0, + "content": "32L21\u000fL2 log(4d/δ). The hyper-parameters√", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 608, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 431, + 620 + ], + "score": 1.0, + "content": "in Estimator 4 are set to the same values as those in Lemma 4.2. Besides, let", + "type": "text" + }, + { + "bbox": [ + 431, + 608, + 486, + 620 + ], + "score": 0.92, + "content": "\\tilde { \\epsilon } = \\sqrt { \\epsilon L _ { 2 } } / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 609, + 505, + 620 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 618, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 146, + 630 + ], + "score": 0.91, + "content": "\\tilde { \\epsilon } _ { 1 } = \\tilde { \\epsilon } r ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 618, + 264, + 632 + ], + "score": 1.0, + "content": "in Algorithm 9. To find an", + "type": "text" + }, + { + "bbox": [ + 264, + 620, + 329, + 631 + ], + "score": 0.65, + "content": "O ( \\epsilon ) { - } S O S P ~ w . p", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 618, + 367, + 632 + ], + "score": 1.0, + "content": ". at least", + "type": "text" + }, + { + "bbox": [ + 368, + 620, + 392, + 630 + ], + "score": 0.83, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 618, + 506, + 632 + ], + "score": 1.0, + "content": ", the runtime complexity is", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 629, + 347, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 343, + 645 + ], + "score": 0.83, + "content": "\\tilde { \\mathcal { O } } ( d \\mathrm { m i n } \\{ n / \\epsilon ^ { 1 . 5 } + n ^ { 0 . 7 5 } / \\epsilon ^ { 1 . 7 5 } , 1 / \\epsilon ^ { 2 . 5 } + \\sqrt { n } / \\epsilon ^ { 2 } \\} \\mathrm { l o g } ( 1 / \\delta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 629, + 347, + 646 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 577, + 508, + 646 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 324, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 658, + 324, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 324, + 672 + ], + "score": 1.0, + "content": "Proof. The proof directly follows from that in Sec. D.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 658, + 324, + 672 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 686, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 686, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 270, + 699 + ], + "score": 1.0, + "content": "We compare the runtime complexity of", + "type": "text" + }, + { + "bbox": [ + 270, + 687, + 301, + 698 + ], + "score": 0.88, + "content": "\\operatorname { S T R } _ { \\mathrm { f r e e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 686, + 320, + 699 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 320, + 687, + 357, + 698 + ], + "score": 0.9, + "content": "{ \\mathrm { S T R } } _ { \\mathrm { f r e e } } +", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 686, + 505, + 699 + ], + "score": 1.0, + "content": "with existing Hessian free methods", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 696, + 504, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 230, + 711 + ], + "score": 1.0, + "content": "in Table 2. 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#sample#feature#sample#feature
a9a32.561123w8a49,749300
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AlgorithmSFOSSO
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AlgorithmRuntime
Hessian-free Cubic (Carmon & Duchi, 2016)Fast-Cubic (Agarwal et al., 2017)Stochastic Cubic (Tripuraneni et al., 2018)SRVRCfree (Zhou & Gu, 2019)（）0dn dn0.75+151.756d *35O(min{ d}）*
STRfree (this paper)STRfree+ (this paper)O(min{dn d dn0.5e1.75，2.752O(min{ dn0.75dn d dn0.51.5+e1.75，2.5+20)
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b/parse/train/JXAyJeYqUkZ/JXAyJeYqUkZ.md new file mode 100644 index 0000000000000000000000000000000000000000..eb40ff8f5cf8798c77e8193600dce7f29e18a987 --- /dev/null +++ b/parse/train/JXAyJeYqUkZ/JXAyJeYqUkZ.md @@ -0,0 +1,248 @@ +# CoAtNet: Marrying Convolution and Attention for All Data Sizes + +Zihang Dai, Hanxiao Liu, Quoc V. Le, Mingxing Tan Google Research, Brain Team {zihangd,hanxiaol,qvl,tanmingxing}@google.com + +# Abstract + +Transformers have attracted increasing interests in computer vision, but they still fall behind state-of-the-art convolutional networks. In this work, we show that while Transformers tend to have larger model capacity, their generalization can be worse than convolutional networks due to the lack of the right inductive bias. To effectively combine the strengths from both architectures, we present CoAtNets (pronounced “coat” nets), a family of hybrid models built from two key insights: (1) depthwise Convolution and self-Attention can be naturally unified via simple relative attention; (2) vertically stacking convolution layers and attention layers in a principled way is surprisingly effective in improving generalization, capacity and efficiency. Experiments show that our CoAtNets achieve state-of-the-art performance under different resource constraints across various datasets: Without extra data, CoAtNet achieves $8 6 . 0 \%$ ImageNet top-1 accuracy; When pre-trained with 13M images from ImageNet-21K, our CoAtNet achieves $8 8 . 5 6 \%$ top-1 accuracy, matching ViT-huge pre-trained with 300M images from JFT-300M while using $2 3 \mathrm { x }$ less data; Notably, when we further scale up CoAtNet with JFT-3B, it achieves $9 0 . 8 8 \%$ top-1 accuracy on ImageNet, establishing a new state-of-the-art result. + +# 1 Introduction + +Since the breakthrough of AlexNet [1], Convolutional Neural Networks (ConvNets) have been the dominating model architecture for computer vision [2, 3, 4, 5]. Meanwhile, with the success of self-attention models like Transformers [6] in natural language processing [7, 8], many previous works have attempted to bring in the power of attention into computer vision [9, 10, 11, 12]. More recently, Vision Transformer (ViT) [13] has shown that with almost1 only vanilla Transformer layers, one could obtain reasonable performance on ImageNet-1K [14] alone. More importantly, when pre-trained on large-scale weakly labeled JFT-300M dataset [15], ViT achieves comparable results to state-of-the-art (SOTA) ConvNets, indicating that Transformer models potentially have higher capacity at scale than ConvNets. + +While ViT has shown impressive results with enormous JFT 300M training images, its performance still falls behind ConvNets in the low data regime. For example, without extra JFT-300M pre-training, the ImageNet accuracy of ViT is still significantly lower than ConvNets with comparable model size [5] (see Table 13). Subsequent works use special regularization and stronger data augmentation to improve the vanilla ViT [16, 17, 18], yet none of these ViT variants could outperform the SOTA convolution-only models on ImageNet classification given the same amount of data and computation [19, 20]. This suggests that vanilla Transformer layers may lack certain desirable inductive biases possessed by ConvNets, and thus require significant amount of data and computational resource to compensate. Not surprisingly, many recent works have been trying to incorporate the inductive biases of ConvNets into Transformer models, by imposing local receptive fields for attention layers [21, 22] or augmenting the attention and FFN layers with implicit or explicit convolutional operations [23, 24, 25]. However, these approaches are either ad-hoc or focused on injecting a particular property, lacking a systematic understanding of the respective roles of convolution and attention when combined. + +In this work, we systematically study the problem of hybridizing convolution and attention from two fundamental aspects in machine learning – generalization and model capacity. Our study shows that convolutional layers tend to have better generalization with faster converging speed thanks to their strong prior of inductive bias, while attention layers have higher model capacity that can benefit from larger datasets. Combining convolutional and attention layers can achieve better generalization and capacity; however, a key challenge here is how to effectively combine them to achieve better trade-offs between accuracy and efficiency. In this paper, we investigate two key insights: First, we observe that the commonly used depthwise convolution can be effectively merged into attention layers with simple relative attention; Second, simply stacking convolutional and attention layers, in a proper way, could be surprisingly effective to achieve better generalization and capacity. Based on these insights, we propose a simple yet effective network architecture named CoAtNet, which enjoys the strengths from both ConvNets and Transformers. + +Our CoAtNet achieves SOTA performances under comparable resource constraints across different data sizes. Specifically, under the low-data regime, CoAtNet inherits the great generalization property of ConvNets thanks to the favorable inductive biases. Moreover, given abundant data, CoAtNet not only enjoys the superior scalability of Transformer models, but also achieves faster convergence and thus improved efficiency. When only ImageNet-1K is used for training, CoAtNet achieves $8 6 . 0 \%$ top-1 accuracy, matching the prior art NFNet [20] under similar computation resource and training conditions. Further, when pre-trained on ImageNet-21K with about 10M images, CoAtNet reaches $8 8 . 5 6 \%$ top-1 accuracy when finetuned on ImageNet-1K, matching the ViT-Huge pre-trained on JFT-300M, a $2 3 \times$ larger dataset. Finally, when JFT-3B is used for pre-training, CoAtNet exhibits better efficiency compared to ViT, and pushes the ImageNet-1K top-1 accuracy to $9 0 . 8 8 \%$ while using $1 . 5 \mathrm { x }$ less computation of the prior art set by ViT-G/14 [26]. + +# 2 Model + +In the section, we focus on the question of how to “optimally” combine the convolution and transformer. Roughly speaking, we decompose the question into two parts: + +1. How to combine the convolution and self-attention within one basic computational block? 2. How to vertically stack different types of computational blocks together to form a complete network? + +The rationale of the decomposition will become clearer as we gradually reveal our design choices. + +# 2.1 Merging Convolution and Self-Attention + +For convolution, we mainly focus on the MBConv block [27] which employs depthwise convolution [28] to capture the spatial interaction. A key reason of this choice is that both the FFN module in Transformer and MBConv employ the design of “inverted bottleneck”, which first expands the channel size of the input by $4 \mathbf { x }$ and later project the the $4 \mathbf { x }$ -wide hidden state back to the original channel size to enable residual connection. + +Besides the similarity of inverted bottleneck, we also notice that both depthwise convolution and self-attention can be expressed as a per-dimension weighted sum of values in a pre-defined receptive field. Specifically, convolution relies on a fixed kernel to gather information from a local receptive field + +$$ +y _ { i } = \sum _ { j \in \mathcal { L } ( i ) } w _ { i - j } \odot x _ { j } \quad \mathrm { ( d e p t h w i s e c o n v o l u t i o n ) } , +$$ + +where $x _ { i } , y _ { i } \in \mathbb { R } ^ { D }$ are the input and output at position $i$ respectively, and $\mathcal { L } ( i )$ denotes a local neighborhood of $i$ , e.g., a 3x3 grid centered at $i$ in image processing. + +In comparison, self-attention allows the receptive field to be the entire spatial locations and computes the weights based on the re-normalized pairwise similarity between the pair $( x _ { i } , x _ { j } )$ : 2 + +$$ +y _ { i } = \sum _ { j \in \mathcal { G } } \underbrace { \frac { \exp { \left( x _ { i } ^ { \top } x _ { j } \right) } } { \sum _ { k \in \mathcal { G } } \exp { \left( x _ { i } ^ { \top } x _ { k } \right) } } } _ { A _ { i , j } } x _ { j } \quad \mathrm { ( s e l f - a t t e n t i o n ) } , +$$ + +where $\mathcal { G }$ indicates the global spatial space. Before getting into the question of how to best combine them, it is worthwhile to compare their relative strengths and weaknesses, which helps to figure out the good properties we hope to retain. + +• First of all, the depthwise convolution kernel $w _ { i - j }$ is an input-independent parameter of static value, while the attention weight $A _ { i , j }$ dynamically depends on the representation of the input. Hence, it is much easier for the self-attention to capture complicated relational interactions between different spatial positions, a property that we desire most when processing high-level concepts. However, the flexibility comes with a risk of easier overfitting, especially when data is limited. • Secondly, notice that given any position pair $( i , j )$ , the corresponding convolution weight $w _ { i - j }$ only cares about the relative shift between them, i.e. $i - j$ , rather than the specific values of $i$ or $j$ . This property is often referred to translation equivalence, which has been found to improve generalization under datasets of limited size [29]. Due to the usage of absolution positional embeddings, standard Transformer (ViT) lacks this property. This partially explains why ConvNets are usually better than Transformers when the dataset is not enormously large. • Finally, the size of the receptive field is one of the most crucial differences between self-attention and convolution. Generally speaking, a larger receptive field provides more contextual information, which could lead to higher model capacity. Hence, the global receptive field has been a key motivation to employ self-attention in vision. However, a large receptive field requires significantly more computation. In the case of global attention, the complexity is quadratic w.r.t. spatial size, which has been a fundamental trade-off in applying self-attention models. + +Table 1: Desirable properties found in convolution or self-attention. + +
PropertiesConvolutionSelf-Attention
Translation Equivariance√
Input-adaptive Weighting√
Global Receptive Field
+ +Given the comparison above, an ideal model should be able to combine the 3 desirable properties in Table 1. With the similar form of depthwise convolution in Eqn. (1) and self-attention in Eqn. (2), a straightforward idea that could achieve this is simply to sum a global static convolution kernel with the adaptive attention matrix, either after or before the Softmax normalization, i.e., + +$$ +y _ { i } ^ { \mathrm { p o s t } } = \sum _ { j \in \mathcal { G } } \left( \frac { \exp \left( x _ { i } ^ { \top } x _ { j } \right) } { \sum _ { k \in \mathcal { G } } \exp \left( x _ { i } ^ { \top } x _ { k } \right) } + w _ { i - j } \right) x _ { j } \ \mathrm { ~ o r ~ } \ y _ { i } ^ { \mathrm { p e } } = \sum _ { j \in \mathcal { G } } \frac { \exp \left( x _ { i } ^ { \top } x _ { j } + w _ { i - j } \right) } { \sum _ { k \in \mathcal { G } } \exp \left( x _ { i } ^ { \top } x _ { k } + w _ { i - k } \right) } x _ { j } . +$$ + +Interestingly, while the idea seems overly simplified, the pre-normalization version $y ^ { \mathrm { p r e } }$ corresponds to a particular variant of relative self-attention [30, 31]. In this case, the attention weight $A _ { i , j }$ is decided jointly by the $w _ { i - j }$ of translation equivariance and the input-adaptive $x _ { i } ^ { \top } x _ { j }$ , which can enjoy both effects depending on their relative magnitudes. Importantly, note that in order to enable the global convolution kernel without blowing up the number of parameters, we have reloaded the notation of $w _ { i - j }$ as a scalar (i.e., $w \in \mathbb { R } ^ { O ( | \mathcal { G } | ) }$ ) rather than a vector in Eqn. (1). Another advantage of the scalar formulation of $w$ is that retrieving $w _ { i - j }$ for all $( i , j )$ is clearly subsumed by computing the pairwise dot-product attention, hence resulting in minimum additional cost (see Appendix A.1). Given the benefits, we will use the Transformer block with the pre-normalization relative attention variant in Eqn. (3) as the key component of the proposed CoAtNet model. + +# 2.2 Vertical Layout Design + +After figuring out a neat way to combine convolution and attention, we next consider how to utilize it to stack an entire network. + +As we have discuss above, the global context has a quadratic complexity w.r.t. the spatial size. Hence, if we directly apply the relative attention in Eqn. (3) to the raw image input, the computation will be excessively slow due to the large number of pixels in any image of common sizes. Hence, to construct a network that is feasible in practice, we have mainly three options: + +(A) Perform some down-sampling to reduce the spatial size and employ the global relative attention after the feature map reaches manageable level. +(B) Enforce local attention, which restricts the global receptive field $\mathcal { G }$ in attention to a local field $\mathcal { L }$ just like in convolution [22, 21]. +(C) Replace the quadratic Softmax attention with certain linear attention variant which only has a linear complexity w.r.t. the spatial size [12, 32, 33]. + +We briefly experimented with option (C) without getting a reasonably good result. For option (B), we found that implementing local attention involves many non-trivial shape formatting operations that requires intensive memory access. On our accelerator of choice (TPU), such operation turns out to be extremely slow [34], which not only defeats the original purpose of speeding up global attention, but also hurts the model capacity. Hence, as some recent work has studied this variant [22, 21], we will focus on option (A) and compare our results with theirs in our empirical study (Section 4). + +For option (A), the down-sampling can be achieved by either (1) a convolution stem with aggressive stride (e.g., stride 16x16) as in ViT or (2) a multi-stage network with gradual pooling as in ConvNets. With these choices, we derive a search space of 5 variants and compare them in controlled experiments. + +• When the ViT Stem is used, we directly stack $L$ Transformer blocks with relative attention, which we denote as $\mathrm { V I T } _ { \mathrm { R E L } }$ . +• When the multi-stage layout is used, we mimic ConvNets to construct a network of 5 stages (S0, S1, S2, S3 & S4), with spatial resolution gradually decreased from S0 to S4. At the beginning of each stage, we always reduce the spatial size by $2 \mathbf { x }$ and increase the number of channels (see Appendix A.1 for the detailed down-sampling implementation). The first stage S0 is a simple 2-layer convolutional Stem and S1 always employs MBConv blocks with squeeze-excitation (SE), as the spatial size is too large for global attention. Starting from S2 through S4, we consider either the MBConv or the Transformer block, with a constraint that convolution stages must appear before Transformer stages. The constraint is based on the prior that convolution is better at processing local patterns that are more common in early stages. This leads to 4 variants with increasingly more Transformer stages, C-C-C-C, C-C-C-T, C-C-T-T and C-T-T-T, where C and T denote Convolution and Transformer respectively. + +To systematically study the design choices, we consider two fundamental aspects generalization capability and model capacity: For generalization, we are interested in the gap between the training loss and the evaluation accuracy. If two models have the same training loss, then the model with higher evaluation accuracy has better generalization capability, since it can generalize better to unseen evaluation dataset. Generalization capability is particularly important to data efficiency when training data size is limited. For model capacity, we measure the ability to fit large training datasets. When training data is abundant and overfitting is not an issue, the model with higher capacity will achieve better final performance after reasonable training steps. Note that, since simply increasing the model size can lead to higher model capacity, to perform a meaningful comparison, we make sure the model sizes of the 5 variants are comparable. + +To compare the generalization and model capacity, we train different variants of hybrid models on ImageNet-1K (1.3M) and JFT $\left( > 3 0 0 \mathbf { M } \right)$ dataset for 300 and 3 epochs respectively, both without any regularization or augmentation. The training loss and evaluation accuracy on both datasets are summarized in Figure 1. + +• From the ImageNet-1K results, a key observation is that, in terms of generalization capability (i.e., gap between train and evaluation metrics), we have + +$$ +\mathrm { C \mathrm { - } C \mathrm { - } C \mathrm { - } C \approx C \mathrm { - } C \mathrm { - } C \mathrm { - } T \ge C \mathrm { - } C \mathrm { - } T \mathrm { - } T > C \mathrm { - } T \mathrm { - } T \mathrm { - } T \gg V \mathrm { I } \mathrm { T } _ { \mathrm { R E L } } . } +$$ + +![](images/8e4cc944c4e098b4c2962d8c1e585910035ad5a220c25df539ba8f057d9c47df.jpg) +Figure 1: Comparison for model generalization and capacity under different data size. For fair comparison, all models have similar parameter size and computational cost. + +Particularly, $\mathrm { V I T } _ { \mathrm { R E L } }$ is significantly worse than variants by a large margin, which we conjecture is related to the lack of proper low-level information processing in its aggressive down-sampling Stem. Among the multi-stage variants, the overall trend is that the more convolution stages the model has, the smaller the generalization gap is. + +• As for model capacity, from the JFT comparison, both the train and evaluation metrics at the end of the training suggest the following ranking: + +$$ +\mathrm { C - C \mathrm { - } T \mathrm { - } T \approx C \mathrm { - } T \mathrm { - } T \mathrm { - } T > V I T _ { R E L } > C \mathrm { - } C \mathrm { - } C \mathrm { - } T > C \mathrm { - } C \mathrm { - } C \mathrm { - } C \mathrm { - } C \mathrm { . } } +$$ + +Importantly, this suggests that simply having more Transformer blocks does NOT necessarily mean higher capacity for visual processing. On one hand, while initially worse, $\mathrm { V I T } _ { \mathrm { R E L } }$ ultimately catch up with the two variants with more MBConv stages, indicating the capacity advantage of Transformer blocks. On the other hand, both C-C-T-T and C-T-T-T clearly outperforming $\mathrm { V I T } _ { \mathrm { R E L } }$ suggest that the ViT stem with an aggressive stride may have lost too much information and hence limit the model capacity. More interestingly, the fact that C-C-T-T $\approx \mathbf { C }$ -T-T-T indicates the for processing low-level information, static local operations like convolution could be as capable as adaptive global attention mechanism, while saving computation and memory usage substantially. + +Finally, to decide between C-C-T-T and C-T-T-T, we conduct another transferability test3 — we finetune the two JFT pre-trained models above on ImageNet-1K for 30 epochs and compare their transfer performances. From Table 2, it turns out that C-C-T-T achieves a clearly better transfer accuracy than C-T-T-T, despite the same pre-training performance. + +Table 2: Transferability test results. + +
MetricC-C-T-TC-T-T-T
Pre-training Precision@1 (JFT)34.4034.36
Transfer Accuracy 224x22482.3981.78
Transfer Accuracy 384x38484.2384.02
+ +Taking generalization, model capacity, transferability and efficiency into consideration, we adapt the C-C-T-T multi-stage layout for CoAtNet. More model details are included in Appendix A.1. + +# 3 Related Work + +Convolutional network building blocks. Convolutional Networks (ConvNets) have been the dominating neural architectures for many computer vision tasks. Traditionally, regular convolutions, such as ResNet blocks [3], are popular in large-scale ConvNets; in contrast, depthwise convolutions [28] are popular in mobile platforms due to its lower computational cost and smaller parameter size [27]. Recent works show that an improved inverted residual bottlenecks (MBConv [27, 35]), which is built upon depthwise convolutions, can achieve both high accuracy and better efficiency [5, 19]. As discussed in Section 2, due to the strong connection between MBConv and Transformer blocks , this paper mostly employs MBConv as convolution building blocks. + +Self-attention and Transformers. With the key ingredients of self-attention, Transformers have been widely adopted for neural language processing and speech understanding. As an early work, stand-alone self-attention network [34] shows self-attention alone can work well for different vision tasks, though with some practical difficulties. Recently, ViT [13] applies a vanilla Transformer to ImageNet classification, and achieves impressive results after pre-training on a large-scale JFT dataset. However, ViT still largely lags behind state-of-the-art ConvNets when training data is limited. Following that, many recent works have been focused on improving vision Transformers for data efficiency and model efficiency. For a more comprehensive review of vision Transformers, we refer readers to the dedicated surveys [36, 37]. + +Relative attention. Under the general name of relative attention, there have been various variants in literature [30, 38, 39, 34, 40, 31]. Generally speaking, we can separate them into two categories: (a) the input-dependent version where the extra relative attention score is a function of the input states $f ( \bar { x } _ { i } , x _ { j } , \bar { i } - j )$ , and (b) the input-independent version $f ( i - j )$ . The variant in CoAtNet belongs to the input-independent version, and is similar to the one used in T5 [31], but unlike T5, we neither share the relative attention parameters across layers nor use the bucketing mechanism. As a benefit of the input independence, obtaining $f ( i - j )$ for all $( i , j )$ pairs is computationally much cheaper than the input-dependent version on TPU. In addition, at inference time, this only needs to be computed once and cached for future use. A recent work [22] also utilizes such an input-independent parameterization, but it restricts the receptive field to a local window. + +Combining convolution and self-attention. The idea of combining convolution and self-attention for vision recognition is not new. A common approach is to augment the ConvNet backbone with explicit self-attention or non-local modules [9, 10, 11, 12], or to replace certain convolution layers with standard self-attention [11] or a more flexible mix of linear attention and convolution [41]. While self-attention usually improves the accuracy, they often come with extra computational cost and hence are often regarded as an add-on to the ConvNets, similar to squeeze-and-excitation [42] module. In comparison, after the success of ViT and ResNet-ViT [13], another popular line of research starts with a Transformer backbone and tries to incorporate explicit convolution or some desirable properties of convolution into the Transformer backbone [25, 24, 23, 22, 21, 43, 44]. + +While our work also belongs to this category, we show that our relative attention instantiation is a natural mixture of depthwise convolution and content-based attention with minimum additional cost. More importantly, starting from the perspectives of generalization and model capacity, we take a systematic approach to the vertical layout design and show how and why different network stages prefer different types of layers. Therefore, compared to models that simply use an off-the-shelf ConvNet as the stem layer, such as ResNet-ViT [13], CoAtNet also scales the Convolution stage (S2) when the overall size increases. On the other hand, compared to models employing local attention [22, 21], CoAtNet consistently uses full attention for S3 & S4 to ensure the model capacity, as S3 occupies the majority of the computation and parameters. + +# 4 Experiments + +In this section, we compare CoAtNet with previous results under comparable settings. For completeness, all the hyper-parameters not mentioned here are included in Appendix A.2. + +# 4.1 Experiment Setting + +CoAtNet model family. To compare with existing models of different sizes, we also design a family of CoAtNet models as summarized in Table 3. Overall, we always double the number of channels from S1 to S4, while ensuring the width of the Stem S0 to be smaller or equal to that of S1. Also, for simplicity, when increasing the depth of the network, we only scale the number of blocks in S2 and S3. + +Evaluation Protocol. Our experiments focus on image classification. To evaluate the performance of the model across different data sizes, we utilize three datasets of increasingly larger sizes, namely ImageNet-1K (1.28M images), ImageNet-21K (12.7M images) and JFT (300M images). Following previous works, we first pre-train our models on each of the three datasets at resolution 224 for 300, 90 and 14 epochs respectively. Then, we finetune the pre-trained models on ImageNet-1K at the desired resolutions for 30 epochs and obtain the corresponding evaluation accuracy. One exception is the ImageNet-1K performance at resolution 224, which can be directly obtained at the end of pre-training. Note that similar to other models utilizing Transformer blocks, directly evaluating models pre-trained on ImageNet-1K at a larger resolution without finetuning usually leads to performance drop. Hence, finetuning is always employed whenever input resolution changes. + +Table 3: L denotes the number of blocks and D denotes the hidden dimension (#channels). For all Conv and MBConv blocks, we always use the kernel size 3. For all Transformer blocks, we set the size of each attention head to 32, following [22]. The expansion rate for the inverted bottleneck is always 4 and the expansion (shrink) rate for the SE is always 0.25. + +
StagesSizeCoAtNet-0CoAtNet-1CoAtNet-2CoAtNet-3CoAtNet-4
S0-Conv1/2L=2 D=64L=2D=64L=2D=128 L=2D=192L=2 D=192
S1-MbConv1/4L=2 D=96L=2D=96L=2 D=128L=2D=192L=2 D=192
S2-MBConv1/8L=3 D=192L=6D=192L=6 D=256L=6D=384L=12 D=384
S3-TFMRel1/16L=5 D=384L=14D=384L=14 D=512L=14D=768L=28 D=768
S4-TFMRel1/32L=2 D=768L=2D=768L=2 D=1024L=2D=1536L=2 D=1536
+ +Data Augmentation & Regularization. In this work, we only consider two widely used data augmentations, namely RandAugment [45] and MixUp [46], and three common techniques, including stochastic depth [47], label smoothing [48] and weight decay [49], to regularize the model. Intuitively, the specific hyper-parameters of the augmentation and regularization methods depend on model size and data scale, where strong regularization is usually applied for larger models and smaller dataset. + +Under the general principle, a complication under the current paradigm is how to adjust the regularization for pre-training and finetuning as data size can change. Specifically, we have an interesting observation that if a certain type of augmentation is entirely disabled during pre-training, simply turning it on during fine-tuning would most likely harm the performance rather than improving. We conjecture this could be related to data distribution shift. As a result, for certain runs of the proposed model, we deliberately apply RandAugment and stochastic depth of a small degree when pre-training on the two larger datasets, ImageNet21-K and JFT. Although such regularization can harm the pre-training metrics, this allows more versatile regularization and augmentation during finetuning, leading to improved down-stream performances. + +# 4.2 Main Results + +![](images/cf1567b93b834a38bb5d149ee37f969799ca7bcd6b3f63af756a7cd673e8c469.jpg) +Figure 2: Accuracy-to-FLOPs scaling curve under ImageNet-1K only setting at $2 2 4 \mathbf { x } 2 2 4$ . + +![](images/f8c09d4a039db4e8c22000a3a3a66bae5de7e541a68ec3f2c6a6234546bf4ac2.jpg) +Figure 3: Accuracy-to-Params scaling curve under ImageNet- $2 1 \mathrm { K } \Rightarrow$ ImageNet-1K setting. + +ImageNet-1K The experiment results with only the ImageNet-1K dataset are shown in Table 4. Under similar conditions, the proposed CoAtNet models not only outperform ViT variants, but also match the best convolution-only architectures, i.e., EfficientNet-V2 and NFNets. Additionally, we also visualize the all results at resolution $2 2 4 \mathbf { x } 2 2 4$ in Fig. 2. As we can see, CoAtNet scales much better than previous model with attention modules. + +Table 4: Model performance on ImageNet. 1K only denotes training on ImageNet-1K only; $2 1 \mathtt { K } + 1 \mathtt { K }$ denotes pre-training on ImageNet-21K and finetuning on ImageNet-1K; PT-RA denotes applying RandAugment during 21K pre-training, and E150 means 150 epochs of 21K pre-training, which is longer than the standard 90 epochs. More results are in Appendix A.3. + +
ModelsEval Size#Params#FLOPsImageNet Top-1 Accuracy
Conv Only1K only21K+1K
EfficientNet-B7600²66M37B84.7-
EfficientNetV2-L480²121M53B85.786.8
NFNet-F34162²255M114.8B85.7=
NFNet-F55442377M289.8B86.01
ViT-Stem TFMDeiT-B384286M55.4B83.1-
ViT-L/16384²304M190.7B-85.3
CaiT-S-36384²68M48.0B85.0
DeepViT-L224²55M12.5B83.1-
Multi-stage TFMSwin-B384²88M47.0B84.286.0
Swin-L384²197M103.9B-86.4
Conv+TFMBotNet-T7384275.1M45.8B84.7-
LambdaResNet-420320²-=84.8
T2T-ViT-24224²64.1M15.0B82.6=
CvT-21384²32M24.9B83.3-
CvT-W243842277M193.2B-87.7
Conv+TFM (ours)CoAtNet-0 CoAtNet-1224²25M4.2B81.6=
224²42M8.4B83.3-
CoAtNet-2 CoAtNet-3224²75M15.7B84.187.1
2242168M34.7B84.587.6
CoAtNet-0384²25M13.4B83.9-
CoAtNet-1384242M27.4B85.1-
CoAtNet-2384²75M49.8B85.787.1
CoAtNet-3384²168M107.4B85.887.6
CoAtNet-4384²275M189.5B-87.9
+ PT-RA384²275M189.5B88.3
+ PT-RA-E1503842275M189.5B88.4
CoAtNet-2512275M96.7B85.987.3
CoAtNet-3512²168M203.1B86.087.9
CoAtNet-4512²275M360.9B-88.1
+ PT-RA512²275M360.9B=88.4
+ PT-RA-E1505122275M360.9B=88.56
+ +ImageNet-21K As we can see from Table 4 and Fig. 3, when ImageNet-21K is used for pretraining, the advantage of CoAtNet becomes more obvious, substantially outperforming all previous models. Notably, the best CoAtNet variant achieves a top-1 accuracy of $8 8 . 5 6 \%$ , matching the ViTH/14 performance of $8 8 . 5 5 \%$ , which requires pre-training the $2 . 3 \mathbf { x }$ larger ViT model on a $2 3 \mathrm { x }$ larger proprietary weakly labeled dataset (JFT) for $2 . 2 \mathbf { x }$ more steps. This marks a dramatic improvement in both data efficiency and computation efficiency. + +JFT Finally, in Table 5, we further evaluate CoAtNet under the large-scale data regime with JFT300M and JFT-3B. Encouragingly, our CoAtNet-4 can almost match the best previous performance with JFT-300M set by NFNet- $\mathrm { F 4 + }$ , while being $2 \mathbf { x }$ more efficient in terms of both TPU training time and parameter count. When we scale up the model to consume similar training resource as NFNet- $. \mathrm { F 4 + }$ , CoAtNet-5 reaches $8 9 . 7 7 \%$ on top-1 accuracy, outperforming previous results under comparable settings. + +Moreover, as we further push the training resource towards the level used by ViT-G/14 and utilize the same JFT-3B dataset of an even larger size [26], with over $4 \mathbf { x }$ less computation, CoAtNet-6 is able to match the performance of $\mathrm { V i T - G } / 1 4$ of $9 0 . 4 5 \%$ , and with $1 . 5 \mathrm { x }$ less computation, CoAtNet-7 achieves $8 9 . 7 7 \%$ on top-1 accuracy $9 0 . 8 8 \%$ , achieving the new state-of-the-art performance. + +Table 5: Performance Comparison on large-scale JFT dataset. TPUv3-core-days denotes the pretraining time, Top-1 Accuracy denotes the finetuned accuracy on ImageNet. Note that the last 3 rows use a larger dataset JFT-3B [26] for pre-training, while others use JFT-300M [15]. See Appendix A.2 for the size details of CoAtNet-5/6/7. †: Down-sampling in the MBConv block is achieved by stride-2 Depthwise Convolution. ⇧: ViT-G/14 computation consumption is read from Fig. 1 of the paper [26]. + +
ModelsEval Size#Params#FLOPsTPUv3-core-daysTop-1 Accuracy
ResNet + ViT-L/163842330M=187.12
ViT-L/165122307M364B0.68K87.76
ViT-H/145182632M1021B2.5K88.55
NFNet-F4+5122527M367B1.86K89.2
CoAtNet-3t3842168M114B0.58K88.52
CoAtNet-3t5122168M214B0.58K88.81
CoAtNet-45122275M361B0.95K89.11
CoAtNet-55122688M812B1.82K89.77
ViT-G/1451821.84B5160B>30K90.45
CoAtNet-651221.47B1521B6.6K90.45
CoAtNet-751222.44B2586B20.1K90.88
+ +# 4.3 Ablation Studies + +In this section, we will ablate our design choices for CoAtNet. + +Firstly, we study the importance of the relative attention from combining convolution and attention into a single computation unit. Specifically, we compare two models, one with the relative attention and the other without, under both the ImageNet-1K alone and ImageNet-21K transfer setting. As we can see from Table 6, when only the ImageNet-1K is used, relative attention clearly outperforms the standard attention, indicating a better generalization. In addition, under the ImageNet-21K transfer setting, the relative attention variant achieves a substantially better transfer accuracy, despite their very close pre-training performances. This suggests the main advantage of relative attention in visual processing is not in higher capacity but in better generalization. + +Table 6: Ablation on relative attention. + +
SetingMetricWith Rel-AttnWithout Rel-Attn
ImageNet-1KAccuracy (2242)84.183.8
Accuracy (3842)85.785.3
ImageNet-21K →ImageNet-1KPre-train Precision@1 (224²)53.052.8
Finetune Accuracy (384²)87.987.4
+ +Table 7: Ablation on architecture layout. + +
SettingModelsLayoutTop-1 Accuracy
ImageNet-1KVO: CoAtNet-2[2,2,6,14，2]84.1
V1: S2← S3[2,2, 2,18,2]83.4
V2: S2→ S3[2,2,8,12,2]84.0
ImageNet-21KVO: CoAtNet-3[2,2,6,14,2]53.0 -→87.6
⇒ImageNet-1KV1: S2 ← S3[2,2,2,18,2]53.0 -→87.4
+ +Secondly, as S2 with MBConv blocks and S3 with relative Transformer blocks occupy most of the computation of the CoAtNet, a question to ask is how to split the computation between S2 (MBConv) and S3 (Transformer) to achieve a good performance. In practice, it boils down to deciding the number of blocks to have in each stage, which we will refer to as “layout” design. For this purpose, we compare a few different layouts that we experimented with in Table 7. + +Table 8: Ablation on head size and normalization type. + +
SettingModelsImage SizeTop-1 Accuracy
ImageNet-1KCoAtNet-2224284.1
Head size: 32 → 64224283.9
Norm type: 1 BN →LN224284.1
ImageNet-21K ⇒ ImageNet-1KCoAtNet-3384287.9
Norm type: BN →→ LN384²87.8
+ +• If we keep the total number of blocks in S2 and S3 fixed and vary the number in each stage, we observe that V0 is a sweet spot between V1 and V2. Basically, having more Transformer blocks in S3 generally leads to better performance until the number of MBConv blocks in S2 is too small to generalize well. + +• To further evaluate whether the sweet spot also holds in the transfer setting, where a higher capacity is often regarded more important, we further compare V0 and V1 under the ImageNet21K transferring to ImageNet-1K setup. Interestingly, despite that V1 and V0 have the same performance during ImageNet-21K pre-training, the transfer accuracy of V1 clearly falls behind V0. Again, this suggests the importance of convolution in achieving good transferability and generalization. + +Lastly, we study two choices of model details, namely the dimension of each attention (default to 32) head as well as the type of normalization (default to BatchNorm) used in MBConv blocks. From Table 8, we can see increasing head size from 32 to 64 can slightly hurt performance, though it actually improves the TPU speed by a significant amount. In practice, this will be a quality-speed trade-off one can make. On the other hand, BatchNorm and LayerNorm have almost the same performance, while BatchNorm is $10 - 2 0 \%$ faster on TPU depending on the per-core batch size. + +# 5 Conclusion + +In this paper, we systematically study the properties of convolutions and Transformers, which leads to a principled way to combine them into a new family of models named CoAtNet. Extensive experiments show that CoAtNet enjoys both good generalization like ConvNets and superior model capacity like Transformers, achieving state-of-the-art performances under different data sizes and computation budgets. + +Note that this paper currently focuses on ImageNet classification for model development. 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Springer, 2016. +[51] Dan Hendrycks and Kevin Gimpel. Gaussian error linear units (gelus). arXiv preprint arXiv:1606.08415, 2016. +[52] Zihang Dai, Guokun Lai, Yiming Yang, and Quoc V Le. Funnel-transformer: Filtering out sequential redundancy for efficient language processing. arXiv preprint arXiv:2006.03236, 2020. \ No newline at end of file diff --git a/parse/train/JXAyJeYqUkZ/JXAyJeYqUkZ_content_list.json b/parse/train/JXAyJeYqUkZ/JXAyJeYqUkZ_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..f733772f0dbe2279236643d34cb3f1f5101bd143 --- /dev/null +++ b/parse/train/JXAyJeYqUkZ/JXAyJeYqUkZ_content_list.json @@ -0,0 +1,1045 @@ +[ + { + "type": "text", + "text": "CoAtNet: Marrying Convolution and Attention for All Data Sizes ", + "text_level": 1, + "bbox": [ + 214, + 122, + 785, + 171 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Zihang Dai, Hanxiao Liu, Quoc V. Le, Mingxing Tan Google Research, Brain Team {zihangd,hanxiaol,qvl,tanmingxing}@google.com ", + "bbox": [ + 307, + 220, + 691, + 263 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 299, + 535, + 315 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Transformers have attracted increasing interests in computer vision, but they still fall behind state-of-the-art convolutional networks. In this work, we show that while Transformers tend to have larger model capacity, their generalization can be worse than convolutional networks due to the lack of the right inductive bias. To effectively combine the strengths from both architectures, we present CoAtNets (pronounced “coat” nets), a family of hybrid models built from two key insights: (1) depthwise Convolution and self-Attention can be naturally unified via simple relative attention; (2) vertically stacking convolution layers and attention layers in a principled way is surprisingly effective in improving generalization, capacity and efficiency. Experiments show that our CoAtNets achieve state-of-the-art performance under different resource constraints across various datasets: Without extra data, CoAtNet achieves $8 6 . 0 \\%$ ImageNet top-1 accuracy; When pre-trained with 13M images from ImageNet-21K, our CoAtNet achieves $8 8 . 5 6 \\%$ top-1 accuracy, matching ViT-huge pre-trained with 300M images from JFT-300M while using $2 3 \\mathrm { x }$ less data; Notably, when we further scale up CoAtNet with JFT-3B, it achieves $9 0 . 8 8 \\%$ top-1 accuracy on ImageNet, establishing a new state-of-the-art result. ", + "bbox": [ + 233, + 332, + 766, + 551 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 176, + 575, + 310, + 593 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Since the breakthrough of AlexNet [1], Convolutional Neural Networks (ConvNets) have been the dominating model architecture for computer vision [2, 3, 4, 5]. Meanwhile, with the success of self-attention models like Transformers [6] in natural language processing [7, 8], many previous works have attempted to bring in the power of attention into computer vision [9, 10, 11, 12]. More recently, Vision Transformer (ViT) [13] has shown that with almost1 only vanilla Transformer layers, one could obtain reasonable performance on ImageNet-1K [14] alone. More importantly, when pre-trained on large-scale weakly labeled JFT-300M dataset [15], ViT achieves comparable results to state-of-the-art (SOTA) ConvNets, indicating that Transformer models potentially have higher capacity at scale than ConvNets. ", + "bbox": [ + 174, + 607, + 825, + 732 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "While ViT has shown impressive results with enormous JFT 300M training images, its performance still falls behind ConvNets in the low data regime. For example, without extra JFT-300M pre-training, the ImageNet accuracy of ViT is still significantly lower than ConvNets with comparable model size [5] (see Table 13). Subsequent works use special regularization and stronger data augmentation to improve the vanilla ViT [16, 17, 18], yet none of these ViT variants could outperform the SOTA convolution-only models on ImageNet classification given the same amount of data and computation [19, 20]. This suggests that vanilla Transformer layers may lack certain desirable inductive biases possessed by ConvNets, and thus require significant amount of data and computational resource to compensate. Not surprisingly, many recent works have been trying to incorporate the inductive biases of ConvNets into Transformer models, by imposing local receptive fields for attention layers [21, 22] or augmenting the attention and FFN layers with implicit or explicit convolutional operations [23, 24, 25]. However, these approaches are either ad-hoc or focused on injecting a particular property, lacking a systematic understanding of the respective roles of convolution and attention when combined. ", + "bbox": [ + 174, + 738, + 826, + 877 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 147 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this work, we systematically study the problem of hybridizing convolution and attention from two fundamental aspects in machine learning – generalization and model capacity. Our study shows that convolutional layers tend to have better generalization with faster converging speed thanks to their strong prior of inductive bias, while attention layers have higher model capacity that can benefit from larger datasets. Combining convolutional and attention layers can achieve better generalization and capacity; however, a key challenge here is how to effectively combine them to achieve better trade-offs between accuracy and efficiency. In this paper, we investigate two key insights: First, we observe that the commonly used depthwise convolution can be effectively merged into attention layers with simple relative attention; Second, simply stacking convolutional and attention layers, in a proper way, could be surprisingly effective to achieve better generalization and capacity. Based on these insights, we propose a simple yet effective network architecture named CoAtNet, which enjoys the strengths from both ConvNets and Transformers. ", + "bbox": [ + 174, + 154, + 825, + 319 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our CoAtNet achieves SOTA performances under comparable resource constraints across different data sizes. Specifically, under the low-data regime, CoAtNet inherits the great generalization property of ConvNets thanks to the favorable inductive biases. Moreover, given abundant data, CoAtNet not only enjoys the superior scalability of Transformer models, but also achieves faster convergence and thus improved efficiency. When only ImageNet-1K is used for training, CoAtNet achieves $8 6 . 0 \\%$ top-1 accuracy, matching the prior art NFNet [20] under similar computation resource and training conditions. Further, when pre-trained on ImageNet-21K with about 10M images, CoAtNet reaches $8 8 . 5 6 \\%$ top-1 accuracy when finetuned on ImageNet-1K, matching the ViT-Huge pre-trained on JFT-300M, a $2 3 \\times$ larger dataset. Finally, when JFT-3B is used for pre-training, CoAtNet exhibits better efficiency compared to ViT, and pushes the ImageNet-1K top-1 accuracy to $9 0 . 8 8 \\%$ while using $1 . 5 \\mathrm { x }$ less computation of the prior art set by ViT-G/14 [26]. ", + "bbox": [ + 173, + 325, + 825, + 478 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Model ", + "text_level": 1, + "bbox": [ + 174, + 505, + 259, + 521 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In the section, we focus on the question of how to “optimally” combine the convolution and transformer. Roughly speaking, we decompose the question into two parts: ", + "bbox": [ + 173, + 540, + 823, + 569 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1. How to combine the convolution and self-attention within one basic computational block? 2. How to vertically stack different types of computational blocks together to form a complete network? ", + "bbox": [ + 200, + 574, + 825, + 622 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The rationale of the decomposition will become clearer as we gradually reveal our design choices. ", + "bbox": [ + 169, + 628, + 816, + 645 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 Merging Convolution and Self-Attention ", + "text_level": 1, + "bbox": [ + 176, + 667, + 493, + 683 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "For convolution, we mainly focus on the MBConv block [27] which employs depthwise convolution [28] to capture the spatial interaction. A key reason of this choice is that both the FFN module in Transformer and MBConv employ the design of “inverted bottleneck”, which first expands the channel size of the input by $4 \\mathbf { x }$ and later project the the $4 \\mathbf { x }$ -wide hidden state back to the original channel size to enable residual connection. ", + "bbox": [ + 173, + 695, + 825, + 765 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Besides the similarity of inverted bottleneck, we also notice that both depthwise convolution and self-attention can be expressed as a per-dimension weighted sum of values in a pre-defined receptive field. Specifically, convolution relies on a fixed kernel to gather information from a local receptive field ", + "bbox": [ + 173, + 771, + 825, + 827 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/6674e066791bbf528f350e89a70d47c627c6a3fb12df1f14900957b317ba58ee.jpg", + "text": "$$\ny _ { i } = \\sum _ { j \\in \\mathcal { L } ( i ) } w _ { i - j } \\odot x _ { j } \\quad \\mathrm { ( d e p t h w i s e c o n v o l u t i o n ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 333, + 832, + 661, + 868 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $x _ { i } , y _ { i } \\in \\mathbb { R } ^ { D }$ are the input and output at position $i$ respectively, and $\\mathcal { L } ( i )$ denotes a local neighborhood of $i$ , e.g., a 3x3 grid centered at $i$ in image processing. ", + "bbox": [ + 174, + 882, + 823, + 912 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In comparison, self-attention allows the receptive field to be the entire spatial locations and computes the weights based on the re-normalized pairwise similarity between the pair $( x _ { i } , x _ { j } )$ : 2 ", + "bbox": [ + 169, + 90, + 823, + 119 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/a9164aac5bb6f55e0bd90b25f31a8559f39a62986021c906fcf5f7116eba58d0.jpg", + "text": "$$\ny _ { i } = \\sum _ { j \\in \\mathcal { G } } \\underbrace { \\frac { \\exp { \\left( x _ { i } ^ { \\top } x _ { j } \\right) } } { \\sum _ { k \\in \\mathcal { G } } \\exp { \\left( x _ { i } ^ { \\top } x _ { k } \\right) } } } _ { A _ { i , j } } x _ { j } \\quad \\mathrm { ( s e l f - a t t e n t i o n ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 333, + 128, + 663, + 188 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\mathcal { G }$ indicates the global spatial space. Before getting into the question of how to best combine them, it is worthwhile to compare their relative strengths and weaknesses, which helps to figure out the good properties we hope to retain. ", + "bbox": [ + 174, + 195, + 825, + 238 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "• First of all, the depthwise convolution kernel $w _ { i - j }$ is an input-independent parameter of static value, while the attention weight $A _ { i , j }$ dynamically depends on the representation of the input. Hence, it is much easier for the self-attention to capture complicated relational interactions between different spatial positions, a property that we desire most when processing high-level concepts. However, the flexibility comes with a risk of easier overfitting, especially when data is limited. • Secondly, notice that given any position pair $( i , j )$ , the corresponding convolution weight $w _ { i - j }$ only cares about the relative shift between them, i.e. $i - j$ , rather than the specific values of $i$ or $j$ . This property is often referred to translation equivalence, which has been found to improve generalization under datasets of limited size [29]. Due to the usage of absolution positional embeddings, standard Transformer (ViT) lacks this property. This partially explains why ConvNets are usually better than Transformers when the dataset is not enormously large. • Finally, the size of the receptive field is one of the most crucial differences between self-attention and convolution. Generally speaking, a larger receptive field provides more contextual information, which could lead to higher model capacity. Hence, the global receptive field has been a key motivation to employ self-attention in vision. However, a large receptive field requires significantly more computation. In the case of global attention, the complexity is quadratic w.r.t. spatial size, which has been a fundamental trade-off in applying self-attention models. ", + "bbox": [ + 173, + 242, + 826, + 486 + ], + "page_idx": 2 + }, + { + "type": "table", + "img_path": "images/d75e2fc9a7e5f7449242faadd95d844ba4bb23d1ab343baa75e9dc6fb7c8645c.jpg", + "table_caption": [ + "Table 1: Desirable properties found in convolution or self-attention. " + ], + "table_footnote": [], + "table_body": "
PropertiesConvolutionSelf-Attention
Translation Equivariance√
Input-adaptive Weighting√
Global Receptive Field
", + "bbox": [ + 285, + 525, + 702, + 595 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Given the comparison above, an ideal model should be able to combine the 3 desirable properties in Table 1. With the similar form of depthwise convolution in Eqn. (1) and self-attention in Eqn. (2), a straightforward idea that could achieve this is simply to sum a global static convolution kernel with the adaptive attention matrix, either after or before the Softmax normalization, i.e., ", + "bbox": [ + 178, + 612, + 821, + 667 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/6ad701e539bf342fce3258bfc8889c519cd941777a414d3170d3f20d5b95a53d.jpg", + "text": "$$\ny _ { i } ^ { \\mathrm { p o s t } } = \\sum _ { j \\in \\mathcal { G } } \\left( \\frac { \\exp \\left( x _ { i } ^ { \\top } x _ { j } \\right) } { \\sum _ { k \\in \\mathcal { G } } \\exp \\left( x _ { i } ^ { \\top } x _ { k } \\right) } + w _ { i - j } \\right) x _ { j } \\ \\mathrm { ~ o r ~ } \\ y _ { i } ^ { \\mathrm { p e } } = \\sum _ { j \\in \\mathcal { G } } \\frac { \\exp \\left( x _ { i } ^ { \\top } x _ { j } + w _ { i - j } \\right) } { \\sum _ { k \\in \\mathcal { G } } \\exp \\left( x _ { i } ^ { \\top } x _ { k } + w _ { i - k } \\right) } x _ { j } .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 676, + 800, + 720 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Interestingly, while the idea seems overly simplified, the pre-normalization version $y ^ { \\mathrm { p r e } }$ corresponds to a particular variant of relative self-attention [30, 31]. In this case, the attention weight $A _ { i , j }$ is decided jointly by the $w _ { i - j }$ of translation equivariance and the input-adaptive $x _ { i } ^ { \\top } x _ { j }$ , which can enjoy both effects depending on their relative magnitudes. Importantly, note that in order to enable the global convolution kernel without blowing up the number of parameters, we have reloaded the notation of $w _ { i - j }$ as a scalar (i.e., $w \\in \\mathbb { R } ^ { O ( | \\mathcal { G } | ) }$ ) rather than a vector in Eqn. (1). Another advantage of the scalar formulation of $w$ is that retrieving $w _ { i - j }$ for all $( i , j )$ is clearly subsumed by computing the pairwise dot-product attention, hence resulting in minimum additional cost (see Appendix A.1). Given the benefits, we will use the Transformer block with the pre-normalization relative attention variant in Eqn. (3) as the key component of the proposed CoAtNet model. ", + "bbox": [ + 173, + 728, + 825, + 871 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 Vertical Layout Design ", + "text_level": 1, + "bbox": [ + 174, + 92, + 372, + 106 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "After figuring out a neat way to combine convolution and attention, we next consider how to utilize it to stack an entire network. ", + "bbox": [ + 173, + 118, + 823, + 146 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "As we have discuss above, the global context has a quadratic complexity w.r.t. the spatial size. Hence, if we directly apply the relative attention in Eqn. (3) to the raw image input, the computation will be excessively slow due to the large number of pixels in any image of common sizes. Hence, to construct a network that is feasible in practice, we have mainly three options: ", + "bbox": [ + 173, + 152, + 825, + 209 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "(A) Perform some down-sampling to reduce the spatial size and employ the global relative attention after the feature map reaches manageable level. \n(B) Enforce local attention, which restricts the global receptive field $\\mathcal { G }$ in attention to a local field $\\mathcal { L }$ just like in convolution [22, 21]. \n(C) Replace the quadratic Softmax attention with certain linear attention variant which only has a linear complexity w.r.t. the spatial size [12, 32, 33]. ", + "bbox": [ + 176, + 215, + 825, + 309 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We briefly experimented with option (C) without getting a reasonably good result. For option (B), we found that implementing local attention involves many non-trivial shape formatting operations that requires intensive memory access. On our accelerator of choice (TPU), such operation turns out to be extremely slow [34], which not only defeats the original purpose of speeding up global attention, but also hurts the model capacity. Hence, as some recent work has studied this variant [22, 21], we will focus on option (A) and compare our results with theirs in our empirical study (Section 4). ", + "bbox": [ + 173, + 315, + 825, + 398 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For option (A), the down-sampling can be achieved by either (1) a convolution stem with aggressive stride (e.g., stride 16x16) as in ViT or (2) a multi-stage network with gradual pooling as in ConvNets. With these choices, we derive a search space of 5 variants and compare them in controlled experiments. ", + "bbox": [ + 174, + 404, + 823, + 446 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "• When the ViT Stem is used, we directly stack $L$ Transformer blocks with relative attention, which we denote as $\\mathrm { V I T } _ { \\mathrm { R E L } }$ . \n• When the multi-stage layout is used, we mimic ConvNets to construct a network of 5 stages (S0, S1, S2, S3 & S4), with spatial resolution gradually decreased from S0 to S4. At the beginning of each stage, we always reduce the spatial size by $2 \\mathbf { x }$ and increase the number of channels (see Appendix A.1 for the detailed down-sampling implementation). The first stage S0 is a simple 2-layer convolutional Stem and S1 always employs MBConv blocks with squeeze-excitation (SE), as the spatial size is too large for global attention. Starting from S2 through S4, we consider either the MBConv or the Transformer block, with a constraint that convolution stages must appear before Transformer stages. The constraint is based on the prior that convolution is better at processing local patterns that are more common in early stages. This leads to 4 variants with increasingly more Transformer stages, C-C-C-C, C-C-C-T, C-C-T-T and C-T-T-T, where C and T denote Convolution and Transformer respectively. ", + "bbox": [ + 171, + 453, + 825, + 642 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To systematically study the design choices, we consider two fundamental aspects generalization capability and model capacity: For generalization, we are interested in the gap between the training loss and the evaluation accuracy. If two models have the same training loss, then the model with higher evaluation accuracy has better generalization capability, since it can generalize better to unseen evaluation dataset. Generalization capability is particularly important to data efficiency when training data size is limited. For model capacity, we measure the ability to fit large training datasets. When training data is abundant and overfitting is not an issue, the model with higher capacity will achieve better final performance after reasonable training steps. Note that, since simply increasing the model size can lead to higher model capacity, to perform a meaningful comparison, we make sure the model sizes of the 5 variants are comparable. ", + "bbox": [ + 173, + 648, + 825, + 787 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To compare the generalization and model capacity, we train different variants of hybrid models on ImageNet-1K (1.3M) and JFT $\\left( > 3 0 0 \\mathbf { M } \\right)$ dataset for 300 and 3 epochs respectively, both without any regularization or augmentation. The training loss and evaluation accuracy on both datasets are summarized in Figure 1. ", + "bbox": [ + 174, + 792, + 825, + 849 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "• From the ImageNet-1K results, a key observation is that, in terms of generalization capability (i.e., gap between train and evaluation metrics), we have ", + "bbox": [ + 169, + 854, + 825, + 883 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/5f09b3609b6f1eadeb4a99c8e756ab7b3480d39c7a84b75a5be78aad8ba17e00.jpg", + "text": "$$\n\\mathrm { C \\mathrm { - } C \\mathrm { - } C \\mathrm { - } C \\approx C \\mathrm { - } C \\mathrm { - } C \\mathrm { - } T \\ge C \\mathrm { - } C \\mathrm { - } T \\mathrm { - } T > C \\mathrm { - } T \\mathrm { - } T \\mathrm { - } T \\gg V \\mathrm { I } \\mathrm { T } _ { \\mathrm { R E L } } . }\n$$", + "text_format": "latex", + "bbox": [ + 305, + 896, + 702, + 912 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/8e4cc944c4e098b4c2962d8c1e585910035ad5a220c25df539ba8f057d9c47df.jpg", + "image_caption": [ + "Figure 1: Comparison for model generalization and capacity under different data size. For fair comparison, all models have similar parameter size and computational cost. " + ], + "image_footnote": [], + "bbox": [ + 178, + 90, + 820, + 234 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Particularly, $\\mathrm { V I T } _ { \\mathrm { R E L } }$ is significantly worse than variants by a large margin, which we conjecture is related to the lack of proper low-level information processing in its aggressive down-sampling Stem. Among the multi-stage variants, the overall trend is that the more convolution stages the model has, the smaller the generalization gap is. ", + "bbox": [ + 186, + 297, + 825, + 354 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "• As for model capacity, from the JFT comparison, both the train and evaluation metrics at the end of the training suggest the following ranking: ", + "bbox": [ + 176, + 356, + 823, + 385 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/db72680f50180e0871b5c7bce1cdd56ab51755cadb71545f3fb276d6f9b0b410.jpg", + "text": "$$\n\\mathrm { C - C \\mathrm { - } T \\mathrm { - } T \\approx C \\mathrm { - } T \\mathrm { - } T \\mathrm { - } T > V I T _ { R E L } > C \\mathrm { - } C \\mathrm { - } C \\mathrm { - } T > C \\mathrm { - } C \\mathrm { - } C \\mathrm { - } C \\mathrm { - } C \\mathrm { . } }\n$$", + "text_format": "latex", + "bbox": [ + 307, + 391, + 702, + 407 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Importantly, this suggests that simply having more Transformer blocks does NOT necessarily mean higher capacity for visual processing. On one hand, while initially worse, $\\mathrm { V I T } _ { \\mathrm { R E L } }$ ultimately catch up with the two variants with more MBConv stages, indicating the capacity advantage of Transformer blocks. On the other hand, both C-C-T-T and C-T-T-T clearly outperforming $\\mathrm { V I T } _ { \\mathrm { R E L } }$ suggest that the ViT stem with an aggressive stride may have lost too much information and hence limit the model capacity. More interestingly, the fact that C-C-T-T $\\approx \\mathbf { C }$ -T-T-T indicates the for processing low-level information, static local operations like convolution could be as capable as adaptive global attention mechanism, while saving computation and memory usage substantially. ", + "bbox": [ + 186, + 415, + 825, + 526 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Finally, to decide between C-C-T-T and C-T-T-T, we conduct another transferability test3 — we finetune the two JFT pre-trained models above on ImageNet-1K for 30 epochs and compare their transfer performances. From Table 2, it turns out that C-C-T-T achieves a clearly better transfer accuracy than C-T-T-T, despite the same pre-training performance. ", + "bbox": [ + 176, + 531, + 825, + 588 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/c21bffeded81866fc6229e3aca114c77d365b7c6effb04ceec185d4089ddb3a2.jpg", + "table_caption": [ + "Table 2: Transferability test results. " + ], + "table_footnote": [], + "table_body": "
MetricC-C-T-TC-T-T-T
Pre-training Precision@1 (JFT)34.4034.36
Transfer Accuracy 224x22482.3981.78
Transfer Accuracy 384x38484.2384.02
", + "bbox": [ + 300, + 612, + 692, + 683 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Taking generalization, model capacity, transferability and efficiency into consideration, we adapt the C-C-T-T multi-stage layout for CoAtNet. More model details are included in Appendix A.1. ", + "bbox": [ + 173, + 696, + 825, + 726 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3 Related Work ", + "text_level": 1, + "bbox": [ + 174, + 744, + 321, + 762 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Convolutional network building blocks. Convolutional Networks (ConvNets) have been the dominating neural architectures for many computer vision tasks. Traditionally, regular convolutions, such as ResNet blocks [3], are popular in large-scale ConvNets; in contrast, depthwise convolutions [28] are popular in mobile platforms due to its lower computational cost and smaller parameter size [27]. Recent works show that an improved inverted residual bottlenecks (MBConv [27, 35]), which is built upon depthwise convolutions, can achieve both high accuracy and better efficiency [5, 19]. As discussed in Section 2, due to the strong connection between MBConv and Transformer blocks , this paper mostly employs MBConv as convolution building blocks. ", + "bbox": [ + 173, + 775, + 826, + 887 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Self-attention and Transformers. With the key ingredients of self-attention, Transformers have been widely adopted for neural language processing and speech understanding. As an early work, stand-alone self-attention network [34] shows self-attention alone can work well for different vision tasks, though with some practical difficulties. Recently, ViT [13] applies a vanilla Transformer to ImageNet classification, and achieves impressive results after pre-training on a large-scale JFT dataset. However, ViT still largely lags behind state-of-the-art ConvNets when training data is limited. Following that, many recent works have been focused on improving vision Transformers for data efficiency and model efficiency. For a more comprehensive review of vision Transformers, we refer readers to the dedicated surveys [36, 37]. ", + "bbox": [ + 174, + 90, + 825, + 215 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Relative attention. Under the general name of relative attention, there have been various variants in literature [30, 38, 39, 34, 40, 31]. Generally speaking, we can separate them into two categories: (a) the input-dependent version where the extra relative attention score is a function of the input states $f ( \\bar { x } _ { i } , x _ { j } , \\bar { i } - j )$ , and (b) the input-independent version $f ( i - j )$ . The variant in CoAtNet belongs to the input-independent version, and is similar to the one used in T5 [31], but unlike T5, we neither share the relative attention parameters across layers nor use the bucketing mechanism. As a benefit of the input independence, obtaining $f ( i - j )$ for all $( i , j )$ pairs is computationally much cheaper than the input-dependent version on TPU. In addition, at inference time, this only needs to be computed once and cached for future use. A recent work [22] also utilizes such an input-independent parameterization, but it restricts the receptive field to a local window. ", + "bbox": [ + 173, + 229, + 825, + 368 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Combining convolution and self-attention. The idea of combining convolution and self-attention for vision recognition is not new. A common approach is to augment the ConvNet backbone with explicit self-attention or non-local modules [9, 10, 11, 12], or to replace certain convolution layers with standard self-attention [11] or a more flexible mix of linear attention and convolution [41]. While self-attention usually improves the accuracy, they often come with extra computational cost and hence are often regarded as an add-on to the ConvNets, similar to squeeze-and-excitation [42] module. In comparison, after the success of ViT and ResNet-ViT [13], another popular line of research starts with a Transformer backbone and tries to incorporate explicit convolution or some desirable properties of convolution into the Transformer backbone [25, 24, 23, 22, 21, 43, 44]. ", + "bbox": [ + 174, + 382, + 825, + 507 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "While our work also belongs to this category, we show that our relative attention instantiation is a natural mixture of depthwise convolution and content-based attention with minimum additional cost. More importantly, starting from the perspectives of generalization and model capacity, we take a systematic approach to the vertical layout design and show how and why different network stages prefer different types of layers. Therefore, compared to models that simply use an off-the-shelf ConvNet as the stem layer, such as ResNet-ViT [13], CoAtNet also scales the Convolution stage (S2) when the overall size increases. On the other hand, compared to models employing local attention [22, 21], CoAtNet consistently uses full attention for S3 & S4 to ensure the model capacity, as S3 occupies the majority of the computation and parameters. ", + "bbox": [ + 174, + 513, + 825, + 638 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 Experiments ", + "text_level": 1, + "bbox": [ + 174, + 657, + 312, + 674 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this section, we compare CoAtNet with previous results under comparable settings. For completeness, all the hyper-parameters not mentioned here are included in Appendix A.2. ", + "bbox": [ + 174, + 688, + 823, + 717 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 Experiment Setting ", + "text_level": 1, + "bbox": [ + 174, + 732, + 348, + 747 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "CoAtNet model family. To compare with existing models of different sizes, we also design a family of CoAtNet models as summarized in Table 3. Overall, we always double the number of channels from S1 to S4, while ensuring the width of the Stem S0 to be smaller or equal to that of S1. Also, for simplicity, when increasing the depth of the network, we only scale the number of blocks in S2 and S3. ", + "bbox": [ + 174, + 757, + 825, + 827 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Evaluation Protocol. Our experiments focus on image classification. To evaluate the performance of the model across different data sizes, we utilize three datasets of increasingly larger sizes, namely ImageNet-1K (1.28M images), ImageNet-21K (12.7M images) and JFT (300M images). Following previous works, we first pre-train our models on each of the three datasets at resolution 224 for 300, 90 and 14 epochs respectively. Then, we finetune the pre-trained models on ImageNet-1K at the desired resolutions for 30 epochs and obtain the corresponding evaluation accuracy. One exception is the ImageNet-1K performance at resolution 224, which can be directly obtained at the end of pre-training. Note that similar to other models utilizing Transformer blocks, directly evaluating models pre-trained on ImageNet-1K at a larger resolution without finetuning usually leads to performance drop. Hence, finetuning is always employed whenever input resolution changes. ", + "bbox": [ + 174, + 842, + 823, + 911 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/84f510eacc36a4509902c6b3ea9d4f6ed24dfc3c9e12f559c758703dee925573.jpg", + "table_caption": [ + "Table 3: L denotes the number of blocks and D denotes the hidden dimension (#channels). For all Conv and MBConv blocks, we always use the kernel size 3. For all Transformer blocks, we set the size of each attention head to 32, following [22]. The expansion rate for the inverted bottleneck is always 4 and the expansion (shrink) rate for the SE is always 0.25. " + ], + "table_footnote": [], + "table_body": "
StagesSizeCoAtNet-0CoAtNet-1CoAtNet-2CoAtNet-3CoAtNet-4
S0-Conv1/2L=2 D=64L=2D=64L=2D=128 L=2D=192L=2 D=192
S1-MbConv1/4L=2 D=96L=2D=96L=2 D=128L=2D=192L=2 D=192
S2-MBConv1/8L=3 D=192L=6D=192L=6 D=256L=6D=384L=12 D=384
S3-TFMRel1/16L=5 D=384L=14D=384L=14 D=512L=14D=768L=28 D=768
S4-TFMRel1/32L=2 D=768L=2D=768L=2 D=1024L=2D=1536L=2 D=1536
", + "bbox": [ + 173, + 154, + 826, + 244 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 261, + 826, + 332 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Data Augmentation & Regularization. In this work, we only consider two widely used data augmentations, namely RandAugment [45] and MixUp [46], and three common techniques, including stochastic depth [47], label smoothing [48] and weight decay [49], to regularize the model. Intuitively, the specific hyper-parameters of the augmentation and regularization methods depend on model size and data scale, where strong regularization is usually applied for larger models and smaller dataset. ", + "bbox": [ + 174, + 347, + 825, + 416 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Under the general principle, a complication under the current paradigm is how to adjust the regularization for pre-training and finetuning as data size can change. Specifically, we have an interesting observation that if a certain type of augmentation is entirely disabled during pre-training, simply turning it on during fine-tuning would most likely harm the performance rather than improving. We conjecture this could be related to data distribution shift. As a result, for certain runs of the proposed model, we deliberately apply RandAugment and stochastic depth of a small degree when pre-training on the two larger datasets, ImageNet21-K and JFT. Although such regularization can harm the pre-training metrics, this allows more versatile regularization and augmentation during finetuning, leading to improved down-stream performances. ", + "bbox": [ + 173, + 422, + 826, + 547 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 Main Results ", + "text_level": 1, + "bbox": [ + 174, + 565, + 305, + 579 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/cf1567b93b834a38bb5d149ee37f969799ca7bcd6b3f63af756a7cd673e8c469.jpg", + "image_caption": [ + "Figure 2: Accuracy-to-FLOPs scaling curve under ImageNet-1K only setting at $2 2 4 \\mathbf { x } 2 2 4$ . " + ], + "image_footnote": [], + "bbox": [ + 179, + 601, + 478, + 785 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/f8c09d4a039db4e8c22000a3a3a66bae5de7e541a68ec3f2c6a6234546bf4ac2.jpg", + "image_caption": [ + "Figure 3: Accuracy-to-Params scaling curve under ImageNet- $2 1 \\mathrm { K } \\Rightarrow$ ImageNet-1K setting. " + ], + "image_footnote": [], + "bbox": [ + 513, + 602, + 813, + 785 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "ImageNet-1K The experiment results with only the ImageNet-1K dataset are shown in Table 4. Under similar conditions, the proposed CoAtNet models not only outperform ViT variants, but also match the best convolution-only architectures, i.e., EfficientNet-V2 and NFNets. Additionally, we also visualize the all results at resolution $2 2 4 \\mathbf { x } 2 2 4$ in Fig. 2. As we can see, CoAtNet scales much better than previous model with attention modules. ", + "bbox": [ + 174, + 842, + 825, + 911 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/07c0ff9e9c5c785f3c33e97ba94f2023e40f027155a317dec4e0af0e0e53d4b3.jpg", + "table_caption": [ + "Table 4: Model performance on ImageNet. 1K only denotes training on ImageNet-1K only; $2 1 \\mathtt { K } + 1 \\mathtt { K }$ denotes pre-training on ImageNet-21K and finetuning on ImageNet-1K; PT-RA denotes applying RandAugment during 21K pre-training, and E150 means 150 epochs of 21K pre-training, which is longer than the standard 90 epochs. More results are in Appendix A.3. " + ], + "table_footnote": [], + "table_body": "
ModelsEval Size#Params#FLOPsImageNet Top-1 Accuracy
Conv Only1K only21K+1K
EfficientNet-B7600²66M37B84.7-
EfficientNetV2-L480²121M53B85.786.8
NFNet-F34162²255M114.8B85.7=
NFNet-F55442377M289.8B86.01
ViT-Stem TFMDeiT-B384286M55.4B83.1-
ViT-L/16384²304M190.7B-85.3
CaiT-S-36384²68M48.0B85.0
DeepViT-L224²55M12.5B83.1-
Multi-stage TFMSwin-B384²88M47.0B84.286.0
Swin-L384²197M103.9B-86.4
Conv+TFMBotNet-T7384275.1M45.8B84.7-
LambdaResNet-420320²-=84.8
T2T-ViT-24224²64.1M15.0B82.6=
CvT-21384²32M24.9B83.3-
CvT-W243842277M193.2B-87.7
Conv+TFM (ours)CoAtNet-0 CoAtNet-1224²25M4.2B81.6=
224²42M8.4B83.3-
CoAtNet-2 CoAtNet-3224²75M15.7B84.187.1
2242168M34.7B84.587.6
CoAtNet-0384²25M13.4B83.9-
CoAtNet-1384242M27.4B85.1-
CoAtNet-2384²75M49.8B85.787.1
CoAtNet-3384²168M107.4B85.887.6
CoAtNet-4384²275M189.5B-87.9
+ PT-RA384²275M189.5B88.3
+ PT-RA-E1503842275M189.5B88.4
CoAtNet-2512275M96.7B85.987.3
CoAtNet-3512²168M203.1B86.087.9
CoAtNet-4512²275M360.9B-88.1
+ PT-RA512²275M360.9B=88.4
+ PT-RA-E1505122275M360.9B=88.56
", + "bbox": [ + 171, + 140, + 843, + 645 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "ImageNet-21K As we can see from Table 4 and Fig. 3, when ImageNet-21K is used for pretraining, the advantage of CoAtNet becomes more obvious, substantially outperforming all previous models. Notably, the best CoAtNet variant achieves a top-1 accuracy of $8 8 . 5 6 \\%$ , matching the ViTH/14 performance of $8 8 . 5 5 \\%$ , which requires pre-training the $2 . 3 \\mathbf { x }$ larger ViT model on a $2 3 \\mathrm { x }$ larger proprietary weakly labeled dataset (JFT) for $2 . 2 \\mathbf { x }$ more steps. This marks a dramatic improvement in both data efficiency and computation efficiency. ", + "bbox": [ + 174, + 681, + 826, + 766 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "JFT Finally, in Table 5, we further evaluate CoAtNet under the large-scale data regime with JFT300M and JFT-3B. Encouragingly, our CoAtNet-4 can almost match the best previous performance with JFT-300M set by NFNet- $\\mathrm { F 4 + }$ , while being $2 \\mathbf { x }$ more efficient in terms of both TPU training time and parameter count. When we scale up the model to consume similar training resource as NFNet- $. \\mathrm { F 4 + }$ , CoAtNet-5 reaches $8 9 . 7 7 \\%$ on top-1 accuracy, outperforming previous results under comparable settings. ", + "bbox": [ + 174, + 792, + 825, + 877 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Moreover, as we further push the training resource towards the level used by ViT-G/14 and utilize the same JFT-3B dataset of an even larger size [26], with over $4 \\mathbf { x }$ less computation, CoAtNet-6 is able to match the performance of $\\mathrm { V i T - G } / 1 4$ of $9 0 . 4 5 \\%$ , and with $1 . 5 \\mathrm { x }$ less computation, CoAtNet-7 achieves $8 9 . 7 7 \\%$ on top-1 accuracy $9 0 . 8 8 \\%$ , achieving the new state-of-the-art performance. ", + "bbox": [ + 174, + 883, + 821, + 911 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/4da8c9ad6e7a2eb2d4e66c521d47f928929362333fe2c47dcf4ccbd3f89040eb.jpg", + "table_caption": [ + "Table 5: Performance Comparison on large-scale JFT dataset. TPUv3-core-days denotes the pretraining time, Top-1 Accuracy denotes the finetuned accuracy on ImageNet. Note that the last 3 rows use a larger dataset JFT-3B [26] for pre-training, while others use JFT-300M [15]. See Appendix A.2 for the size details of CoAtNet-5/6/7. †: Down-sampling in the MBConv block is achieved by stride-2 Depthwise Convolution. ⇧: ViT-G/14 computation consumption is read from Fig. 1 of the paper [26]. " + ], + "table_footnote": [], + "table_body": "
ModelsEval Size#Params#FLOPsTPUv3-core-daysTop-1 Accuracy
ResNet + ViT-L/163842330M=187.12
ViT-L/165122307M364B0.68K87.76
ViT-H/145182632M1021B2.5K88.55
NFNet-F4+5122527M367B1.86K89.2
CoAtNet-3t3842168M114B0.58K88.52
CoAtNet-3t5122168M214B0.58K88.81
CoAtNet-45122275M361B0.95K89.11
CoAtNet-55122688M812B1.82K89.77
ViT-G/1451821.84B5160B>30K90.45
CoAtNet-651221.47B1521B6.6K90.45
CoAtNet-751222.44B2586B20.1K90.88
", + "bbox": [ + 173, + 156, + 834, + 358 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 372, + 823, + 401 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.3 Ablation Studies ", + "text_level": 1, + "bbox": [ + 174, + 416, + 328, + 431 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this section, we will ablate our design choices for CoAtNet. ", + "bbox": [ + 174, + 441, + 581, + 457 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Firstly, we study the importance of the relative attention from combining convolution and attention into a single computation unit. Specifically, we compare two models, one with the relative attention and the other without, under both the ImageNet-1K alone and ImageNet-21K transfer setting. As we can see from Table 6, when only the ImageNet-1K is used, relative attention clearly outperforms the standard attention, indicating a better generalization. In addition, under the ImageNet-21K transfer setting, the relative attention variant achieves a substantially better transfer accuracy, despite their very close pre-training performances. This suggests the main advantage of relative attention in visual processing is not in higher capacity but in better generalization. ", + "bbox": [ + 173, + 462, + 825, + 574 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/00ceb486f1a8197ca19a9f75354b351e896ee887a8c0a4bb90f5cf09fdd07a32.jpg", + "table_caption": [ + "Table 6: Ablation on relative attention. " + ], + "table_footnote": [], + "table_body": "
SetingMetricWith Rel-AttnWithout Rel-Attn
ImageNet-1KAccuracy (2242)84.183.8
Accuracy (3842)85.785.3
ImageNet-21K →ImageNet-1KPre-train Precision@1 (224²)53.052.8
Finetune Accuracy (384²)87.987.4
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SettingModelsLayoutTop-1 Accuracy
ImageNet-1KVO: CoAtNet-2[2,2,6,14，2]84.1
V1: S2← S3[2,2, 2,18,2]83.4
V2: S2→ S3[2,2,8,12,2]84.0
ImageNet-21KVO: CoAtNet-3[2,2,6,14,2]53.0 -→87.6
⇒ImageNet-1KV1: S2 ← S3[2,2,2,18,2]53.0 -→87.4
", + "bbox": [ + 227, + 723, + 766, + 828 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Secondly, as S2 with MBConv blocks and S3 with relative Transformer blocks occupy most of the computation of the CoAtNet, a question to ask is how to split the computation between S2 (MBConv) and S3 (Transformer) to achieve a good performance. In practice, it boils down to deciding the number of blocks to have in each stage, which we will refer to as “layout” design. For this purpose, we compare a few different layouts that we experimented with in Table 7. ", + "bbox": [ + 174, + 842, + 826, + 911 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/622bb1939958732b44fe6505260003f83b893fe0388e4f1de344e047e3c173b0.jpg", + "table_caption": [ + "Table 8: Ablation on head size and normalization type. " + ], + "table_footnote": [], + "table_body": "
SettingModelsImage SizeTop-1 Accuracy
ImageNet-1KCoAtNet-2224284.1
Head size: 32 → 64224283.9
Norm type: 1 BN →LN224284.1
ImageNet-21K ⇒ ImageNet-1KCoAtNet-3384287.9
Norm type: BN →→ LN384²87.8
", + "bbox": [ + 225, + 99, + 767, + 208 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "• If we keep the total number of blocks in S2 and S3 fixed and vary the number in each stage, we observe that V0 is a sweet spot between V1 and V2. Basically, having more Transformer blocks in S3 generally leads to better performance until the number of MBConv blocks in S2 is too small to generalize well. ", + "bbox": [ + 174, + 233, + 823, + 289 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "• To further evaluate whether the sweet spot also holds in the transfer setting, where a higher capacity is often regarded more important, we further compare V0 and V1 under the ImageNet21K transferring to ImageNet-1K setup. Interestingly, despite that V1 and V0 have the same performance during ImageNet-21K pre-training, the transfer accuracy of V1 clearly falls behind V0. Again, this suggests the importance of convolution in achieving good transferability and generalization. ", + "bbox": [ + 173, + 291, + 825, + 373 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Lastly, we study two choices of model details, namely the dimension of each attention (default to 32) head as well as the type of normalization (default to BatchNorm) used in MBConv blocks. From Table 8, we can see increasing head size from 32 to 64 can slightly hurt performance, though it actually improves the TPU speed by a significant amount. In practice, this will be a quality-speed trade-off one can make. On the other hand, BatchNorm and LayerNorm have almost the same performance, while BatchNorm is $10 - 2 0 \\%$ faster on TPU depending on the per-core batch size. ", + "bbox": [ + 174, + 381, + 825, + 463 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5 Conclusion ", + "text_level": 1, + "bbox": [ + 174, + 483, + 299, + 500 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In this paper, we systematically study the properties of convolutions and Transformers, which leads to a principled way to combine them into a new family of models named CoAtNet. Extensive experiments show that CoAtNet enjoys both good generalization like ConvNets and superior model capacity like Transformers, achieving state-of-the-art performances under different data sizes and computation budgets. ", + "bbox": [ + 174, + 513, + 825, + 583 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Note that this paper currently focuses on ImageNet classification for model development. However, we believe our approach is applicable to broader applications like object detection and semantic segmentation. We will leave them for future work. 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In this work, we show that", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 284, + 470, + 296 + ], + "spans": [ + { + "bbox": [ + 141, + 284, + 470, + 296 + ], + "score": 1.0, + "content": "while Transformers tend to have larger model capacity, their generalization can be", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 295, + 469, + 307 + ], + "spans": [ + { + "bbox": [ + 141, + 295, + 469, + 307 + ], + "score": 1.0, + "content": "worse than convolutional networks due to the lack of the right inductive bias. To", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 306, + 470, + 319 + ], + "spans": [ + { + "bbox": [ + 141, + 306, + 470, + 319 + ], + "score": 1.0, + "content": "effectively combine the strengths from both architectures, we present CoAtNets", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 142, + 317, + 470, + 329 + ], + "spans": [ + { + "bbox": [ + 142, + 317, + 470, + 329 + ], + "score": 1.0, + "content": "(pronounced “coat” nets), a family of hybrid models built from two key insights:", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 327, + 469, + 340 + ], + "spans": [ + { + "bbox": [ + 141, + 327, + 469, + 340 + ], + "score": 1.0, + "content": "(1) depthwise Convolution and self-Attention can be naturally unified via simple", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 339, + 469, + 351 + ], + "spans": [ + { + "bbox": [ + 141, + 339, + 469, + 351 + ], + "score": 1.0, + "content": "relative attention; (2) vertically stacking convolution layers and attention layers in", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 350, + 470, + 362 + ], + "spans": [ + { + "bbox": [ + 141, + 350, + 470, + 362 + ], + "score": 1.0, + "content": "a principled way is surprisingly effective in improving generalization, capacity and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 360, + 470, + 373 + ], + "spans": [ + { + "bbox": [ + 141, + 360, + 470, + 373 + ], + "score": 1.0, + "content": "efficiency. 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More", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "recently, Vision Transformer (ViT) [13] has shown that with almost1 only vanilla Transformer layers,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "score": 1.0, + "content": "one could obtain reasonable performance on ImageNet-1K [14] alone. 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For example, without extra JFT-300M pre-training,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 607, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 506, + 620 + ], + "score": 1.0, + "content": "the ImageNet accuracy of ViT is still significantly lower than ConvNets with comparable model", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "size [5] (see Table 13). Subsequent works use special regularization and stronger data augmentation", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 629, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 506, + 642 + ], + "score": 1.0, + "content": "to improve the vanilla ViT [16, 17, 18], yet none of these ViT variants could outperform the SOTA", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "convolution-only models on ImageNet classification given the same amount of data and computa-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "tion [19, 20]. 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To", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 306, + 470, + 319 + ], + "spans": [ + { + "bbox": [ + 141, + 306, + 470, + 319 + ], + "score": 1.0, + "content": "effectively combine the strengths from both architectures, we present CoAtNets", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 142, + 317, + 470, + 329 + ], + "spans": [ + { + "bbox": [ + 142, + 317, + 470, + 329 + ], + "score": 1.0, + "content": "(pronounced “coat” nets), a family of hybrid models built from two key insights:", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 327, + 469, + 340 + ], + "spans": [ + { + "bbox": [ + 141, + 327, + 469, + 340 + ], + "score": 1.0, + "content": "(1) depthwise Convolution and self-Attention can be naturally unified via simple", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 339, + 469, + 351 + ], + "spans": [ + { + "bbox": [ + 141, + 339, + 469, + 351 + ], + "score": 1.0, + "content": "relative attention; (2) vertically stacking convolution layers and attention layers in", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 350, + 470, + 362 + ], + "spans": [ + { + "bbox": [ + 141, + 350, + 470, + 362 + ], + "score": 1.0, + "content": "a principled way is surprisingly effective in improving generalization, capacity and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 360, + 470, + 373 + ], + "spans": [ + { + "bbox": [ + 141, + 360, + 470, + 373 + ], + "score": 1.0, + "content": "efficiency. 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More importantly, when", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "pre-trained on large-scale weakly labeled JFT-300M dataset [15], ViT achieves comparable results", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 557, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 506, + 571 + ], + "score": 1.0, + "content": "to state-of-the-art (SOTA) ConvNets, indicating that Transformer models potentially have higher", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 569, + 238, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 238, + 581 + ], + "score": 1.0, + "content": "capacity at scale than ConvNets.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 482, + 506, + 581 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 585, + 506, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 584, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 506, + 598 + ], + "score": 1.0, + "content": "While ViT has shown impressive results with enormous JFT 300M training images, its performance", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 594, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 610 + ], + "score": 1.0, + "content": "still falls behind ConvNets in the low data regime. For example, without extra JFT-300M pre-training,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 607, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 506, + 620 + ], + "score": 1.0, + "content": "the ImageNet accuracy of ViT is still significantly lower than ConvNets with comparable model", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "size [5] (see Table 13). Subsequent works use special regularization and stronger data augmentation", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 629, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 506, + 642 + ], + "score": 1.0, + "content": "to improve the vanilla ViT [16, 17, 18], yet none of these ViT variants could outperform the SOTA", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "convolution-only models on ImageNet classification given the same amount of data and computa-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "tion [19, 20]. This suggests that vanilla Transformer layers may lack certain desirable inductive biases", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 661, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 506, + 675 + ], + "score": 1.0, + "content": "possessed by ConvNets, and thus require significant amount of data and computational resource", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 673, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 506, + 685 + ], + "score": 1.0, + "content": "to compensate. Not surprisingly, many recent works have been trying to incorporate the induc-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 683, + 505, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 505, + 696 + ], + "score": 1.0, + "content": "tive biases of ConvNets into Transformer models, by imposing local receptive fields for attention", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "layers [21, 22] or augmenting the attention and FFN layers with implicit or explicit convolutional", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "operations [23, 24, 25]. However, these approaches are either ad-hoc or focused on injecting a", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "particular property, lacking a systematic understanding of the respective roles of convolution and", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 212, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 212, + 117 + ], + "score": 1.0, + "content": "attention when combined.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 584, + 506, + 696 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "layers [21, 22] or augmenting the attention and FFN layers with implicit or explicit convolutional", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "operations [23, 24, 25]. However, these approaches are either ad-hoc or focused on injecting a", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "particular property, lacking a systematic understanding of the respective roles of convolution and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 212, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 212, + 117 + ], + "score": 1.0, + "content": "attention when combined.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 122, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 105, + 122, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 505, + 135 + ], + "score": 1.0, + "content": "In this work, we systematically study the problem of hybridizing convolution and attention from", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 133, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 505, + 145 + ], + "score": 1.0, + "content": "two fundamental aspects in machine learning – generalization and model capacity. Our study shows", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "that convolutional layers tend to have better generalization with faster converging speed thanks to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 155, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 505, + 167 + ], + "score": 1.0, + "content": "their strong prior of inductive bias, while attention layers have higher model capacity that can benefit", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "from larger datasets. Combining convolutional and attention layers can achieve better generalization", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 177, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 506, + 189 + ], + "score": 1.0, + "content": "and capacity; however, a key challenge here is how to effectively combine them to achieve better", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "trade-offs between accuracy and efficiency. In this paper, we investigate two key insights: First, we", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "observe that the commonly used depthwise convolution can be effectively merged into attention", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 210, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 506, + 222 + ], + "score": 1.0, + "content": "layers with simple relative attention; Second, simply stacking convolutional and attention layers, in a", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "proper way, could be surprisingly effective to achieve better generalization and capacity. Based on", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "score": 1.0, + "content": "these insights, we propose a simple yet effective network architecture named CoAtNet, which enjoys", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 242, + 318, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 318, + 254 + ], + "score": 1.0, + "content": "the strengths from both ConvNets and Transformers.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 258, + 505, + 379 + ], + "lines": [ + { + "bbox": [ + 106, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "Our CoAtNet achieves SOTA performances under comparable resource constraints across different", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "data sizes. Specifically, under the low-data regime, CoAtNet inherits the great generalization property", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 280, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 505, + 292 + ], + "score": 1.0, + "content": "of ConvNets thanks to the favorable inductive biases. Moreover, given abundant data, CoAtNet not", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "score": 1.0, + "content": "only enjoys the superior scalability of Transformer models, but also achieves faster convergence and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 301, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 477, + 314 + ], + "score": 1.0, + "content": "thus improved efficiency. When only ImageNet-1K is used for training, CoAtNet achieves", + "type": "text" + }, + { + "bbox": [ + 477, + 302, + 505, + 313 + ], + "score": 0.86, + "content": "8 6 . 0 \\%", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 312, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 326 + ], + "score": 1.0, + "content": "top-1 accuracy, matching the prior art NFNet [20] under similar computation resource and training", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 323, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 337 + ], + "score": 1.0, + "content": "conditions. Further, when pre-trained on ImageNet-21K with about 10M images, CoAtNet reaches", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 139, + 345 + ], + "score": 0.87, + "content": "8 8 . 5 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "top-1 accuracy when finetuned on ImageNet-1K, matching the ViT-Huge pre-trained on", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 345, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 161, + 358 + ], + "score": 1.0, + "content": "JFT-300M, a", + "type": "text" + }, + { + "bbox": [ + 161, + 345, + 181, + 356 + ], + "score": 0.88, + "content": "2 3 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 345, + 506, + 358 + ], + "score": 1.0, + "content": "larger dataset. Finally, when JFT-3B is used for pre-training, CoAtNet exhibits", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 445, + 369 + ], + "score": 1.0, + "content": "better efficiency compared to ViT, and pushes the ImageNet-1K top-1 accuracy to", + "type": "text" + }, + { + "bbox": [ + 446, + 357, + 479, + 367 + ], + "score": 0.87, + "content": "9 0 . 8 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "while", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 367, + 369, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 131, + 380 + ], + "score": 1.0, + "content": "using", + "type": "text" + }, + { + "bbox": [ + 131, + 367, + 150, + 378 + ], + "score": 0.67, + "content": "1 . 5 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 367, + 369, + 380 + ], + "score": 1.0, + "content": "less computation of the prior art set by ViT-G/14 [26].", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 107, + 400, + 159, + 413 + ], + "lines": [ + { + "bbox": [ + 104, + 398, + 160, + 416 + ], + "spans": [ + { + "bbox": [ + 104, + 398, + 160, + 416 + ], + "score": 1.0, + "content": "2 Model", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 428, + 504, + 451 + ], + "lines": [ + { + "bbox": [ + 106, + 429, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 440 + ], + "score": 1.0, + "content": "In the section, we focus on the question of how to “optimally” combine the convolution and trans-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 439, + 388, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 388, + 452 + ], + "score": 1.0, + "content": "former. 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This", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 114, + 273, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 114, + 273, + 505, + 285 + ], + "score": 1.0, + "content": "property is often referred to translation equivalence, which has been found to improve generalization", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 114, + 284, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 114, + 284, + 505, + 295 + ], + "score": 1.0, + "content": "under datasets of limited size [29]. Due to the usage of absolution positional embeddings, standard", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 113, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 113, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "Transformer (ViT) lacks this property. 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PropertiesConvolutionSelf-Attention
Translation Equivariance√
Input-adaptive Weighting√
Global Receptive Field
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With the similar form of depthwise convolution in Eqn. (1) and self-attention in Eqn. 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In this case, the attention weight", + "type": "text" + }, + { + "bbox": [ + 476, + 589, + 494, + 601 + ], + "score": 0.91, + "content": "A _ { i , j }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 587, + 506, + 603 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 600, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 200, + 614 + ], + "score": 1.0, + "content": "decided jointly by the", + "type": "text" + }, + { + "bbox": [ + 200, + 603, + 222, + 613 + ], + "score": 0.88, + "content": "w _ { i - j }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 600, + 432, + 614 + ], + "score": 1.0, + "content": "of translation equivariance and the input-adaptive", + "type": "text" + }, + { + "bbox": [ + 432, + 600, + 455, + 613 + ], + "score": 0.92, + "content": "x _ { i } ^ { \\top } x _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 600, + 505, + 614 + ], + "score": 1.0, + "content": ", which can", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 611, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 624 + ], + "score": 1.0, + "content": "enjoy both effects depending on their relative magnitudes. Importantly, note that in order to enable", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "the global convolution kernel without blowing up the number of parameters, we have reloaded the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 632, + 507, + 648 + ], + "spans": [ + { + "bbox": [ + 104, + 632, + 151, + 648 + ], + "score": 1.0, + "content": "notation of", + "type": "text" + }, + { + "bbox": [ + 151, + 636, + 173, + 647 + ], + "score": 0.88, + "content": "w _ { i - j }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 632, + 237, + 648 + ], + "score": 1.0, + "content": "as a scalar (i.e.,", + "type": "text" + }, + { + "bbox": [ + 237, + 633, + 288, + 645 + ], + "score": 0.91, + "content": "w \\in \\mathbb { R } ^ { O ( | \\mathcal { G } | ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 632, + 507, + 648 + ], + "score": 1.0, + "content": ") rather than a vector in Eqn. 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Another advantage of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 644, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 210, + 659 + ], + "score": 1.0, + "content": "the scalar formulation of", + "type": "text" + }, + { + "bbox": [ + 210, + 647, + 218, + 655 + ], + "score": 0.8, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 644, + 289, + 659 + ], + "score": 1.0, + "content": "is that retrieving", + "type": "text" + }, + { + "bbox": [ + 289, + 647, + 311, + 658 + ], + "score": 0.89, + "content": "w _ { i - j }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 644, + 340, + 659 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 340, + 646, + 361, + 658 + ], + "score": 0.92, + "content": "( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 644, + 505, + 659 + ], + "score": 1.0, + "content": "is clearly subsumed by computing", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 657, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 506, + 668 + ], + "score": 1.0, + "content": "the pairwise dot-product attention, hence resulting in minimum additional cost (see Appendix A.1).", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 668, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 505, + 680 + ], + "score": 1.0, + "content": "Given the benefits, we will use the Transformer block with the pre-normalization relative attention", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 678, + 403, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 403, + 690 + ], + "score": 1.0, + "content": "variant in Eqn. (3) as the key component of the proposed CoAtNet model.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 40.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 701, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 117, + 699, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 117, + 699, + 506, + 714 + ], + "score": 1.0, + "content": "2 To simplify the presentation, we deliberately omit the multi-head query, key and value projections for now.", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 711, + 366, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 366, + 723 + ], + "score": 1.0, + "content": "In the actual implementation, we always use the multi-head projections.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 72, + 504, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "In comparison, self-attention allows the receptive field to be the entire spatial locations and computes", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 82, + 452, + 97 + ], + "spans": [ + { + "bbox": [ + 104, + 82, + 411, + 97 + ], + "score": 1.0, + "content": "the weights based on the re-normalized pairwise similarity between the pair", + "type": "text" + }, + { + "bbox": [ + 411, + 83, + 442, + 96 + ], + "score": 0.92, + "content": "( x _ { i } , x _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 82, + 452, + 97 + ], + "score": 1.0, + "content": ": 2", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 104, + 72, + 505, + 97 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 102, + 406, + 149 + ], + "lines": [ + { + "bbox": [ + 204, + 102, + 406, + 149 + ], + "spans": [ + { + "bbox": [ + 204, + 102, + 406, + 149 + ], + "score": 0.94, + "content": "y _ { i } = \\sum _ { j \\in \\mathcal { G } } \\underbrace { \\frac { \\exp { \\left( x _ { i } ^ { \\top } x _ { j } \\right) } } { \\sum _ { k \\in \\mathcal { G } } \\exp { \\left( x _ { i } ^ { \\top } x _ { k } \\right) } } } _ { A _ { i , j } } x _ { j } \\quad \\mathrm { ( s e l f - a t t e n t i o n ) } ,", + "type": "interline_equation", + "image_path": "a9164aac5bb6f55e0bd90b25f31a8559f39a62986021c906fcf5f7116eba58d0.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 204, + 102, + 406, + 117.66666666666667 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 204, + 117.66666666666667, + 406, + 133.33333333333334 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 204, + 133.33333333333334, + 406, + 149.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 155, + 505, + 189 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 133, + 168 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 156, + 141, + 166 + ], + "score": 0.84, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 154, + 505, + 168 + ], + "score": 1.0, + "content": "indicates the global spatial space. 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This", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 114, + 273, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 114, + 273, + 505, + 285 + ], + "score": 1.0, + "content": "property is often referred to translation equivalence, which has been found to improve generalization", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 114, + 284, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 114, + 284, + 505, + 295 + ], + "score": 1.0, + "content": "under datasets of limited size [29]. Due to the usage of absolution positional embeddings, standard", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 113, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 113, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "Transformer (ViT) lacks this property. 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PropertiesConvolutionSelf-Attention
Translation Equivariance√
Input-adaptive Weighting√
Global Receptive Field
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With the similar form of depthwise convolution in Eqn. (1) and self-attention in Eqn. (2), a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 518 + ], + "score": 1.0, + "content": "straightforward idea that could achieve this is simply to sum a global static convolution kernel with", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 518, + 439, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 439, + 530 + ], + "score": 1.0, + "content": "the adaptive attention matrix, either after or before the Softmax normalization, i.e.,", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5, + "bbox_fs": [ + 106, + 484, + 506, + 530 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 536, + 490, + 571 + ], + "lines": [ + { + "bbox": [ + 111, + 536, + 490, + 571 + ], + "spans": [ + { + "bbox": [ + 111, + 536, + 490, + 571 + ], + "score": 0.92, + "content": "y _ { i } ^ { \\mathrm { p o s t } } = \\sum _ { j \\in \\mathcal { G } } \\left( \\frac { \\exp \\left( x _ { i } ^ { \\top } x _ { j } \\right) } { \\sum _ { k \\in \\mathcal { G } } \\exp \\left( x _ { i } ^ { \\top } x _ { k } \\right) } + w _ { i - j } \\right) x _ { j } \\ \\mathrm { ~ o r ~ } \\ y _ { i } ^ { \\mathrm { p e } } = \\sum _ { j \\in \\mathcal { G } } \\frac { \\exp \\left( x _ { i } ^ { \\top } x _ { j } + w _ { i - j } \\right) } { \\sum _ { k \\in \\mathcal { G } } \\exp \\left( x _ { i } ^ { \\top } x _ { k } + w _ { i - k } \\right) } x _ { j } .", + "type": "interline_equation", + "image_path": "6ad701e539bf342fce3258bfc8889c519cd941777a414d3170d3f20d5b95a53d.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 111, + 536, + 490, + 547.6666666666666 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 111, + 547.6666666666666, + 490, + 559.3333333333333 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 111, + 559.3333333333333, + 490, + 570.9999999999999 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 577, + 505, + 690 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 437, + 590 + ], + "score": 1.0, + "content": "Interestingly, while the idea seems overly simplified, the pre-normalization version", + "type": "text" + }, + { + "bbox": [ + 438, + 578, + 453, + 589 + ], + "score": 0.89, + "content": "y ^ { \\mathrm { p r e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 576, + 506, + 590 + ], + "score": 1.0, + "content": "corresponds", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 587, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 104, + 587, + 476, + 603 + ], + "score": 1.0, + "content": "to a particular variant of relative self-attention [30, 31]. In this case, the attention weight", + "type": "text" + }, + { + "bbox": [ + 476, + 589, + 494, + 601 + ], + "score": 0.91, + "content": "A _ { i , j }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 587, + 506, + 603 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 600, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 200, + 614 + ], + "score": 1.0, + "content": "decided jointly by the", + "type": "text" + }, + { + "bbox": [ + 200, + 603, + 222, + 613 + ], + "score": 0.88, + "content": "w _ { i - j }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 600, + 432, + 614 + ], + "score": 1.0, + "content": "of translation equivariance and the input-adaptive", + "type": "text" + }, + { + "bbox": [ + 432, + 600, + 455, + 613 + ], + "score": 0.92, + "content": "x _ { i } ^ { \\top } x _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 600, + 505, + 614 + ], + "score": 1.0, + "content": ", which can", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 611, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 624 + ], + "score": 1.0, + "content": "enjoy both effects depending on their relative magnitudes. Importantly, note that in order to enable", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "the global convolution kernel without blowing up the number of parameters, we have reloaded the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 632, + 507, + 648 + ], + "spans": [ + { + "bbox": [ + 104, + 632, + 151, + 648 + ], + "score": 1.0, + "content": "notation of", + "type": "text" + }, + { + "bbox": [ + 151, + 636, + 173, + 647 + ], + "score": 0.88, + "content": "w _ { i - j }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 632, + 237, + 648 + ], + "score": 1.0, + "content": "as a scalar (i.e.,", + "type": "text" + }, + { + "bbox": [ + 237, + 633, + 288, + 645 + ], + "score": 0.91, + "content": "w \\in \\mathbb { R } ^ { O ( | \\mathcal { G } | ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 632, + 507, + 648 + ], + "score": 1.0, + "content": ") rather than a vector in Eqn. (1). Another advantage of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 644, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 210, + 659 + ], + "score": 1.0, + "content": "the scalar formulation of", + "type": "text" + }, + { + "bbox": [ + 210, + 647, + 218, + 655 + ], + "score": 0.8, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 644, + 289, + 659 + ], + "score": 1.0, + "content": "is that retrieving", + "type": "text" + }, + { + "bbox": [ + 289, + 647, + 311, + 658 + ], + "score": 0.89, + "content": "w _ { i - j }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 644, + 340, + 659 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 340, + 646, + 361, + 658 + ], + "score": 0.92, + "content": "( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 644, + 505, + 659 + ], + "score": 1.0, + "content": "is clearly subsumed by computing", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 657, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 506, + 668 + ], + "score": 1.0, + "content": "the pairwise dot-product attention, hence resulting in minimum additional cost (see Appendix A.1).", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 668, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 505, + 680 + ], + "score": 1.0, + "content": "Given the benefits, we will use the Transformer block with the pre-normalization relative attention", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 678, + 403, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 403, + 690 + ], + "score": 1.0, + "content": "variant in Eqn. (3) as the key component of the proposed CoAtNet model.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 40.5, + "bbox_fs": [ + 104, + 576, + 507, + 690 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 73, + 228, + 84 + ], + "lines": [ + { + "bbox": [ + 105, + 71, + 229, + 87 + ], + "spans": [ + { + "bbox": [ + 105, + 71, + 229, + 87 + ], + "score": 1.0, + "content": "2.2 Vertical Layout Design", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 94, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "After figuring out a neat way to combine convolution and attention, we next consider how to utilize it", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 214, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 214, + 117 + ], + "score": 1.0, + "content": "to stack an entire network.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 121, + 505, + 166 + ], + "lines": [ + { + "bbox": [ + 104, + 121, + 507, + 135 + ], + "spans": [ + { + "bbox": [ + 104, + 121, + 507, + 135 + ], + "score": 1.0, + "content": "As we have discuss above, the global context has a quadratic complexity w.r.t. the spatial size. Hence,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "if we directly apply the relative attention in Eqn. (3) to the raw image input, the computation will", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "be excessively slow due to the large number of pixels in any image of common sizes. Hence, to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 416, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 416, + 167 + ], + "score": 1.0, + "content": "construct a network that is feasible in practice, we have mainly three options:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 108, + 171, + 505, + 245 + ], + "lines": [ + { + "bbox": [ + 107, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 107, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "(A) Perform some down-sampling to reduce the spatial size and employ the global relative attention", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 126, + 181, + 318, + 194 + ], + "spans": [ + { + "bbox": [ + 126, + 181, + 318, + 194 + ], + "score": 1.0, + "content": "after the feature map reaches manageable level.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 196, + 504, + 209 + ], + "spans": [ + { + "bbox": [ + 107, + 196, + 380, + 209 + ], + "score": 1.0, + "content": "(B) Enforce local attention, which restricts the global receptive field", + "type": "text" + }, + { + "bbox": [ + 381, + 197, + 389, + 207 + ], + "score": 0.82, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 196, + 495, + 209 + ], + "score": 1.0, + "content": "in attention to a local field", + "type": "text" + }, + { + "bbox": [ + 496, + 197, + 504, + 207 + ], + "score": 0.62, + "content": "\\mathcal { L }", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 125, + 208, + 257, + 219 + ], + "spans": [ + { + "bbox": [ + 125, + 208, + 257, + 219 + ], + "score": 1.0, + "content": "just like in convolution [22, 21].", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 222, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 506, + 235 + ], + "score": 1.0, + "content": "(C) Replace the quadratic Softmax attention with certain linear attention variant which only has a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 126, + 232, + 334, + 246 + ], + "spans": [ + { + "bbox": [ + 126, + 232, + 334, + 246 + ], + "score": 1.0, + "content": "linear complexity w.r.t. the spatial size [12, 32, 33].", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 250, + 505, + 316 + ], + "lines": [ + { + "bbox": [ + 105, + 249, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 505, + 263 + ], + "score": 1.0, + "content": "We briefly experimented with option (C) without getting a reasonably good result. For option (B), we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 261, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 506, + 273 + ], + "score": 1.0, + "content": "found that implementing local attention involves many non-trivial shape formatting operations that", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "requires intensive memory access. On our accelerator of choice (TPU), such operation turns out to be", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 283, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 505, + 295 + ], + "score": 1.0, + "content": "extremely slow [34], which not only defeats the original purpose of speeding up global attention, but", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 293, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 306 + ], + "score": 1.0, + "content": "also hurts the model capacity. Hence, as some recent work has studied this variant [22, 21], we will", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 305, + 469, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 469, + 316 + ], + "score": 1.0, + "content": "focus on option (A) and compare our results with theirs in our empirical study (Section 4).", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 320, + 504, + 354 + ], + "lines": [ + { + "bbox": [ + 105, + 320, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 334 + ], + "score": 1.0, + "content": "For option (A), the down-sampling can be achieved by either (1) a convolution stem with aggressive", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 332, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 505, + 344 + ], + "score": 1.0, + "content": "stride (e.g., stride 16x16) as in ViT or (2) a multi-stage network with gradual pooling as in ConvNets.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "score": 1.0, + "content": "With these choices, we derive a search space of 5 variants and compare them in controlled experiments.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 105, + 359, + 505, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 298, + 371 + ], + "score": 1.0, + "content": "• When the ViT Stem is used, we directly stack", + "type": "text" + }, + { + "bbox": [ + 298, + 360, + 306, + 369 + ], + "score": 0.72, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "Transformer blocks with relative attention, which", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 113, + 369, + 204, + 383 + ], + "spans": [ + { + "bbox": [ + 113, + 369, + 168, + 383 + ], + "score": 1.0, + "content": "we denote as", + "type": "text" + }, + { + "bbox": [ + 169, + 370, + 199, + 381 + ], + "score": 0.87, + "content": "\\mathrm { V I T } _ { \\mathrm { R E L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 369, + 204, + 383 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "• When the multi-stage layout is used, we mimic ConvNets to construct a network of 5 stages (S0,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 112, + 394, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 112, + 394, + 506, + 410 + ], + "score": 1.0, + "content": "S1, S2, S3 & S4), with spatial resolution gradually decreased from S0 to S4. At the beginning", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 113, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 113, + 406, + 322, + 419 + ], + "score": 1.0, + "content": "of each stage, we always reduce the spatial size by", + "type": "text" + }, + { + "bbox": [ + 322, + 407, + 335, + 417 + ], + "score": 0.57, + "content": "2 \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "and increase the number of channels (see", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 114, + 417, + 371, + 430 + ], + "spans": [ + { + "bbox": [ + 114, + 417, + 371, + 430 + ], + "score": 1.0, + "content": "Appendix A.1 for the detailed down-sampling implementation).", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 114, + 432, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 114, + 432, + 505, + 445 + ], + "score": 1.0, + "content": "The first stage S0 is a simple 2-layer convolutional Stem and S1 always employs MBConv blocks", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 114, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 114, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "with squeeze-excitation (SE), as the spatial size is too large for global attention. Starting from", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 115, + 455, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 115, + 455, + 505, + 466 + ], + "score": 1.0, + "content": "S2 through S4, we consider either the MBConv or the Transformer block, with a constraint that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 114, + 466, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 114, + 466, + 506, + 477 + ], + "score": 1.0, + "content": "convolution stages must appear before Transformer stages. The constraint is based on the prior", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 114, + 476, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 114, + 476, + 505, + 488 + ], + "score": 1.0, + "content": "that convolution is better at processing local patterns that are more common in early stages. This", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 113, + 487, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 113, + 487, + 506, + 499 + ], + "score": 1.0, + "content": "leads to 4 variants with increasingly more Transformer stages, C-C-C-C, C-C-C-T, C-C-T-T and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 114, + 497, + 421, + 511 + ], + "spans": [ + { + "bbox": [ + 114, + 497, + 421, + 511 + ], + "score": 1.0, + "content": "C-T-T-T, where C and T denote Convolution and Transformer respectively.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 514, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "To systematically study the design choices, we consider two fundamental aspects generalization", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "capability and model capacity: For generalization, we are interested in the gap between the training", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 537, + 504, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 504, + 548 + ], + "score": 1.0, + "content": "loss and the evaluation accuracy. If two models have the same training loss, then the model with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "score": 1.0, + "content": "higher evaluation accuracy has better generalization capability, since it can generalize better to unseen", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 556, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 104, + 556, + 506, + 572 + ], + "score": 1.0, + "content": "evaluation dataset. Generalization capability is particularly important to data efficiency when training", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "score": 1.0, + "content": "data size is limited. For model capacity, we measure the ability to fit large training datasets. When", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 579, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 506, + 592 + ], + "score": 1.0, + "content": "training data is abundant and overfitting is not an issue, the model with higher capacity will achieve", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "better final performance after reasonable training steps. Note that, since simply increasing the model", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 601, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 104, + 601, + 506, + 614 + ], + "score": 1.0, + "content": "size can lead to higher model capacity, to perform a meaningful comparison, we make sure the model", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 613, + 261, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 261, + 624 + ], + "score": 1.0, + "content": "sizes of the 5 variants are comparable.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 107, + 628, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "To compare the generalization and model capacity, we train different variants of hybrid models on", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 639, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 235, + 653 + ], + "score": 1.0, + "content": "ImageNet-1K (1.3M) and JFT", + "type": "text" + }, + { + "bbox": [ + 235, + 640, + 272, + 651 + ], + "score": 0.76, + "content": "\\left( > 3 0 0 \\mathbf { M } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 639, + 505, + 653 + ], + "score": 1.0, + "content": "dataset for 300 and 3 epochs respectively, both without", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "any regularization or augmentation. The training loss and evaluation accuracy on both datasets are", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 662, + 207, + 674 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 207, + 674 + ], + "score": 1.0, + "content": "summarized in Figure 1.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5 + }, + { + "type": "text", + "bbox": [ + 104, + 677, + 505, + 700 + ], + "lines": [ + { + "bbox": [ + 104, + 676, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 104, + 676, + 506, + 691 + ], + "score": 1.0, + "content": "• From the ImageNet-1K results, a key observation is that, in terms of generalization capability (i.e.,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 113, + 688, + 321, + 702 + ], + "spans": [ + { + "bbox": [ + 113, + 688, + 321, + 702 + ], + "score": 1.0, + "content": "gap between train and evaluation metrics), we have", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49.5 + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 710, + 430, + 723 + ], + "lines": [ + { + "bbox": [ + 187, + 710, + 430, + 723 + ], + "spans": [ + { + "bbox": [ + 187, + 710, + 430, + 723 + ], + "score": 0.79, + "content": "\\mathrm { C \\mathrm { - } C \\mathrm { - } C \\mathrm { - } C \\approx C \\mathrm { - } C \\mathrm { - } C \\mathrm { - } T \\ge C \\mathrm { - } C \\mathrm { - } T \\mathrm { - } T > C \\mathrm { - } T \\mathrm { - } T \\mathrm { - } T \\gg V \\mathrm { I } \\mathrm { T } _ { \\mathrm { R E L } } . }", + "type": "interline_equation", + "image_path": "5f09b3609b6f1eadeb4a99c8e756ab7b3480d39c7a84b75a5be78aad8ba17e00.jpg" + } + ] + } + ], + "index": 51, + "virtual_lines": [ + { + "bbox": [ + 187, + 710, + 430, + 723 + ], + "spans": [], + "index": 51 + } + ] + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 73, + 228, + 84 + ], + "lines": [ + { + "bbox": [ + 105, + 71, + 229, + 87 + ], + "spans": [ + { + "bbox": [ + 105, + 71, + 229, + 87 + ], + "score": 1.0, + "content": "2.2 Vertical Layout Design", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 94, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "After figuring out a neat way to combine convolution and attention, we next consider how to utilize it", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 214, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 214, + 117 + ], + "score": 1.0, + "content": "to stack an entire network.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 94, + 506, + 117 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 121, + 505, + 166 + ], + "lines": [ + { + "bbox": [ + 104, + 121, + 507, + 135 + ], + "spans": [ + { + "bbox": [ + 104, + 121, + 507, + 135 + ], + "score": 1.0, + "content": "As we have discuss above, the global context has a quadratic complexity w.r.t. the spatial size. Hence,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "if we directly apply the relative attention in Eqn. (3) to the raw image input, the computation will", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "be excessively slow due to the large number of pixels in any image of common sizes. Hence, to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 416, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 416, + 167 + ], + "score": 1.0, + "content": "construct a network that is feasible in practice, we have mainly three options:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5, + "bbox_fs": [ + 104, + 121, + 507, + 167 + ] + }, + { + "type": "list", + "bbox": [ + 108, + 171, + 505, + 245 + ], + "lines": [ + { + "bbox": [ + 107, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 107, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "(A) Perform some down-sampling to reduce the spatial size and employ the global relative attention", + "type": "text" + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 126, + 181, + 318, + 194 + ], + "spans": [ + { + "bbox": [ + 126, + 181, + 318, + 194 + ], + "score": 1.0, + "content": "after the feature map reaches manageable level.", + "type": "text" + } + ], + "index": 8, + "is_list_end_line": true + }, + { + "bbox": [ + 107, + 196, + 504, + 209 + ], + "spans": [ + { + "bbox": [ + 107, + 196, + 380, + 209 + ], + "score": 1.0, + "content": "(B) Enforce local attention, which restricts the global receptive field", + "type": "text" + }, + { + "bbox": [ + 381, + 197, + 389, + 207 + ], + "score": 0.82, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 196, + 495, + 209 + ], + "score": 1.0, + "content": "in attention to a local field", + "type": "text" + }, + { + "bbox": [ + 496, + 197, + 504, + 207 + ], + "score": 0.62, + "content": "\\mathcal { L }", + "type": "inline_equation" + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 125, + 208, + 257, + 219 + ], + "spans": [ + { + "bbox": [ + 125, + 208, + 257, + 219 + ], + "score": 1.0, + "content": "just like in convolution [22, 21].", + "type": "text" + } + ], + "index": 10, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 222, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 506, + 235 + ], + "score": 1.0, + "content": "(C) Replace the quadratic Softmax attention with certain linear attention variant which only has a", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 126, + 232, + 334, + 246 + ], + "spans": [ + { + "bbox": [ + 126, + 232, + 334, + 246 + ], + "score": 1.0, + "content": "linear complexity w.r.t. the spatial size [12, 32, 33].", + "type": "text" + } + ], + "index": 12, + "is_list_end_line": true + } + ], + "index": 9.5, + "bbox_fs": [ + 106, + 171, + 506, + 246 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 250, + 505, + 316 + ], + "lines": [ + { + "bbox": [ + 105, + 249, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 505, + 263 + ], + "score": 1.0, + "content": "We briefly experimented with option (C) without getting a reasonably good result. For option (B), we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 261, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 506, + 273 + ], + "score": 1.0, + "content": "found that implementing local attention involves many non-trivial shape formatting operations that", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "requires intensive memory access. On our accelerator of choice (TPU), such operation turns out to be", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 283, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 505, + 295 + ], + "score": 1.0, + "content": "extremely slow [34], which not only defeats the original purpose of speeding up global attention, but", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 293, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 306 + ], + "score": 1.0, + "content": "also hurts the model capacity. Hence, as some recent work has studied this variant [22, 21], we will", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 305, + 469, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 469, + 316 + ], + "score": 1.0, + "content": "focus on option (A) and compare our results with theirs in our empirical study (Section 4).", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 249, + 506, + 316 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 320, + 504, + 354 + ], + "lines": [ + { + "bbox": [ + 105, + 320, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 334 + ], + "score": 1.0, + "content": "For option (A), the down-sampling can be achieved by either (1) a convolution stem with aggressive", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 332, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 505, + 344 + ], + "score": 1.0, + "content": "stride (e.g., stride 16x16) as in ViT or (2) a multi-stage network with gradual pooling as in ConvNets.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "score": 1.0, + "content": "With these choices, we derive a search space of 5 variants and compare them in controlled experiments.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 320, + 505, + 355 + ] + }, + { + "type": "list", + "bbox": [ + 105, + 359, + 505, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 298, + 371 + ], + "score": 1.0, + "content": "• When the ViT Stem is used, we directly stack", + "type": "text" + }, + { + "bbox": [ + 298, + 360, + 306, + 369 + ], + "score": 0.72, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "Transformer blocks with relative attention, which", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 113, + 369, + 204, + 383 + ], + "spans": [ + { + "bbox": [ + 113, + 369, + 168, + 383 + ], + "score": 1.0, + "content": "we denote as", + "type": "text" + }, + { + "bbox": [ + 169, + 370, + 199, + 381 + ], + "score": 0.87, + "content": "\\mathrm { V I T } _ { \\mathrm { R E L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 369, + 204, + 383 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "• When the multi-stage layout is used, we mimic ConvNets to construct a network of 5 stages (S0,", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 112, + 394, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 112, + 394, + 506, + 410 + ], + "score": 1.0, + "content": "S1, S2, S3 & S4), with spatial resolution gradually decreased from S0 to S4. At the beginning", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 113, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 113, + 406, + 322, + 419 + ], + "score": 1.0, + "content": "of each stage, we always reduce the spatial size by", + "type": "text" + }, + { + "bbox": [ + 322, + 407, + 335, + 417 + ], + "score": 0.57, + "content": "2 \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "and increase the number of channels (see", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 114, + 417, + 371, + 430 + ], + "spans": [ + { + "bbox": [ + 114, + 417, + 371, + 430 + ], + "score": 1.0, + "content": "Appendix A.1 for the detailed down-sampling implementation).", + "type": "text" + } + ], + "index": 27, + "is_list_end_line": true + }, + { + "bbox": [ + 114, + 432, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 114, + 432, + 505, + 445 + ], + "score": 1.0, + "content": "The first stage S0 is a simple 2-layer convolutional Stem and S1 always employs MBConv blocks", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 114, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 114, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "with squeeze-excitation (SE), as the spatial size is too large for global attention. Starting from", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 115, + 455, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 115, + 455, + 505, + 466 + ], + "score": 1.0, + "content": "S2 through S4, we consider either the MBConv or the Transformer block, with a constraint that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 114, + 466, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 114, + 466, + 506, + 477 + ], + "score": 1.0, + "content": "convolution stages must appear before Transformer stages. The constraint is based on the prior", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 114, + 476, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 114, + 476, + 505, + 488 + ], + "score": 1.0, + "content": "that convolution is better at processing local patterns that are more common in early stages. This", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 113, + 487, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 113, + 487, + 506, + 499 + ], + "score": 1.0, + "content": "leads to 4 variants with increasingly more Transformer stages, C-C-C-C, C-C-C-T, C-C-T-T and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 114, + 497, + 421, + 511 + ], + "spans": [ + { + "bbox": [ + 114, + 497, + 421, + 511 + ], + "score": 1.0, + "content": "C-T-T-T, where C and T denote Convolution and Transformer respectively.", + "type": "text" + } + ], + "index": 34, + "is_list_end_line": true + } + ], + "index": 28, + "bbox_fs": [ + 105, + 358, + 506, + 511 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 514, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "To systematically study the design choices, we consider two fundamental aspects generalization", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "capability and model capacity: For generalization, we are interested in the gap between the training", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 537, + 504, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 504, + 548 + ], + "score": 1.0, + "content": "loss and the evaluation accuracy. If two models have the same training loss, then the model with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "score": 1.0, + "content": "higher evaluation accuracy has better generalization capability, since it can generalize better to unseen", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 556, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 104, + 556, + 506, + 572 + ], + "score": 1.0, + "content": "evaluation dataset. Generalization capability is particularly important to data efficiency when training", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "score": 1.0, + "content": "data size is limited. For model capacity, we measure the ability to fit large training datasets. When", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 579, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 506, + 592 + ], + "score": 1.0, + "content": "training data is abundant and overfitting is not an issue, the model with higher capacity will achieve", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "better final performance after reasonable training steps. Note that, since simply increasing the model", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 601, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 104, + 601, + 506, + 614 + ], + "score": 1.0, + "content": "size can lead to higher model capacity, to perform a meaningful comparison, we make sure the model", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 613, + 261, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 261, + 624 + ], + "score": 1.0, + "content": "sizes of the 5 variants are comparable.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 39.5, + "bbox_fs": [ + 104, + 514, + 506, + 624 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 628, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "To compare the generalization and model capacity, we train different variants of hybrid models on", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 639, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 235, + 653 + ], + "score": 1.0, + "content": "ImageNet-1K (1.3M) and JFT", + "type": "text" + }, + { + "bbox": [ + 235, + 640, + 272, + 651 + ], + "score": 0.76, + "content": "\\left( > 3 0 0 \\mathbf { M } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 639, + 505, + 653 + ], + "score": 1.0, + "content": "dataset for 300 and 3 epochs respectively, both without", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "any regularization or augmentation. The training loss and evaluation accuracy on both datasets are", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 662, + 207, + 674 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 207, + 674 + ], + "score": 1.0, + "content": "summarized in Figure 1.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 629, + 505, + 674 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 677, + 505, + 700 + ], + "lines": [ + { + "bbox": [ + 104, + 676, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 104, + 676, + 506, + 691 + ], + "score": 1.0, + "content": "• From the ImageNet-1K results, a key observation is that, in terms of generalization capability (i.e.,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 113, + 688, + 321, + 702 + ], + "spans": [ + { + "bbox": [ + 113, + 688, + 321, + 702 + ], + "score": 1.0, + "content": "gap between train and evaluation metrics), we have", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49.5, + "bbox_fs": [ + 104, + 676, + 506, + 702 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 710, + 430, + 723 + ], + "lines": [ + { + "bbox": [ + 187, + 710, + 430, + 723 + ], + "spans": [ + { + "bbox": [ + 187, + 710, + 430, + 723 + ], + "score": 0.79, + "content": "\\mathrm { C \\mathrm { - } C \\mathrm { - } C \\mathrm { - } C \\approx C \\mathrm { - } C \\mathrm { - } C \\mathrm { - } T \\ge C \\mathrm { - } C \\mathrm { - } T \\mathrm { - } T > C \\mathrm { - } T \\mathrm { - } T \\mathrm { - } T \\gg V \\mathrm { I } \\mathrm { T } _ { \\mathrm { R E L } } . }", + "type": "interline_equation", + "image_path": "5f09b3609b6f1eadeb4a99c8e756ab7b3480d39c7a84b75a5be78aad8ba17e00.jpg" + } + ] + } + ], + "index": 51, + "virtual_lines": [ + { + "bbox": [ + 187, + 710, + 430, + 723 + ], + "spans": [], + "index": 51 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 72, + 502, + 186 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 72, + 502, + 186 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 72, + 502, + 186 + ], + "spans": [ + { + "bbox": [ + 109, + 72, + 502, + 186 + ], + "score": 0.971, + "type": "image", + "image_path": "8e4cc944c4e098b4c2962d8c1e585910035ad5a220c25df539ba8f057d9c47df.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 72, + 502, + 110.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 110.0, + 502, + 148.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 148.0, + 502, + 186.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 192, + 505, + 215 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "Figure 1: Comparison for model generalization and capacity under different data size. For fair", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 203, + 411, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 411, + 216 + ], + "score": 1.0, + "content": "comparison, all models have similar parameter size and computational cost.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 114, + 236, + 505, + 281 + ], + "lines": [ + { + "bbox": [ + 114, + 236, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 114, + 236, + 167, + 250 + ], + "score": 1.0, + "content": "Particularly,", + "type": "text" + }, + { + "bbox": [ + 167, + 237, + 198, + 248 + ], + "score": 0.89, + "content": "\\mathrm { V I T } _ { \\mathrm { R E L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 236, + 505, + 250 + ], + "score": 1.0, + "content": "is significantly worse than variants by a large margin, which we conjecture", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 113, + 247, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 113, + 247, + 506, + 261 + ], + "score": 1.0, + "content": "is related to the lack of proper low-level information processing in its aggressive down-sampling", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 114, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 114, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "Stem. Among the multi-stage variants, the overall trend is that the more convolution stages the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 114, + 270, + 309, + 282 + ], + "spans": [ + { + "bbox": [ + 114, + 270, + 309, + 282 + ], + "score": 1.0, + "content": "model has, the smaller the generalization gap is.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 108, + 282, + 504, + 305 + ], + "lines": [ + { + "bbox": [ + 106, + 280, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 506, + 295 + ], + "score": 1.0, + "content": "• As for model capacity, from the JFT comparison, both the train and evaluation metrics at the end", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 113, + 291, + 298, + 307 + ], + "spans": [ + { + "bbox": [ + 113, + 291, + 298, + 307 + ], + "score": 1.0, + "content": "of the training suggest the following ranking:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "interline_equation", + "bbox": [ + 188, + 310, + 430, + 323 + ], + "lines": [ + { + "bbox": [ + 188, + 310, + 430, + 323 + ], + "spans": [ + { + "bbox": [ + 188, + 310, + 430, + 323 + ], + "score": 0.82, + "content": "\\mathrm { C - C \\mathrm { - } T \\mathrm { - } T \\approx C \\mathrm { - } T \\mathrm { - } T \\mathrm { - } T > V I T _ { R E L } > C \\mathrm { - } C \\mathrm { - } C \\mathrm { - } T > C \\mathrm { - } C \\mathrm { - } C \\mathrm { - } C \\mathrm { - } C \\mathrm { . } }", + "type": "interline_equation", + "image_path": "db72680f50180e0871b5c7bce1cdd56ab51755cadb71545f3fb276d6f9b0b410.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 188, + 310, + 430, + 323 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 114, + 329, + 505, + 417 + ], + "lines": [ + { + "bbox": [ + 114, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 114, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "Importantly, this suggests that simply having more Transformer blocks does NOT necessarily", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 115, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 115, + 341, + 431, + 352 + ], + "score": 1.0, + "content": "mean higher capacity for visual processing. On one hand, while initially worse,", + "type": "text" + }, + { + "bbox": [ + 432, + 340, + 461, + 351 + ], + "score": 0.89, + "content": "\\mathrm { V I T } _ { \\mathrm { R E L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 341, + 505, + 352 + ], + "score": 1.0, + "content": "ultimately", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 114, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 114, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "catch up with the two variants with more MBConv stages, indicating the capacity advantage of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 113, + 359, + 504, + 376 + ], + "spans": [ + { + "bbox": [ + 113, + 359, + 473, + 376 + ], + "score": 1.0, + "content": "Transformer blocks. On the other hand, both C-C-T-T and C-T-T-T clearly outperforming", + "type": "text" + }, + { + "bbox": [ + 473, + 362, + 504, + 373 + ], + "score": 0.88, + "content": "\\mathrm { V I T } _ { \\mathrm { R E L } }", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 113, + 372, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 113, + 372, + 506, + 385 + ], + "score": 1.0, + "content": "suggest that the ViT stem with an aggressive stride may have lost too much information and hence", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 113, + 382, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 113, + 382, + 387, + 396 + ], + "score": 1.0, + "content": "limit the model capacity. More interestingly, the fact that C-C-T-T", + "type": "text" + }, + { + "bbox": [ + 387, + 384, + 407, + 394 + ], + "score": 0.37, + "content": "\\approx \\mathbf { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 382, + 506, + 396 + ], + "score": 1.0, + "content": "-T-T-T indicates the for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 113, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 113, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "processing low-level information, static local operations like convolution could be as capable as", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 114, + 405, + 504, + 418 + ], + "spans": [ + { + "bbox": [ + 114, + 405, + 504, + 418 + ], + "score": 1.0, + "content": "adaptive global attention mechanism, while saving computation and memory usage substantially.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 108, + 421, + 505, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "Finally, to decide between C-C-T-T and C-T-T-T, we conduct another transferability test3 — we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 432, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 446 + ], + "score": 1.0, + "content": "finetune the two JFT pre-trained models above on ImageNet-1K for 30 epochs and compare their", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 444, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 505, + 456 + ], + "score": 1.0, + "content": "transfer performances. From Table 2, it turns out that C-C-T-T achieves a clearly better transfer", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 455, + 379, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 379, + 468 + ], + "score": 1.0, + "content": "accuracy than C-T-T-T, despite the same pre-training performance.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5 + }, + { + "type": "table", + "bbox": [ + 184, + 485, + 424, + 541 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 234, + 474, + 376, + 484 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 232, + 471, + 378, + 487 + ], + "spans": [ + { + "bbox": [ + 232, + 471, + 378, + 487 + ], + "score": 1.0, + "content": "Table 2: Transferability test results.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "table_body", + "bbox": [ + 184, + 485, + 424, + 541 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 184, + 485, + 424, + 541 + ], + "spans": [ + { + "bbox": [ + 184, + 485, + 424, + 541 + ], + "score": 0.978, + "html": "
MetricC-C-T-TC-T-T-T
Pre-training Precision@1 (JFT)34.4034.36
Transfer Accuracy 224x22482.3981.78
Transfer Accuracy 384x38484.2384.02
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More model details are included in Appendix A.1.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "title", + "bbox": [ + 107, + 590, + 197, + 604 + ], + "lines": [ + { + "bbox": [ + 104, + 589, + 198, + 605 + ], + "spans": [ + { + "bbox": [ + 104, + 589, + 198, + 605 + ], + "score": 1.0, + "content": "3 Related Work", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 614, + 506, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 614, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 506, + 628 + ], + "score": 1.0, + "content": "Convolutional network building blocks. Convolutional Networks (ConvNets) have been the dom-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "inating neural architectures for many computer vision tasks. Traditionally, regular convolutions, such", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "score": 1.0, + "content": "as ResNet blocks [3], are popular in large-scale ConvNets; in contrast, depthwise convolutions [28]", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 648, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 506, + 660 + ], + "score": 1.0, + "content": "are popular in mobile platforms due to its lower computational cost and smaller parameter size [27].", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 659, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 506, + 671 + ], + "score": 1.0, + "content": "Recent works show that an improved inverted residual bottlenecks (MBConv [27, 35]), which is", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 668, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 506, + 682 + ], + "score": 1.0, + "content": "built upon depthwise convolutions, can achieve both high accuracy and better efficiency [5, 19]. As", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 681, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 506, + 693 + ], + "score": 1.0, + "content": "discussed in Section 2, due to the strong connection between MBConv and Transformer blocks , this", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 692, + 363, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 363, + 704 + ], + "score": 1.0, + "content": "paper mostly employs MBConv as convolution building blocks.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 117, + 711, + 462, + 722 + ], + "lines": [ + { + "bbox": [ + 119, + 710, + 463, + 724 + ], + "spans": [ + { + "bbox": [ + 119, + 710, + 463, + 724 + ], + "score": 1.0, + "content": "3 Rigorously speaking, this test examines not only the transferability but also the generalization.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 72, + 502, + 186 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 72, + 502, + 186 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 72, + 502, + 186 + ], + "spans": [ + { + "bbox": [ + 109, + 72, + 502, + 186 + ], + "score": 0.971, + "type": "image", + "image_path": "8e4cc944c4e098b4c2962d8c1e585910035ad5a220c25df539ba8f057d9c47df.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 72, + 502, + 110.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 110.0, + 502, + 148.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 148.0, + 502, + 186.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 192, + 505, + 215 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "Figure 1: Comparison for model generalization and capacity under different data size. For fair", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 203, + 411, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 411, + 216 + ], + "score": 1.0, + "content": "comparison, all models have similar parameter size and computational cost.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 114, + 236, + 505, + 281 + ], + "lines": [ + { + "bbox": [ + 114, + 236, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 114, + 236, + 167, + 250 + ], + "score": 1.0, + "content": "Particularly,", + "type": "text" + }, + { + "bbox": [ + 167, + 237, + 198, + 248 + ], + "score": 0.89, + "content": "\\mathrm { V I T } _ { \\mathrm { R E L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 236, + 505, + 250 + ], + "score": 1.0, + "content": "is significantly worse than variants by a large margin, which we conjecture", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 113, + 247, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 113, + 247, + 506, + 261 + ], + "score": 1.0, + "content": "is related to the lack of proper low-level information processing in its aggressive down-sampling", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 114, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 114, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "Stem. Among the multi-stage variants, the overall trend is that the more convolution stages the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 114, + 270, + 309, + 282 + ], + "spans": [ + { + "bbox": [ + 114, + 270, + 309, + 282 + ], + "score": 1.0, + "content": "model has, the smaller the generalization gap is.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5, + "bbox_fs": [ + 113, + 236, + 506, + 282 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 282, + 504, + 305 + ], + "lines": [ + { + "bbox": [ + 106, + 280, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 506, + 295 + ], + "score": 1.0, + "content": "• As for model capacity, from the JFT comparison, both the train and evaluation metrics at the end", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 113, + 291, + 298, + 307 + ], + "spans": [ + { + "bbox": [ + 113, + 291, + 298, + 307 + ], + "score": 1.0, + "content": "of the training suggest the following ranking:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 106, + 280, + 506, + 307 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 188, + 310, + 430, + 323 + ], + "lines": [ + { + "bbox": [ + 188, + 310, + 430, + 323 + ], + "spans": [ + { + "bbox": [ + 188, + 310, + 430, + 323 + ], + "score": 0.82, + "content": "\\mathrm { C - C \\mathrm { - } T \\mathrm { - } T \\approx C \\mathrm { - } T \\mathrm { - } T \\mathrm { - } T > V I T _ { R E L } > C \\mathrm { - } C \\mathrm { - } C \\mathrm { - } T > C \\mathrm { - } C \\mathrm { - } C \\mathrm { - } C \\mathrm { - } C \\mathrm { . } }", + "type": "interline_equation", + "image_path": "db72680f50180e0871b5c7bce1cdd56ab51755cadb71545f3fb276d6f9b0b410.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 188, + 310, + 430, + 323 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 114, + 329, + 505, + 417 + ], + "lines": [ + { + "bbox": [ + 114, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 114, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "Importantly, this suggests that simply having more Transformer blocks does NOT necessarily", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 115, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 115, + 341, + 431, + 352 + ], + "score": 1.0, + "content": "mean higher capacity for visual processing. On one hand, while initially worse,", + "type": "text" + }, + { + "bbox": [ + 432, + 340, + 461, + 351 + ], + "score": 0.89, + "content": "\\mathrm { V I T } _ { \\mathrm { R E L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 341, + 505, + 352 + ], + "score": 1.0, + "content": "ultimately", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 114, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 114, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "catch up with the two variants with more MBConv stages, indicating the capacity advantage of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 113, + 359, + 504, + 376 + ], + "spans": [ + { + "bbox": [ + 113, + 359, + 473, + 376 + ], + "score": 1.0, + "content": "Transformer blocks. On the other hand, both C-C-T-T and C-T-T-T clearly outperforming", + "type": "text" + }, + { + "bbox": [ + 473, + 362, + 504, + 373 + ], + "score": 0.88, + "content": "\\mathrm { V I T } _ { \\mathrm { R E L } }", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 113, + 372, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 113, + 372, + 506, + 385 + ], + "score": 1.0, + "content": "suggest that the ViT stem with an aggressive stride may have lost too much information and hence", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 113, + 382, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 113, + 382, + 387, + 396 + ], + "score": 1.0, + "content": "limit the model capacity. More interestingly, the fact that C-C-T-T", + "type": "text" + }, + { + "bbox": [ + 387, + 384, + 407, + 394 + ], + "score": 0.37, + "content": "\\approx \\mathbf { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 382, + 506, + 396 + ], + "score": 1.0, + "content": "-T-T-T indicates the for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 113, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 113, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "processing low-level information, static local operations like convolution could be as capable as", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 114, + 405, + 504, + 418 + ], + "spans": [ + { + "bbox": [ + 114, + 405, + 504, + 418 + ], + "score": 1.0, + "content": "adaptive global attention mechanism, while saving computation and memory usage substantially.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5, + "bbox_fs": [ + 113, + 329, + 506, + 418 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 421, + 505, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "Finally, to decide between C-C-T-T and C-T-T-T, we conduct another transferability test3 — we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 432, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 446 + ], + "score": 1.0, + "content": "finetune the two JFT pre-trained models above on ImageNet-1K for 30 epochs and compare their", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 444, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 505, + 456 + ], + "score": 1.0, + "content": "transfer performances. From Table 2, it turns out that C-C-T-T achieves a clearly better transfer", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 455, + 379, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 379, + 468 + ], + "score": 1.0, + "content": "accuracy than C-T-T-T, despite the same pre-training performance.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 421, + 506, + 468 + ] + }, + { + "type": "table", + "bbox": [ + 184, + 485, + 424, + 541 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 234, + 474, + 376, + 484 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 232, + 471, + 378, + 487 + ], + "spans": [ + { + "bbox": [ + 232, + 471, + 378, + 487 + ], + "score": 1.0, + "content": "Table 2: Transferability test results.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "table_body", + "bbox": [ + 184, + 485, + 424, + 541 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 184, + 485, + 424, + 541 + ], + "spans": [ + { + "bbox": [ + 184, + 485, + 424, + 541 + ], + "score": 0.978, + "html": "
MetricC-C-T-TC-T-T-T
Pre-training Precision@1 (JFT)34.4034.36
Transfer Accuracy 224x22482.3981.78
Transfer Accuracy 384x38484.2384.02
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More model details are included in Appendix A.1.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 106, + 552, + 504, + 576 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 590, + 197, + 604 + ], + "lines": [ + { + "bbox": [ + 104, + 589, + 198, + 605 + ], + "spans": [ + { + "bbox": [ + 104, + 589, + 198, + 605 + ], + "score": 1.0, + "content": "3 Related Work", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 614, + 506, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 614, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 506, + 628 + ], + "score": 1.0, + "content": "Convolutional network building blocks. Convolutional Networks (ConvNets) have been the dom-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "inating neural architectures for many computer vision tasks. Traditionally, regular convolutions, such", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "score": 1.0, + "content": "as ResNet blocks [3], are popular in large-scale ConvNets; in contrast, depthwise convolutions [28]", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 648, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 506, + 660 + ], + "score": 1.0, + "content": "are popular in mobile platforms due to its lower computational cost and smaller parameter size [27].", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 659, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 506, + 671 + ], + "score": 1.0, + "content": "Recent works show that an improved inverted residual bottlenecks (MBConv [27, 35]), which is", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 668, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 506, + 682 + ], + "score": 1.0, + "content": "built upon depthwise convolutions, can achieve both high accuracy and better efficiency [5, 19]. As", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 681, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 506, + 693 + ], + "score": 1.0, + "content": "discussed in Section 2, due to the strong connection between MBConv and Transformer blocks , this", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 692, + 363, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 363, + 704 + ], + "score": 1.0, + "content": "paper mostly employs MBConv as convolution building blocks.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 614, + 506, + 704 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "Self-attention and Transformers. With the key ingredients of self-attention, Transformers have", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "been widely adopted for neural language processing and speech understanding. As an early work,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "stand-alone self-attention network [34] shows self-attention alone can work well for different vision", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "tasks, though with some practical difficulties. Recently, ViT [13] applies a vanilla Transformer", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "score": 1.0, + "content": "to ImageNet classification, and achieves impressive results after pre-training on a large-scale JFT", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 127, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 506, + 139 + ], + "score": 1.0, + "content": "dataset. However, ViT still largely lags behind state-of-the-art ConvNets when training data is limited.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "Following that, many recent works have been focused on improving vision Transformers for data", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 149, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 506, + 161 + ], + "score": 1.0, + "content": "efficiency and model efficiency. For a more comprehensive review of vision Transformers, we refer", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 160, + 272, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 272, + 172 + ], + "score": 1.0, + "content": "readers to the dedicated surveys [36, 37].", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 182, + 505, + 292 + ], + "lines": [ + { + "bbox": [ + 106, + 182, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 505, + 195 + ], + "score": 1.0, + "content": "Relative attention. Under the general name of relative attention, there have been various variants", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 192, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 104, + 192, + 506, + 207 + ], + "score": 1.0, + "content": "in literature [30, 38, 39, 34, 40, 31]. Generally speaking, we can separate them into two categories:", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "(a) the input-dependent version where the extra relative attention score is a function of the input", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 215, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 132, + 227 + ], + "score": 1.0, + "content": "states", + "type": "text" + }, + { + "bbox": [ + 133, + 215, + 196, + 228 + ], + "score": 0.93, + "content": "f ( \\bar { x } _ { i } , x _ { j } , \\bar { i } - j )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 215, + 361, + 227 + ], + "score": 1.0, + "content": ", and (b) the input-independent version", + "type": "text" + }, + { + "bbox": [ + 361, + 215, + 397, + 227 + ], + "score": 0.93, + "content": "f ( i - j )", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 215, + 506, + 227 + ], + "score": 1.0, + "content": ". The variant in CoAtNet", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "score": 1.0, + "content": "belongs to the input-independent version, and is similar to the one used in T5 [31], but unlike T5, we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 237, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 506, + 250 + ], + "score": 1.0, + "content": "neither share the relative attention parameters across layers nor use the bucketing mechanism. As", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 294, + 261 + ], + "score": 1.0, + "content": "a benefit of the input independence, obtaining", + "type": "text" + }, + { + "bbox": [ + 294, + 248, + 329, + 260 + ], + "score": 0.93, + "content": "f ( i - j )", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 248, + 358, + 261 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 358, + 248, + 379, + 260 + ], + "score": 0.92, + "content": "( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 248, + 505, + 261 + ], + "score": 1.0, + "content": "pairs is computationally much", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 259, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 272 + ], + "score": 1.0, + "content": "cheaper than the input-dependent version on TPU. In addition, at inference time, this only needs to be", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 271, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 505, + 282 + ], + "score": 1.0, + "content": "computed once and cached for future use. A recent work [22] also utilizes such an input-independent", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 281, + 384, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 384, + 293 + ], + "score": 1.0, + "content": "parameterization, but it restricts the receptive field to a local window.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 505, + 402 + ], + "lines": [ + { + "bbox": [ + 106, + 304, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 304, + 505, + 316 + ], + "score": 1.0, + "content": "Combining convolution and self-attention. The idea of combining convolution and self-attention", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "for vision recognition is not new. A common approach is to augment the ConvNet backbone with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "explicit self-attention or non-local modules [9, 10, 11, 12], or to replace certain convolution layers", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "with standard self-attention [11] or a more flexible mix of linear attention and convolution [41]. While", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "self-attention usually improves the accuracy, they often come with extra computational cost and hence", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "score": 1.0, + "content": "are often regarded as an add-on to the ConvNets, similar to squeeze-and-excitation [42] module. In", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "comparison, after the success of ViT and ResNet-ViT [13], another popular line of research starts with", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "a Transformer backbone and tries to incorporate explicit convolution or some desirable properties of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 390, + 392, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 392, + 403 + ], + "score": 1.0, + "content": "convolution into the Transformer backbone [25, 24, 23, 22, 21, 43, 44].", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 407, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 106, + 408, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 506, + 420 + ], + "score": 1.0, + "content": "While our work also belongs to this category, we show that our relative attention instantiation is a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 419, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 506, + 430 + ], + "score": 1.0, + "content": "natural mixture of depthwise convolution and content-based attention with minimum additional cost.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 429, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 506, + 442 + ], + "score": 1.0, + "content": "More importantly, starting from the perspectives of generalization and model capacity, we take a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 439, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 454 + ], + "score": 1.0, + "content": "systematic approach to the vertical layout design and show how and why different network stages", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "prefer different types of layers. Therefore, compared to models that simply use an off-the-shelf", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 460, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 476 + ], + "score": 1.0, + "content": "ConvNet as the stem layer, such as ResNet-ViT [13], CoAtNet also scales the Convolution stage", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "(S2) when the overall size increases. On the other hand, compared to models employing local", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 482, + 507, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 507, + 498 + ], + "score": 1.0, + "content": "attention [22, 21], CoAtNet consistently uses full attention for S3 & S4 to ensure the model capacity,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 495, + 362, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 362, + 508 + ], + "score": 1.0, + "content": "as S3 occupies the majority of the computation and parameters.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 107, + 521, + 191, + 534 + ], + "lines": [ + { + "bbox": [ + 104, + 519, + 193, + 537 + ], + "spans": [ + { + "bbox": [ + 104, + 519, + 193, + 537 + ], + "score": 1.0, + "content": "4 Experiments", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 545, + 504, + 568 + ], + "lines": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "In this section, we compare CoAtNet with previous results under comparable settings. For complete-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 556, + 430, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 430, + 568 + ], + "score": 1.0, + "content": "ness, all the hyper-parameters not mentioned here are included in Appendix A.2.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5 + }, + { + "type": "title", + "bbox": [ + 107, + 580, + 213, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 578, + 215, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 215, + 596 + ], + "score": 1.0, + "content": "4.1 Experiment Setting", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 655 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "CoAtNet model family. To compare with existing models of different sizes, we also design a", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "family of CoAtNet models as summarized in Table 3. Overall, we always double the number of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 623, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 506, + 635 + ], + "score": 1.0, + "content": "channels from S1 to S4, while ensuring the width of the Stem S0 to be smaller or equal to that of S1.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "Also, for simplicity, when increasing the depth of the network, we only scale the number of blocks in", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 644, + 152, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 152, + 656 + ], + "score": 1.0, + "content": "S2 and S3.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "Evaluation Protocol. Our experiments focus on image classification. To evaluate the performance", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "score": 1.0, + "content": "of the model across different data sizes, we utilize three datasets of increasingly larger sizes, namely", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 687, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 703 + ], + "score": 1.0, + "content": "ImageNet-1K (1.28M images), ImageNet-21K (12.7M images) and JFT (300M images). Following", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "previous works, we first pre-train our models on each of the three datasets at resolution 224 for 300, 90", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "and 14 epochs respectively. Then, we finetune the pre-trained models on ImageNet-1K at the desired", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "Self-attention and Transformers. With the key ingredients of self-attention, Transformers have", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "been widely adopted for neural language processing and speech understanding. As an early work,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "stand-alone self-attention network [34] shows self-attention alone can work well for different vision", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "tasks, though with some practical difficulties. Recently, ViT [13] applies a vanilla Transformer", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "score": 1.0, + "content": "to ImageNet classification, and achieves impressive results after pre-training on a large-scale JFT", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 127, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 506, + 139 + ], + "score": 1.0, + "content": "dataset. However, ViT still largely lags behind state-of-the-art ConvNets when training data is limited.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "Following that, many recent works have been focused on improving vision Transformers for data", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 149, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 506, + 161 + ], + "score": 1.0, + "content": "efficiency and model efficiency. For a more comprehensive review of vision Transformers, we refer", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 160, + 272, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 272, + 172 + ], + "score": 1.0, + "content": "readers to the dedicated surveys [36, 37].", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 73, + 506, + 172 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 182, + 505, + 292 + ], + "lines": [ + { + "bbox": [ + 106, + 182, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 505, + 195 + ], + "score": 1.0, + "content": "Relative attention. Under the general name of relative attention, there have been various variants", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 192, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 104, + 192, + 506, + 207 + ], + "score": 1.0, + "content": "in literature [30, 38, 39, 34, 40, 31]. Generally speaking, we can separate them into two categories:", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "(a) the input-dependent version where the extra relative attention score is a function of the input", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 215, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 132, + 227 + ], + "score": 1.0, + "content": "states", + "type": "text" + }, + { + "bbox": [ + 133, + 215, + 196, + 228 + ], + "score": 0.93, + "content": "f ( \\bar { x } _ { i } , x _ { j } , \\bar { i } - j )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 215, + 361, + 227 + ], + "score": 1.0, + "content": ", and (b) the input-independent version", + "type": "text" + }, + { + "bbox": [ + 361, + 215, + 397, + 227 + ], + "score": 0.93, + "content": "f ( i - j )", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 215, + 506, + 227 + ], + "score": 1.0, + "content": ". The variant in CoAtNet", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "score": 1.0, + "content": "belongs to the input-independent version, and is similar to the one used in T5 [31], but unlike T5, we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 237, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 506, + 250 + ], + "score": 1.0, + "content": "neither share the relative attention parameters across layers nor use the bucketing mechanism. As", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 294, + 261 + ], + "score": 1.0, + "content": "a benefit of the input independence, obtaining", + "type": "text" + }, + { + "bbox": [ + 294, + 248, + 329, + 260 + ], + "score": 0.93, + "content": "f ( i - j )", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 248, + 358, + 261 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 358, + 248, + 379, + 260 + ], + "score": 0.92, + "content": "( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 248, + 505, + 261 + ], + "score": 1.0, + "content": "pairs is computationally much", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 259, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 272 + ], + "score": 1.0, + "content": "cheaper than the input-dependent version on TPU. In addition, at inference time, this only needs to be", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 271, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 505, + 282 + ], + "score": 1.0, + "content": "computed once and cached for future use. A recent work [22] also utilizes such an input-independent", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 281, + 384, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 384, + 293 + ], + "score": 1.0, + "content": "parameterization, but it restricts the receptive field to a local window.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13.5, + "bbox_fs": [ + 104, + 182, + 506, + 293 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 505, + 402 + ], + "lines": [ + { + "bbox": [ + 106, + 304, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 304, + 505, + 316 + ], + "score": 1.0, + "content": "Combining convolution and self-attention. 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While", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "self-attention usually improves the accuracy, they often come with extra computational cost and hence", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "score": 1.0, + "content": "are often regarded as an add-on to the ConvNets, similar to squeeze-and-excitation [42] module. In", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "comparison, after the success of ViT and ResNet-ViT [13], another popular line of research starts with", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "a Transformer backbone and tries to incorporate explicit convolution or some desirable properties of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 390, + 392, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 392, + 403 + ], + "score": 1.0, + "content": "convolution into the Transformer backbone [25, 24, 23, 22, 21, 43, 44].", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 304, + 506, + 403 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 407, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 106, + 408, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 506, + 420 + ], + "score": 1.0, + "content": "While our work also belongs to this category, we show that our relative attention instantiation is a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 419, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 506, + 430 + ], + "score": 1.0, + "content": "natural mixture of depthwise convolution and content-based attention with minimum additional cost.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 429, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 506, + 442 + ], + "score": 1.0, + "content": "More importantly, starting from the perspectives of generalization and model capacity, we take a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 439, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 454 + ], + "score": 1.0, + "content": "systematic approach to the vertical layout design and show how and why different network stages", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "prefer different types of layers. 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For complete-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 556, + 430, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 430, + 568 + ], + "score": 1.0, + "content": "ness, all the hyper-parameters not mentioned here are included in Appendix A.2.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 545, + 505, + 568 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 580, + 213, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 578, + 215, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 215, + 596 + ], + "score": 1.0, + "content": "4.1 Experiment Setting", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 655 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "CoAtNet model family. 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ModelsEval Size#Params#FLOPsImageNet Top-1 Accuracy
Conv Only1K only21K+1K
EfficientNet-B7600²66M37B84.7-
EfficientNetV2-L480²121M53B85.786.8
NFNet-F34162²255M114.8B85.7=
NFNet-F55442377M289.8B86.01
ViT-Stem TFMDeiT-B384286M55.4B83.1-
ViT-L/16384²304M190.7B-85.3
CaiT-S-36384²68M48.0B85.0
DeepViT-L224²55M12.5B83.1-
Multi-stage TFMSwin-B384²88M47.0B84.286.0
Swin-L384²197M103.9B-86.4
Conv+TFMBotNet-T7384275.1M45.8B84.7-
LambdaResNet-420320²-=84.8
T2T-ViT-24224²64.1M15.0B82.6=
CvT-21384²32M24.9B83.3-
CvT-W243842277M193.2B-87.7
Conv+TFM (ours)CoAtNet-0 CoAtNet-1224²25M4.2B81.6=
224²42M8.4B83.3-
CoAtNet-2 CoAtNet-3224²75M15.7B84.187.1
2242168M34.7B84.587.6
CoAtNet-0384²25M13.4B83.9-
CoAtNet-1384242M27.4B85.1-
CoAtNet-2384²75M49.8B85.787.1
CoAtNet-3384²168M107.4B85.887.6
CoAtNet-4384²275M189.5B-87.9
+ PT-RA384²275M189.5B88.3
+ PT-RA-E1503842275M189.5B88.4
CoAtNet-2512275M96.7B85.987.3
CoAtNet-3512²168M203.1B86.087.9
CoAtNet-4512²275M360.9B-88.1
+ PT-RA512²275M360.9B=88.4
+ PT-RA-E1505122275M360.9B=88.56
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ModelsEval Size#Params#FLOPsImageNet Top-1 Accuracy
Conv Only1K only21K+1K
EfficientNet-B7600²66M37B84.7-
EfficientNetV2-L480²121M53B85.786.8
NFNet-F34162²255M114.8B85.7=
NFNet-F55442377M289.8B86.01
ViT-Stem TFMDeiT-B384286M55.4B83.1-
ViT-L/16384²304M190.7B-85.3
CaiT-S-36384²68M48.0B85.0
DeepViT-L224²55M12.5B83.1-
Multi-stage TFMSwin-B384²88M47.0B84.286.0
Swin-L384²197M103.9B-86.4
Conv+TFMBotNet-T7384275.1M45.8B84.7-
LambdaResNet-420320²-=84.8
T2T-ViT-24224²64.1M15.0B82.6=
CvT-21384²32M24.9B83.3-
CvT-W243842277M193.2B-87.7
Conv+TFM (ours)CoAtNet-0 CoAtNet-1224²25M4.2B81.6=
224²42M8.4B83.3-
CoAtNet-2 CoAtNet-3224²75M15.7B84.187.1
2242168M34.7B84.587.6
CoAtNet-0384²25M13.4B83.9-
CoAtNet-1384242M27.4B85.1-
CoAtNet-2384²75M49.8B85.787.1
CoAtNet-3384²168M107.4B85.887.6
CoAtNet-4384²275M189.5B-87.9
+ PT-RA384²275M189.5B88.3
+ PT-RA-E1503842275M189.5B88.4
CoAtNet-2512275M96.7B85.987.3
CoAtNet-3512²168M203.1B86.087.9
CoAtNet-4512²275M360.9B-88.1
+ PT-RA512²275M360.9B=88.4
+ PT-RA-E1505122275M360.9B=88.56
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This marks a dramatic improvement in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 594, + 300, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 300, + 609 + ], + "score": 1.0, + "content": "both data efficiency and computation efficiency.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 540, + 507, + 609 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 628, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 629, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 506, + 641 + ], + "score": 1.0, + "content": "JFT Finally, in Table 5, we further evaluate CoAtNet under the large-scale data regime with JFT-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "300M and JFT-3B. 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When we scale up the model to consume similar training resource as", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 672, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 137, + 685 + ], + "score": 1.0, + "content": "NFNet-", + "type": "text" + }, + { + "bbox": [ + 137, + 673, + 155, + 683 + ], + "score": 0.82, + "content": ". \\mathrm { F 4 + }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 672, + 240, + 685 + ], + "score": 1.0, + "content": ", CoAtNet-5 reaches", + "type": "text" + }, + { + "bbox": [ + 240, + 673, + 273, + 683 + ], + "score": 0.87, + "content": "8 9 . 7 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 672, + 506, + 685 + ], + "score": 1.0, + "content": "on top-1 accuracy, outperforming previous results under", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 682, + 191, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 191, + 698 + ], + "score": 1.0, + "content": "comparable settings.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 629, + 506, + 698 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 700, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "Moreover, as we further push the training resource towards the level used by ViT-G/14 and utilize the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 337, + 723 + ], + "score": 1.0, + "content": "same JFT-3B dataset of an even larger size [26], with over", + "type": "text" + }, + { + "bbox": [ + 337, + 711, + 349, + 721 + ], + "score": 0.39, + "content": "4 \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "less computation, CoAtNet-6 is able to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 210, + 308 + ], + "score": 1.0, + "content": "match the performance of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 210, + 295, + 248, + 306 + ], + "score": 0.25, + "content": "\\mathrm { V i T - G } / 1 4", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 249, + 295, + 260, + 308 + ], + "score": 1.0, + "content": "of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 261, + 295, + 292, + 306 + ], + "score": 0.89, + "content": "9 0 . 4 5 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 292, + 295, + 332, + 308 + ], + "score": 1.0, + "content": ", and with", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 333, + 295, + 351, + 306 + ], + "score": 0.74, + "content": "1 . 5 \\mathrm { x }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 352, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "less computation, CoAtNet-7 achieves", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 306, + 441, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 138, + 317 + ], + "score": 0.86, + "content": "8 9 . 7 7 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 139, + 306, + 213, + 319 + ], + "score": 1.0, + "content": "on top-1 accuracy", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 213, + 306, + 245, + 317 + ], + "score": 0.87, + "content": "9 0 . 8 8 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 245, + 306, + 441, + 319 + ], + "score": 1.0, + "content": ", achieving the new state-of-the-art performance.", + "type": "text", + "cross_page": true + } + ], + "index": 9 + } + ], + "index": 19.5, + "bbox_fs": [ + 106, + 699, + 505, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 124, + 511, + 284 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 67, + 506, + 123 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 66, + 506, + 79 + ], + "spans": [ + { + "bbox": [ + 106, + 66, + 506, + 79 + ], + "score": 1.0, + "content": "Table 5: Performance Comparison on large-scale JFT dataset. 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See Appendix A.2", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 99, + 506, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 506, + 113 + ], + "score": 1.0, + "content": "for the size details of CoAtNet-5/6/7. †: Down-sampling in the MBConv block is achieved by stride-2", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 110, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 106, + 110, + 505, + 124 + ], + "score": 1.0, + "content": "Depthwise Convolution. ⇧: ViT-G/14 computation consumption is read from Fig. 1 of the paper [26].", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "table_body", + "bbox": [ + 106, + 124, + 511, + 284 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 124, + 511, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 124, + 511, + 284 + ], + "score": 0.983, + "html": "
ModelsEval Size#Params#FLOPsTPUv3-core-daysTop-1 Accuracy
ResNet + ViT-L/163842330M=187.12
ViT-L/165122307M364B0.68K87.76
ViT-H/145182632M1021B2.5K88.55
NFNet-F4+5122527M367B1.86K89.2
CoAtNet-3t3842168M114B0.58K88.52
CoAtNet-3t5122168M214B0.58K88.81
CoAtNet-45122275M361B0.95K89.11
CoAtNet-55122688M812B1.82K89.77
ViT-G/1451821.84B5160B>30K90.45
CoAtNet-651221.47B1521B6.6K90.45
CoAtNet-751222.44B2586B20.1K90.88
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Specifically, we compare two models, one with the relative attention", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "and the other without, under both the ImageNet-1K alone and ImageNet-21K transfer setting. As we", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 400, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 505, + 412 + ], + "score": 1.0, + "content": "can see from Table 6, when only the ImageNet-1K is used, relative attention clearly outperforms the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 411, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 506, + 423 + ], + "score": 1.0, + "content": "standard attention, indicating a better generalization. In addition, under the ImageNet-21K transfer", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 421, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 506, + 434 + ], + "score": 1.0, + "content": "setting, the relative attention variant achieves a substantially better transfer accuracy, despite their", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 433, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 444 + ], + "score": 1.0, + "content": "very close pre-training performances. This suggests the main advantage of relative attention in visual", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 444, + 362, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 362, + 456 + ], + "score": 1.0, + "content": "processing is not in higher capacity but in better generalization.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5 + }, + { + "type": "table", + "bbox": [ + 118, + 473, + 490, + 548 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 228, + 462, + 383, + 473 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 226, + 460, + 384, + 474 + ], + "spans": [ + { + "bbox": [ + 226, + 460, + 384, + 474 + ], + "score": 1.0, + "content": "Table 6: Ablation on relative attention.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "table_body", + "bbox": [ + 118, + 473, + 490, + 548 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 118, + 473, + 490, + 548 + ], + "spans": [ + { + "bbox": [ + 118, + 473, + 490, + 548 + ], + "score": 0.981, + "html": "
SetingMetricWith Rel-AttnWithout Rel-Attn
ImageNet-1KAccuracy (2242)84.183.8
Accuracy (3842)85.785.3
ImageNet-21K →ImageNet-1KPre-train Precision@1 (224²)53.052.8
Finetune Accuracy (384²)87.987.4
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SettingModelsLayoutTop-1 Accuracy
ImageNet-1KVO: CoAtNet-2[2,2,6,14，2]84.1
V1: S2← S3[2,2, 2,18,2]83.4
V2: S2→ S3[2,2,8,12,2]84.0
ImageNet-21KVO: CoAtNet-3[2,2,6,14,2]53.0 -→87.6
⇒ImageNet-1KV1: S2 ← S3[2,2,2,18,2]53.0 -→87.4
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ModelsEval Size#Params#FLOPsTPUv3-core-daysTop-1 Accuracy
ResNet + ViT-L/163842330M=187.12
ViT-L/165122307M364B0.68K87.76
ViT-H/145182632M1021B2.5K88.55
NFNet-F4+5122527M367B1.86K89.2
CoAtNet-3t3842168M114B0.58K88.52
CoAtNet-3t5122168M214B0.58K88.81
CoAtNet-45122275M361B0.95K89.11
CoAtNet-55122688M812B1.82K89.77
ViT-G/1451821.84B5160B>30K90.45
CoAtNet-651221.47B1521B6.6K90.45
CoAtNet-751222.44B2586B20.1K90.88
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SetingMetricWith Rel-AttnWithout Rel-Attn
ImageNet-1KAccuracy (2242)84.183.8
Accuracy (3842)85.785.3
ImageNet-21K →ImageNet-1KPre-train Precision@1 (224²)53.052.8
Finetune Accuracy (384²)87.987.4
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SettingModelsLayoutTop-1 Accuracy
ImageNet-1KVO: CoAtNet-2[2,2,6,14，2]84.1
V1: S2← S3[2,2, 2,18,2]83.4
V2: S2→ S3[2,2,8,12,2]84.0
ImageNet-21KVO: CoAtNet-3[2,2,6,14,2]53.0 -→87.6
⇒ImageNet-1KV1: S2 ← S3[2,2,2,18,2]53.0 -→87.4
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SettingModelsImage SizeTop-1 Accuracy
ImageNet-1KCoAtNet-2224284.1
Head size: 32 → 64224283.9
Norm type: 1 BN →LN224284.1
ImageNet-21K ⇒ ImageNet-1KCoAtNet-3384287.9
Norm type: BN →→ LN384²87.8
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Basically, having more Transformer blocks in", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 114, + 207, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 114, + 207, + 506, + 219 + ], + "score": 1.0, + "content": "S3 generally leads to better performance until the number of MBConv blocks in S2 is too small to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 114, + 219, + 181, + 229 + ], + "spans": [ + { + "bbox": [ + 114, + 219, + 181, + 229 + ], + "score": 1.0, + "content": "generalize well.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 106, + 231, + 505, + 296 + ], + "lines": [ + { + "bbox": [ + 105, + 230, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 243 + ], + "score": 1.0, + "content": "• To further evaluate whether the sweet spot also holds in the transfer setting, where a higher", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 114, + 241, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 114, + 241, + 506, + 254 + ], + "score": 1.0, + "content": "capacity is often regarded more important, we further compare V0 and V1 under the ImageNet-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 114, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 114, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "21K transferring to ImageNet-1K setup. Interestingly, despite that V1 and V0 have the same", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 113, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 113, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "performance during ImageNet-21K pre-training, the transfer accuracy of V1 clearly falls behind", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 113, + 273, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 113, + 273, + 506, + 288 + ], + "score": 1.0, + "content": "V0. Again, this suggests the importance of convolution in achieving good transferability and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 113, + 286, + 177, + 297 + ], + "spans": [ + { + "bbox": [ + 113, + 286, + 177, + 297 + ], + "score": 1.0, + "content": "generalization.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 505, + 367 + ], + "lines": [ + { + "bbox": [ + 106, + 302, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 313 + ], + "score": 1.0, + "content": "Lastly, we study two choices of model details, namely the dimension of each attention (default to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "32) head as well as the type of normalization (default to BatchNorm) used in MBConv blocks. From", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "score": 1.0, + "content": "Table 8, we can see increasing head size from 32 to 64 can slightly hurt performance, though it", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "actually improves the TPU speed by a significant amount. In practice, this will be a quality-speed", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 344, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 505, + 358 + ], + "score": 1.0, + "content": "trade-off one can make. On the other hand, BatchNorm and LayerNorm have almost the same", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 355, + 493, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 245, + 369 + ], + "score": 1.0, + "content": "performance, while BatchNorm is", + "type": "text" + }, + { + "bbox": [ + 245, + 356, + 283, + 367 + ], + "score": 0.88, + "content": "10 - 2 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 355, + 493, + 369 + ], + "score": 1.0, + "content": "faster on TPU depending on the per-core batch size.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16.5 + }, + { + "type": "title", + "bbox": [ + 107, + 383, + 183, + 396 + ], + "lines": [ + { + "bbox": [ + 104, + 380, + 185, + 399 + ], + "spans": [ + { + "bbox": [ + 104, + 380, + 185, + 399 + ], + "score": 1.0, + "content": "5 Conclusion", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 407, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 407, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 420 + ], + "score": 1.0, + "content": "In this paper, we systematically study the properties of convolutions and Transformers, which leads", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 418, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 431 + ], + "score": 1.0, + "content": "to a principled way to combine them into a new family of models named CoAtNet. Extensive", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "experiments show that CoAtNet enjoys both good generalization like ConvNets and superior model", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "capacity like Transformers, achieving state-of-the-art performances under different data sizes and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 452, + 195, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 195, + 465 + ], + "score": 1.0, + "content": "computation budgets.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 468, + 504, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 481 + ], + "score": 1.0, + "content": "Note that this paper currently focuses on ImageNet classification for model development. However,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "we believe our approach is applicable to broader applications like object detection and semantic", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 490, + 310, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 310, + 501 + ], + "score": 1.0, + "content": "segmentation. We will leave them for future work.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 742, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "spans": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 138, + 79, + 470, + 165 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 195, + 68, + 415, + 78 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 194, + 65, + 416, + 81 + ], + "spans": [ + { + "bbox": [ + 194, + 65, + 416, + 81 + ], + "score": 1.0, + "content": "Table 8: Ablation on head size and normalization type.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 138, + 79, + 470, + 165 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 138, + 79, + 470, + 165 + ], + "spans": [ + { + "bbox": [ + 138, + 79, + 470, + 165 + ], + "score": 0.983, + "html": "
SettingModelsImage SizeTop-1 Accuracy
ImageNet-1KCoAtNet-2224284.1
Head size: 32 → 64224283.9
Norm type: 1 BN →LN224284.1
ImageNet-21K ⇒ ImageNet-1KCoAtNet-3384287.9
Norm type: BN →→ LN384²87.8
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Extensive", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "experiments show that CoAtNet enjoys both good generalization like ConvNets and superior model", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "capacity like Transformers, achieving state-of-the-art performances under different data sizes and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 452, + 195, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 195, + 465 + ], + "score": 1.0, + "content": "computation budgets.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 407, + 506, + 465 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 468, + 504, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 481 + ], + "score": 1.0, + "content": "Note that this paper currently focuses on ImageNet classification for model development. However,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "we believe our approach is applicable to broader applications like object detection and semantic", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 490, + 310, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 310, + 501 + ], + "score": 1.0, + "content": "segmentation. 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PropertiesConvolutionSelf-Attention
Translation Equivariance√
Input-adaptive Weighting√
Global Receptive Field
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MetricC-C-T-TC-T-T-T
Pre-training Precision@1 (JFT)34.4034.36
Transfer Accuracy 224x22482.3981.78
Transfer Accuracy 384x38484.2384.02
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ModelsEval Size#Params#FLOPsImageNet Top-1 Accuracy
Conv Only1K only21K+1K
EfficientNet-B7600²66M37B84.7-
EfficientNetV2-L480²121M53B85.786.8
NFNet-F34162²255M114.8B85.7=
NFNet-F55442377M289.8B86.01
ViT-Stem TFMDeiT-B384286M55.4B83.1-
ViT-L/16384²304M190.7B-85.3
CaiT-S-36384²68M48.0B85.0
DeepViT-L224²55M12.5B83.1-
Multi-stage TFMSwin-B384²88M47.0B84.286.0
Swin-L384²197M103.9B-86.4
Conv+TFMBotNet-T7384275.1M45.8B84.7-
LambdaResNet-420320²-=84.8
T2T-ViT-24224²64.1M15.0B82.6=
CvT-21384²32M24.9B83.3-
CvT-W243842277M193.2B-87.7
Conv+TFM (ours)CoAtNet-0 CoAtNet-1224²25M4.2B81.6=
224²42M8.4B83.3-
CoAtNet-2 CoAtNet-3224²75M15.7B84.187.1
2242168M34.7B84.587.6
CoAtNet-0384²25M13.4B83.9-
CoAtNet-1384242M27.4B85.1-
CoAtNet-2384²75M49.8B85.787.1
CoAtNet-3384²168M107.4B85.887.6
CoAtNet-4384²275M189.5B-87.9
+ PT-RA384²275M189.5B88.3
+ PT-RA-E1503842275M189.5B88.4
CoAtNet-2512275M96.7B85.987.3
CoAtNet-3512²168M203.1B86.087.9
CoAtNet-4512²275M360.9B-88.1
+ PT-RA512²275M360.9B=88.4
+ PT-RA-E1505122275M360.9B=88.56
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ModelsEval Size#Params#FLOPsTPUv3-core-daysTop-1 Accuracy
ResNet + ViT-L/163842330M=187.12
ViT-L/165122307M364B0.68K87.76
ViT-H/145182632M1021B2.5K88.55
NFNet-F4+5122527M367B1.86K89.2
CoAtNet-3t3842168M114B0.58K88.52
CoAtNet-3t5122168M214B0.58K88.81
CoAtNet-45122275M361B0.95K89.11
CoAtNet-55122688M812B1.82K89.77
ViT-G/1451821.84B5160B>30K90.45
CoAtNet-651221.47B1521B6.6K90.45
CoAtNet-751222.44B2586B20.1K90.88
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SetingMetricWith Rel-AttnWithout Rel-Attn
ImageNet-1KAccuracy (2242)84.183.8
Accuracy (3842)85.785.3
ImageNet-21K →ImageNet-1KPre-train Precision@1 (224²)53.052.8
Finetune Accuracy (384²)87.987.4
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SettingModelsLayoutTop-1 Accuracy
ImageNet-1KVO: CoAtNet-2[2,2,6,14，2]84.1
V1: S2← S3[2,2, 2,18,2]83.4
V2: S2→ S3[2,2,8,12,2]84.0
ImageNet-21KVO: CoAtNet-3[2,2,6,14,2]53.0 -→87.6
⇒ImageNet-1KV1: S2 ← S3[2,2,2,18,2]53.0 -→87.4
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SettingModelsImage SizeTop-1 Accuracy
ImageNet-1KCoAtNet-2224284.1
Head size: 32 → 64224283.9
Norm type: 1 BN →LN224284.1
ImageNet-21K ⇒ ImageNet-1KCoAtNet-3384287.9
Norm type: BN →→ LN384²87.8
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b/parse/train/LyjH88yV7F/LyjH88yV7F.md @@ -0,0 +1,215 @@ +# Learning Graph Models for Retrosynthesis Prediction + +Vignesh Ram Somnath1 + +Charlotte Bunne1 + +Connor W. Coley2 + +Andreas Krause1 Regina Barzilay3 + +1Department of Computer Science, ETH 2Department of Chemical Engineering, MIT 3Computer Science and Artificial Intelligence Lab, MIT 1{vsomnath, bunnec, krausea}@ethz.ch, 2ccoley@mit.edu, 3regina@csail.mit.edu + +# Abstract + +Retrosynthesis prediction is a fundamental problem in organic synthesis, where the task is to identify precursor molecules that can be used to synthesize a target molecule. A key consideration in building neural models for this task is aligning model design with strategies adopted by chemists. Building on this viewpoint, this paper introduces a graph-based approach that capitalizes on the idea that the graph topology of precursor molecules is largely unaltered during a chemical reaction. The model first predicts the set of graph edits transforming the target into incomplete molecules called synthons. Next, the model learns to expand synthons into complete molecules by attaching relevant leaving groups. This decomposition simplifies the architecture, making its predictions more interpretable, and also amenable to manual correction. Our model achieves a top-1 accuracy of $5 3 . 7 \%$ , outperforming previous template-free and semi-template-based methods. + +# 1 Introduction + +Retrosynthesis prediction, first formalized by E. J. Corey [Corey, 1991] is a fundamental problem in organic synthesis that attempts to identify a series of chemical transformations for synthesizing a target molecule. In the single-step formulation, the task is to identify a set of reactant molecules given a target. Beyond simple reactions, many practical tasks involving complex organic molecules are difficult even for expert chemists. As a result, substantial experimental exploration is needed to cover for deficiencies of analytical approaches. This has motivated interest in computer-assisted retrosynthesis [Corey and Wipke, 1969], with a recent surge in machine learning methods [Chen et al., 2019, Coley et al., 2017b, Dai et al., 2019, Zheng et al., 2019, Genheden et al., 2020]. + +Computationally, the main challenge is how to explore the combinatorial space of reactions that can yield the target molecule. Largely, previous methods for retrosynthesis prediction can be divided into template-based [Coley et al., 2017b, Dai et al., 2019, Segler and Waller, 2017] and template-free [Chen et al., 2019, Zheng et al., 2019] approaches. Template-based methods match a target molecule against a large set of templates, which are molecular subgraph patterns that highlight changes during a chemical reaction. Despite their interpretability, these methods fail to generalize to new reactions. Template-free methods bypass templates by learning a direct mapping from the SMILES [Weininger, 1988] representations of the product to reactants. Despite their greater generalization potential, these methods generate reactant SMILES character by character, increasing generation complexity. + +![](images/e988f2e4a9f13ae6fa41934806006c24c87183942b7fd7f824274aee3a6237f9.jpg) +Figure 1: Overview of Our Approach. a. Edit Prediction. We train a model to learn a distribution over possible graph edits. In this case, the correct edit corresponds to breaking the bond marked in red. Applying this edit produces two synthons. b. Synthon Completion. Another model is trained to pick candidate leaving groups (blue) for each synthon from a discrete vocabulary, which are then attached to produce the final reactants. + +Another important consideration in building retrosynthesis models is aligning model design with strategies adopted by expert chemists. These strategies are influenced by fundamental properties of chemical reactions, independent of complexity level: (i.) the product atoms are always a subset of the reactant atoms1, and (ii.) the molecular graph topology is largely unaltered from products to reactants. For example, in the standard retrosynthesis dataset, only $6 . 3 \%$ of the atoms in the product undergo any change in connectivity. + +This consideration has received more attention in recent semi-template-based methods [Shi et al., 2020, Yan et al., 2020], that generate reactants from a product in two stages: (i.) first identify intermediate molecules called synthons, (ii.) and then complete synthons into reactants by sequential generation of atoms or SMILES characters.. Our model GRAPHRETRO also uses a similar workflow. However, we avoid sequential generation for completing synthons by instead selecting subgraphs called leaving groups from a precomputed vocabulary. This vocabulary is constructed during preprocessing by extracting subgraphs that differ between a synthon and the corresponding reactant. The vocabulary has a small size (170 for USPTO-50k) indicating remarkable redundancy, while covering $9 9 . 7 \%$ of the test set. Operating at the level of these subgraphs greatly reduces the complexity of reactant generation, with improved empirical performance. This formulation also simplifies our architecture, and makes our predictions more transparent, interpretable and amenable to manual correction. + +The benchmark dataset for evaluating retrosynthesis models is USPTO-50k [Schneider et al., 2016], which consists of 50000 reactions across 10 reaction classes. The dataset contains an unexpected shortcut towards predicting the edit, in that the product atom with atom-mapping 1 is part of the edit in $7 5 \%$ of the cases, allowing predictions that depend on the position of the atom to overestimate performance. We canonicalize the product SMILES and remap the existing dataset, thereby removing the shortcut. On this remapped dataset, GRAPHRETRO achieves a top-1 accuracy of $5 3 . 7 \%$ when the reaction class is not known, outperforming both template-free and semi-template-based methods. + +# 2 Related Work + +Retrosynthesis Prediction Existing machine learning methods for retrosynthesis prediction can be divided into template-based, template-free and recent semi-template-based approaches. + +Template-Based: Templates are either hand-crafted by experts [Hartenfeller et al., 2011, Szymkuc´ et al., 2016], or extracted algorithmically from large databases Coley et al. [2017a], Law et al. [2009]. Exhaustively applying large template sets is expensive due to the involved subgraph matching procedure. Template-based methods therefore utilize different ways of prioritizing templates, by either learning a conditional distribution over the template set [Segler and Waller, 2017], ranking templates based on molecular similarities to precedent reactions [Coley et al., 2017b] or directly modelling the joint distribution of templates and reactants using logic variables [Dai et al., 2019]. Despite their interpretability, these methods fail to generalize outside their rule set. + +Template-Free: Template-free methods [Liu et al., 2017, Zheng et al., 2019, Chen et al., 2019] learn a direct transformation from products to reactants using architectures from neural machine translation and a string based representation of molecules called SMILES [Weininger, 1988]. Linearizing molecules as strings does not utilize the inherently rich chemical structure. In addition, the reactant SMILES are generated from scratch, character by character. Attempts have been made to improve validity by adding a syntax correcter [Zheng et al., 2019] and a mixture model to improve diversity of suggestions [Chen et al., 2019], but the performance remains worse than [Dai et al., 2019] on the standard retrosynthesis dataset. Sun et al. [2021] formulate retrosynthesis using energy-based models, with additional parameterizations and loss terms to enforce the duality between forward (reaction prediction) and backward (retrosynthesis) prediction. + +Semi-Template-Based: Our work is closely related to recently proposed semi-template-based methods [Shi et al., 2020, Yan et al., 2020], which first identify synthons and then expand synthons into reactants through sequential generation using either a graph generative model [Shi et al., 2020] or a Transformer [Yan et al., 2020]. To reduce the complexity of reactant generation, we instead complete synthons using subgraphs called leaving groups selected from a precomputed vocabulary. This allows us to view synthon completion as a classification problem instead of a generative one. We also utilize the dependency graph between possible edits, and update edit predictions using a message passing network (MPN) [Gilmer et al., 2017] on this graph. Both innovations together yield a $4 . 8 \%$ and $3 . 3 \%$ performance improvement respectively over previous semi-template-based methods. + +Reaction Center Identification The reaction center covers a small number of participating atoms involved in the reaction. Our work is also related to models that predict reaction outcomes by learning to rank atom pairs based on their likelihood to be in the reaction center [Coley et al., 2019, Jin et al., 2017]. The task of identifying the reaction center is related to the step of deriving the synthons in our formulation. Our work departs from [Coley et al., 2019, Jin et al., 2017] as we utilize the property that new bond formations occur rarely $( \sim 0 . 1 \% )$ from products to synthons, allowing us to predict a score only for existing bonds and atoms and reduce prediction complexity from $O ( N ^ { \bar { 2 } } )$ to $O ( N )$ . We also utilize the dependency graph between possible edits, and update edit predictions using a MPN on this graph. + +Utilizing Substructures Substructures have been utilized in various tasks from sentence generation by fusing phrases to molecule generation and optimization [Jin et al., 2018, 2020]. Our work is closely related to [Jin et al., 2020] which uses precomputed substructures as building blocks for property-conditioned molecule generation. However, instead of precomputing, synthons —analogous building blocks for reactants— are indirectly learnt during training. + +# 3 Model Design + +Our approach leverages the property that graph topology is largely unaltered from products to reactants. To achieve this, we first derive suitable building blocks from the product called synthons, and then complete them into valid reactants by adding specific functionalities called leaving groups. These derivations, called edits, are characterized by modifications to bonds or hydrogen counts on atoms. We first train a neural network to predict a score for possible edits (Section 3.1). The edit with the highest score is then applied to the product to obtain synthons. Since the number of unique leaving groups are small, we model leaving group selection as a classification problem over a precomputed vocabulary (Section 3.2). To produce candidate reactants, we attach the predicted leaving group to the corresponding synthon through chemically constrained rules. The overall process is outlined in Figure 1. Before describing the two modules, we introduce relevant preliminaries that set the background for the remainder of the paper. + +Retrosynthesis Prediction A retrosynthesis pair $R$ is described by a pair of molecular graphs $( \mathcal { G } _ { p } , \mathcal { G } _ { r } )$ , where $\mathcal { G } _ { p }$ are the products and $\mathcal { G } _ { r }$ the reactants. A molecular graph is described as $\mathcal { G } = \mathbf { \bar { \rho } } ( \mathcal { V } , \mathcal { E } )$ with atoms $\nu$ as nodes and bonds $\mathcal { E }$ as edges. Prior work has focused on the single product case, while reactants can have multiple connected components, i.e. $\mathcal { G } _ { r } = \{ \mathcal { G } _ { r _ { c } } \} _ { c = 1 } ^ { C }$ . Retrosynthesis pairs are atom-mapped so that each product atom has a unique corresponding reactant atom. The retrosynthesis task then, is to infer $\{ \mathcal { G } _ { r _ { c } } \} _ { c = 1 } ^ { C }$ given $\mathcal { G } _ { p }$ . + +Edits Edits consist of (i.) atom pairs $\left\{ \left( a _ { i } , a _ { j } \right) \right\}$ where the bond type changes from products to reactants, and (ii.) atoms $\left\{ { a } _ { i } \right\}$ where the number of hydrogens attached to the atom change from products to reactants. We denote the set of edits by $E$ . Since retrosynthesis pairs in the training set are atom-mapped, edits can be automatically identified by comparing the atoms and atom pairs in the product to their corresponding reactant counterparts. + +Synthons and Leaving Groups Applying edits $E$ to the product $\mathcal { G } _ { p }$ results in incomplete molecules called synthons. Synthons are analogous to rationales or building blocks, which are expanded into valid reactants by adding specific functionalities called leaving groups that are responsible for its reactivity. We denote synthons by $\mathcal { G } _ { s }$ and leaving groups by $\mathcal { G } _ { l }$ . We further assume that synthons and leaving groups have the same number of connected components as the reactants, i.e $\mathcal { G } _ { s } \doteq \{ \mathcal { G } _ { s _ { c } } \} _ { c = 1 } ^ { C }$ and $\mathcal { G } _ { l } = \{ \mathcal { G } _ { l _ { c } } ^ { \star } \} _ { c = 1 } ^ { C }$ . This assumption holds for $9 9 . 9 7 \%$ reactions in the training set. + +Formally, our model generates reactants by first predicting the set of edits $E$ that transform $\mathcal { G } _ { p }$ into $\mathcal { G } _ { s }$ , followed by predicting a leaving group $\mathcal { G } _ { l _ { c } }$ to attach to each synthon $\mathcal { G } _ { s _ { c } }$ . The model is defined as + +$$ +P ( \mathcal G _ { r } | \mathcal G _ { p } ) = \sum _ { E , \mathcal G _ { l } } P ( E | \mathcal G _ { p } ) P ( \mathcal G _ { l } | \mathcal G _ { p } , \mathcal G _ { s } ) , +$$ + +where $\mathcal { G } _ { s } , \mathcal { G } _ { r }$ are deterministic given $E , { \mathcal { G } } _ { l }$ , and $\mathcal { G } _ { p }$ . + +# 3.1 Edit Prediction + +For a given retrosynthesis pair $R = ( \mathcal G _ { p } , \mathcal G _ { r } )$ , we predict an edit score only for existing bonds and atoms, instead of every atom pair as in [Coley et al., 2019, Jin et al., 2017]. This choice is motivated by the low frequency $( \sim 0 . 1 \% )$ of new bond formations in the training set examples. Coupled with the sparsity of molecular graphs, this reduces the prediction complexity from $O ( N ^ { 2 } )$ to $O ( N )$ for a product with $N$ atoms. Our edit prediction model has variants tailored to single and multiple edit prediction. Since $9 5 \%$ of the training set consists of single edit examples, the remainder of this section describes the setup for single edit prediction. A detailed description of our multiple edit prediction model can be found in Appendix ??. + +Each bond $( u , v )$ in $\mathcal { G } _ { p }$ is associated with a label $y _ { u v k } \in \{ 0 , 1 \}$ indicating whether its bond type $k$ has changed from the products to reactants. Each atom $u$ is associated with a label $y _ { u } \in \{ 0 , \bar { 1 } \}$ indicating a change in hydrogen count. We predict edit scores using representations that are learnt using a graph encoder. + +Graph Encoder To obtain atom representations, we use a variant of the message passing network (MPN) described in [Gilmer et al., 2017]. Each atom $u$ has a feature vector $\mathbf { x } _ { u }$ indicating its atom type, degree and other properties. Each bond $( u , v )$ has a feature vector $\mathbf { x } _ { u v }$ indicating its aromaticity, bond type and ring membership. For simplicity, we denote the encoding process by $\mathrm { M P N } ( \cdot )$ and describe architectural details in Appendix ??. The MPN computes atom representations $\{ \mathbf { c } _ { u } | u \in \mathcal { G } \}$ via + +$$ +\{ \mathbf { c } _ { u } \} = \mathrm { M P N } ( \mathcal { G } , \{ \mathbf { x } _ { u } \} , \{ \mathbf { x } _ { u v } \} _ { v \in \mathcal { N } ( u ) } ) , +$$ + +where $\mathcal { N } ( u )$ denotes the neighbors of atom $u$ . The graph representation $\mathbf { c } _ { \mathcal { G } }$ is an aggregation of atom representations, i.e. $\mathbf { c } _ { \mathcal { G } } = \dot { \sum _ { { u } \in \mathcal { V } } } \mathbf { c } _ { u }$ . When $\mathcal { G }$ has connected components $\left\{ { \mathcal { G } } _ { i } \right\}$ , we get a set of graph representations $\left\{ \mathbf { c } _ { \mathcal { G } _ { i } } \right\}$ . For a bond $( u , v )$ , we define its representation $\mathbf { c } _ { u v } = ( \operatorname { A B S } ( \mathbf { c } _ { u } , \mathbf { c } _ { v } ) | | \mathbf { c } _ { u } + \mathbf { c } _ { v } )$ , where ABS denotes absolute difference and $| |$ refers to concatenation. This ensures our representations are permutation invariant. These representations are then used to predict atom and bond edit scores using corresponding neural networks, + +$$ +\begin{array} { r } { \boldsymbol { s } _ { u } = \mathbf { u _ { a } } ^ { T } \boldsymbol { \tau } ( \mathbf { W _ { a } } \mathbf { c } _ { u } + b ) \quad } \\ { \boldsymbol { s } _ { u v k } = \mathbf { u _ { k } } ^ { T } \boldsymbol { \tau } ( \mathbf { W _ { k } } \mathbf { c } _ { u v } + b _ { k } ) , } \end{array} +$$ + +where $\tau ( \cdot )$ is the ReLU activation function. + +Updating Bond Edit Scores Unlike a typical classification problem where the labels are independent, edits can have possible dependencies between each other. For example, bonds part of a stable system such as an aromatic ring have a greater tendency to remain unchanged (label 0). We attempt to leverage such dependencies to update initial edit scores. To this end, we build a graph with bonds $( u , v )$ as nodes, and introduce an edge between bonds sharing an atom. We use another $\mathrm { M P N } ( \cdot )$ on this graph to learn aggregated neighborhood messages $\mathbf { m } _ { u v }$ , and update the edit scores $s _ { u v k }$ in a manner similar to how LSTMs update representations, + +$$ +\begin{array} { r l } & { f _ { u v k } = \sigma ( \mathbf { W _ { k x } ^ { f } } \mathbf { x } _ { u v } + \mathbf { W _ { k m } ^ { f } } \mathbf { m } _ { u v } ) } \\ & { i _ { u v k } = \sigma ( \mathbf { W _ { k x } ^ { i } } \mathbf { x } _ { u v } + \mathbf { W _ { k m } ^ { i } } \mathbf { m } _ { u v } ) } \\ & { \tilde { m } _ { u v k } = \mathbf { u _ { m } } \tau ( \mathbf { W _ { k x } ^ { m } } \mathbf { x } _ { u v } + \mathbf { W _ { k m } ^ { m } } \mathbf { m } _ { u v } ) } \\ & { \tilde { s } _ { u v k } = f _ { u v k } \cdot s _ { u v k } + i _ { u v k } \cdot \tilde { m } _ { u v k } . } \end{array} +$$ + +Training We train by minimizing the cross-entropy loss over possible bond and atom edits + +$$ +\mathcal { L } _ { e } = - \sum _ { ( \mathcal { G } _ { p } , E ) } \left( \sum _ { ( ( u , v ) , k ) \in E } y _ { u v k } \mathrm { l o g } ( \widetilde s _ { u v k } ) + \sum _ { u \in E } y _ { u } \mathrm { l o g } ( s _ { u } ) \right) . +$$ + +The cross-entropy loss enforces the model to learn a distribution over possible edits instead of reasoning about each edit independently, as with the binary cross entropy loss used in [Jin et al., 2017, Coley et al., 2019]. + +# 3.2 Synthon Completion + +Synthons are completed into valid reactants by adding specific functionalities called leaving groups. This involves two complementary tasks: (i.) selecting the appropriate leaving group, and (ii.) attaching the leaving group to the synthon. As ground truth leaving groups are not directly provided, we extract the leaving groups and construct a vocabulary $\mathcal { X }$ of unique leaving groups during preprocessing. + +The vocabulary has a limited size ( $| \mathcal { X } | = 1 7 0$ for a standard dataset with 50, 000 examples, and 72000 synthons) indicating the redundancy of leaving groups used in accomplishing retrosynthetic transformations. This redundancy also allows us to formulate leaving group selection as a classification problem over $\mathcal { X }$ , while retaining the ability to generate diverse reactants using different combinations of leaving groups. + +Vocabulary Construction Before constructing the vocabulary, we align connected components of synthon and reactant graphs by comparing atom mapping overlaps. Using aligned pairs $\mathcal { G } _ { s _ { c } } =$ $( \gamma _ { s _ { c } } , \mathcal { E } _ { s _ { c } } )$ and $\mathcal { G } _ { r _ { c } } = ( \nu _ { r _ { c } } , \mathcal { E } _ { r _ { c } } )$ as input, the leaving group vocabulary $\mathcal { X }$ is constructed by extracting subgraphs $\mathcal { G } _ { l _ { c } } = ( \nu _ { l _ { c } } , \mathcal { E } _ { l _ { c } } )$ such that $\smash { \gamma _ { l _ { c } } = \gamma _ { r _ { c } } \setminus \gamma _ { s _ { c } } }$ . Atoms $\left\{ { a } _ { i } \right\}$ in the leaving groups that attach to synthons are marked with a special symbol. We also add three tokens to $\mathcal { X }$ namely START, which indicates the start of synthon completion, END, which indicates that there is no leaving group to add and PAD, which is used to handle variable numbers of synthon components in a minibatch. + +Leaving Group Selection For synthon component $c \leq C$ , where $C$ is the number of connected components in the synthon graph, we use three inputs for leaving group selection – the product representation $\mathbf { c } _ { \mathcal { G } _ { p } }$ , the synthon component representation $\mathbf { c } _ { \mathcal { G } _ { s _ { c } } }$ , and the leaving group representation for the previous synthon component, ${ \bf e } _ { l _ { c - 1 } }$ . The product and synthon representations are learnt using the $\mathrm { M P N } ( \cdot )$ . For each $x _ { i } \in { \mathcal { X } }$ , representations can be learnt by either training independent embedding vectors (ind) or by treating each $x _ { i }$ as a subgraph and using the $\mathrm { M P N } ( \cdot )$ (shared). In the shared setting, we use the same $\mathrm { M P N } ( \cdot )$ as the product and synthons. + +The leaving group probabilities are then computed by combining $\mathbf { c } _ { \mathcal { G } _ { p } } , \mathbf { c } _ { \mathcal { G } _ { s _ { c } } }$ and $\mathbf { e } _ { l _ { c - 1 } }$ via a single layer neural network and softmax function + +$$ +\hat { q } _ { l _ { c } } = \mathrm { s o f t m a x } \left( \mathbf { U } \tau \left( \mathbf { W } _ { 1 } \mathbf { c } _ { \mathcal { G } _ { p } } + \mathbf { W } _ { 2 } \mathbf { c } _ { \mathcal { G } _ { s _ { c } } } + \mathbf { W } _ { 3 } \mathbf { e } _ { l _ { \left( c - 1 \right) } } \right) \right) , +$$ + +where $\hat { q } _ { l _ { c } }$ is distribution learnt over $\mathcal { X }$ . Using the representation of the previous leaving group ${ \bf e } _ { l _ { c - 1 } }$ allows the model to understand combinations of leaving groups that generate the desired product from the reactants. We also include the product representation $\mathbf { c } _ { \mathcal { G } _ { p } }$ as the synthon graphs are derived from the product graph. + +Training For step $c$ , given the one hot encoding of the true leaving group $q _ { l _ { c } }$ , we minimize the cross-entropy loss + +$$ +\mathcal { L } _ { s } = \sum _ { c = 1 } ^ { C } \mathcal { L } ( \hat { q } _ { l _ { c } } , q _ { l _ { c } } ) . +$$ + +Training utilizes teacher-forcing [Williams and Zipser, 1989] so that the model makes predictions given correct histories. During inference, at every step, we use the representation of leaving group from the previous step with the highest predicted probability. + +Leaving Group Attachment Attaching leaving groups to synthons is a deterministic process and not learnt during training. The task involves identification of the type of bonds to add between attaching atoms in the leaving group (marked during vocabulary construction), and the atom(s) participating in the edit. These bonds can be inferred by applying the valency constraint, which determines the maximum number of neighbors for each atom. The attachment process does not modify any stereochemistry. Given synthons and leaving groups, the attachment process has a $100 \%$ accuracy. The detailed procedure is described in Appendix ??. + +# 3.3 Inference + +Inference is performed using beam search with a log-likelihood scoring function. For a beam width $n$ , we select $n$ edits with highest scores and apply them to the product to obtain $n$ synthons, where each synthon can consist of multiple connected components. The synthons form the nodes for beam search. Each node maintains a cumulative score by aggregating the log-likelihoods of the edit and predicted leaving groups. Leaving group inference starts with a connected component for each synthon, and selects $n$ leaving groups with highest log-likelihoods. From the $n ^ { 2 }$ possibilities, we select $n$ nodes with the highest cumulative scores. This process is repeated until all nodes have a leaving group predicted for each synthon component. + +# 4 Evaluation + +Evaluating retrosynthesis models is challenging as multiple sets of reactants can be generated from the same product. To deal with this, previous works [Coley et al., 2017b, Dai et al., 2019] evaluate the ability of the model to recover retrosynthetic strategies recorded in the dataset. + +Data We use the benchmark dataset USPTO-50k [Schneider et al., 2016] for all our experiments. The dataset contains 50, 000 atom-mapped reactions across 10 reaction classes. We use the same dataset version and splits as provided by [Dai et al., 2019]. The USPTO-50k dataset contains a shortcut in that the product atom with atom-mapping 1 is part of the edit in $\sim 7 5 \%$ of the cases. If the product SMILES is not canonicalized, predictions utilizing operations that depend on the position of the atom or bond will be able to use the shortcut, and overestimate performance. We canonicalize the product SMILES, and reassign atom-mappings to the reactant atoms based on the canonical ordering, which removes the shortcut. Details on the remapping procedure can be found in Appendix ??. + +Evaluation We use the top- $\mathbf { \nabla } \cdot n$ accuracy $( n = 1 , 3 , 5 , 1 0 )$ ) as our evaluation metric, defined as the fraction of examples where the recorded reactants are suggested by the model with rank $\leq n$ . Following prior work [Coley et al., 2017b, Zheng et al., 2019, Dai et al., 2019], we compute the accuracy by comparing the canonical SMILES of predicted reactants to the ground truth. Atom-mapping is excluded from this comparison, but stereochemistry, which describes the relative orientation of atoms in the molecule, is retained. The evaluation is carried out for two settings, with the reaction class being known or unknown. + +Table 1: Top- $n$ exact match accuracy. Best values within each section are highlighted in bold. + +
ModelTop-n Accuracy (%)
Reaction class knownReaction class unknown
1351013510
Template-Based
RETROSIM [Coley et al.,2017b]52.973.881.288.137.354.763.374.1
NEURALSYM [Segler and Waller,2017]55.376.081.485.144.465.372.478.9
GLN [Dai et ai., 2019]64.279.185.290.052.569.075.683.7
DUALTB [Sun et al.,2021]67.784.888.992.055.274.680.586.9
Template-Free
SCROP [Zheng et al.,2019]59.074.878.181.143.760.065.268.7
LV-TRANSFORMER [Chen et al.,2019]----40.565.172.879.4
DUALTF [Sun et al., 2021]65.781.984.785.953.670.774.677.0
Semi-Template-Based
G2Gs [Shi et al.,2020]61.081.386.088.748.967.672.575.5
RETROXPERT [Yan et al.,2020]62.175.878.580.950.461.162.363.4
GRAPHRETRO (ours)63.981.585.288.153.768.372.275.5
+ +Baselines For evaluating overall performance, we compare GRAPHRETRO to nine baselines — four template-based, three template-free, and two semi-template-based methods. These include: + +Template-Based: RETROSIM Coley et al. [2017b] ranks templates for a given target molecule by computing molecular similarities to precedent reactions. NEURALSYM [Segler and Waller, 2017] trains a model to rank templates given a target molecule. GLN [Dai et al., 2019] models the joint distribution of templates and reactants in a hierarchical fashion using logic variables. DUALTB [Sun et al., 2021] uses an energy-based model formulation for retrosynthesis, with additional parameterizations and loss terms to enforce the duality between forward (reaction prediction) and backward (retrosynthesis prediction). Inference is carried out using reactant candidates obtained by applying an extracted template set to the products. + +Template-Free: SCROP [Zheng et al., 2019], LV-TRANSFORMER [Chen et al., 2019] and DUALTF [Sun et al., 2021] use the Transformer architecture [Vaswani et al., 2017] to output reactant SMILES given a product SMILES. To improve the validity of their suggestions, SCROP include a second Transformer that functions as a syntax correcter. LV-TRANSFORMER uses a latent variable mixture model to improve diversity of suggestions. DUALTF utilizes additional parameterizations and loss terms to enforce the duality between forward (reaction prediction) and backward (retrosynthesis prediction). + +Semi-Template-Based: G2GS [Shi et al., 2020] and RETROXPERT [Yan et al., 2020] first identify synthons, and then expand the synthons into reactants by either sequential generation of atoms and bonds (G2Gs), or using the Transformer architecture (RETROXPERT). The training dataset for the Transformer in [Yan et al., 2020] is augmented with incorrectly predicted synthons with the goal of learning a correction mechanism. + +Results for NEURALSYM are taken from [Dai et al., 2019]. The authors in [Yan et al., 2020] report their performance being affected by the dataset leakage2. Thus, we use the most recent results from their website on the canonicalized dataset. For remaining baselines, we directly use the values reported in their paper. For the synthon completion module, we use the ind configuration given its better empirical performance. + +# 4.1 Overall Performance + +Reaction class unknown As shown in Table 1, when the reaction class is unknown, GRAPHRETRO outperforms G2GS by $4 . 8 \%$ and and RETROXPERT by $3 . 3 \%$ in top-1 accuracy. Performance improvements are also seen for larger $n$ , except for $n = 5$ . Barring DUALTB, the top-1 accuracy is also better than other template-free and template-based methods. For larger $n$ , one reason for lower top- $n$ accuracies than most template-based methods is that templates already contain combinations of leaving group patterns. In contrast, our model learns to discover these during training. A second hypothesis to this end is that simply adding log-likelihood scores from edit prediction and synthon completion models may be suboptimal and bias the beam search in the direction of the more dominating term. We leave it to future work to investigate scoring functions that rank the attachment. + +Reaction class known When the reaction class is known, GRAPHRETRO outperforms G2GS and RETROXPERT by a margin of $3 \%$ and $2 \%$ respectively in top-1 accuracy. GRAPHRETRO also outperforms all the template-free methods in top- $^ n$ accuracy. for GRAPHRETRO are also better than most template-based and template-free methods. When the reaction class is known, RETROSIM and GLN restrict template sets corresponding to the reaction class, thus improving performance. The increased edit prediction performance (Section 4.2) for GRAPHRETRO helps outweigh this factor, achieving comparable or better performance till $n = 5$ . + +# 4.2 Individual Module Performance + +To gain more insight into the working of GRAPHRETRO, we evaluate the top- $\boldsymbol { n }$ accuracy $\mathbf { \nabla } _ { n } =$ $1 , 2 , 3 , 5 )$ of edit prediction and synthon completion modules, along with corresponding ablation studies, with results shown in Table 2. + +Edit Prediction For the edit prediction module, we compare the true edit(s) to top- $\boldsymbol { n }$ edits predicted by the model. We also consider two ablation studies, one where we directly use the initial edit scores without updating them, and the other where we predict edits using atom-pairs instead of existing bonds and atoms. Both design choices lead to improvements in performance, as shown in Table 2. We hypothesize that the larger improvement compared to edit prediction using atom-pairs is due to the easier optimization procedure, with lesser imbalance between labels 1 and 0. + +Synthon Completion For evaluating the synthon completion module, we first apply the true edits to obtain synthons, and compare the true leaving groups to top- $\mathbf { \nabla } \cdot n$ leaving groups predicted by the model. We test the performance of both the ind and shared configurations. Both configurations perform similarly, and are able to identify $\sim 9 7 \%$ (close to its upper bound of $9 9 . 7 \%$ ) of the true leaving groups in its top-5 choices, when the reaction class is known. The top-1, 3 and 5 accuracies of the synthon completion for unknown reaction classes for G2Gs are $6 1 . 1 \%$ , $8 1 . 5 \%$ and $8 6 . 7 \%$ respectively, while ours are $7 5 . 6 \%$ , $9 2 . 5 \%$ and $9 6 . 1 \%$ , indicating a $10 \%$ performance improvement using a classification formulation over the generative one adopted by G2Gs. + +Table 2: Performance Study of edit prediction and synthon completion modules + +
SettingTop-n Accuracy (%)
Reaction class knownReaction class unknown
12351235
Edit Prediction84.692.293.794.570.885.189.592.7
- without edit score updates84.392.193.794.570.184.889.492.6
- predicting on atom pairs81.989.590.992.168.683.288.391.8
Synthon Completion (ind)77.489.594.297.675.687.492.596.1
Synthon Completion (shared)76.989.693.997.474.987.792.996.3
+ +# 4.3 Example Predictions + +In Figure 2, we visualize the model predictions and the ground truth for three cases. Figure 2a shows an example where the model identifies both the edits and leaving groups correctly. In Figure 2b, the correct edit is identified but the predicted leaving groups are incorrect. We hypothesize this is due to the fact that in the training set, leaving groups attaching to the carbonyl carbon $\scriptstyle ( \mathbf { C } = \mathbf { O } )$ are small (e.g. -OH, - $\mathrm { - N H _ { 2 } }$ , halides). The true leaving group in this example, however, is large. The model is unable to reason about this and predicts the small leaving group -I. In Figure 2c, the model identifies the edit and consequently the leaving group incorrectly. This highlights a limitation of our model. If the edit is predicted incorrectly, the model cannot suggest the true precursors. + +# 4.4 Limitations + +The simplified and interpretable construction of GRAPHRETRO comes with certain limitations. First, the overall performance of the model is limited by the performance of the edit prediction step. If the predicted edit is incorrect, the true reactants cannot be salvaged. This limitation is partly remedied by our model design, that allows for user intervention to correct the edit. Second, our method is reliant on atom-mapping for extracting edits and leaving groups. Extracting edits directly based on substructure matching currently suffer from false positives, and heuristics to correct for these result in correct edits in only ${ \sim } 9 0 \%$ of the cases. Third, our formulation assumes that we have as many synthons as reactants, which is violated in some reactions. We leave it to future work to extend the model to realize a single reactant from multiple synthons, and introduce more chemically meaningful edit correction mechanisms. + +![](images/9df65c90e84b75ac55b8dc08692726ee93cf492df4b05d29c79c8c792f98fccd.jpg) +Figure 2: Example Predictions. The true edit and incorrect edit (if any) are highlighted in green and red respectively. The true and predicted leaving groups are highlighted in blue. a. Correctly predicted example by the model. b. Correctly predicted edit but incorrectly predicted leaving groups. c. Incorrectly predicted edit and leaving group. + +# 5 Conclusion + +Previous methods for single-step retrosynthesis either restrict prediction to a template set, are insensitive to molecular graph structure or generate molecules from scratch. We address these shortcomings by introducing a graph-based semi-template-based model inspired by a chemist’s workflow, enhancing the interpretability of retrosynthesis models. Given a target molecule, we first identify synthetic building blocks (synthons) which are then realized into valid reactants, thus avoiding molecule generation from scratch. Our model outperforms previous semi-template-methods by significant margins on the benchmark dataset. Future work aims to extend the model to realize a single reactant from multiple synthons, and introduce more chemically meaningful components to improve the synergy between such tools for retrosynthesis prediction and a practitioner’s expertise. + +# Acknowledgements + +This research was supported by the Machine Learning for Pharmaceutical Discovery and Synthesis Consortium at MIT. V.R.S. was also supported by the Zeno Karl Schindler Foundation. C.B. was supported by the Swiss National Science Foundation under the National Center of Competence in Research (NCCR) Catalysis under grant agreement 51NF40 180544. We thank the Leonhard scientific computing cluster at ETH Zürich for providing computational resources. + +# References + +B. Chen, T. Shen, T. S. Jaakkola, and R. Barzilay. Learning to Make Generalizable and Diverse Predictions for Retrosynthesis. In Submission, 2019. +C. W. Coley, R. Barzilay, T. S. Jaakkola, W. H. Green, and K. F. Jensen. Prediction of Organic Reaction Outcomes Using Machine Learning. In ACS Central Science. ACS Publications, 2017a. +C. W. Coley, L. Rogers, W. H. Green, and K. F. Jensen. Computer-Assisted Retrosynthesis Based on Molecular Similarity. ACS Central Science, 3, 2017b. +C. W. Coley, W. Jin, L. Rogers, T. F. Jamison, T. S. Jaakkola, W. H. Green, R. Barzilay, and K. F. Jensen. A graph-convolutional neural network model for the prediction of chemical reactivity. Chemical Science, 10, 2019. +E. Corey and W. T. Wipke. Computer-assisted design of complex organic syntheses. Science, 166 (3902):178–192, 1969. +E. J. Corey. The Logic of Chemical Synthesis: Multistep Synthesis of Complex Carbogenic Molecules (Nobel Lecture). Angewandte Chemie International Edition, 30, 1991. +H. Dai, C. Li, C. Coley, B. Dai, and L. Song. Retrosynthesis Prediction with Conditional Graph Logic Network. In Advances in Neural Information Processing Systems (NeurIPS), volume 32, 2019. +S. Genheden, A. Thakkar, V. Chadimová, J.-L. Reymond, O. Engkvist, and E. Bjerrum. Aizynthfinder: a fast, robust and flexible open-source software for retrosynthetic planning. Journal of cheminformatics, 12(1):1–9, 2020. +J. Gilmer, S. S. Schoenholz, P. F. Riley, O. Vinyals, and G. E. Dahl. Neural Message Passing for Quantum Chemistry. In International Conference on Machine Learning (ICML), volume 70, 2017. +M. Hartenfeller, M. Eberle, P. Meier, C. Nieto-Oberhuber, K.-H. Altmann, G. Schneider, E. Jacoby, and S. Renner. A Collection of Robust Organic Synthesis Reactions for In Silico Molecule Design. 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Predicting Retrosynthetic Reactions using SelfCorrected Transformer Neural Networks. In Journal of Chemical Information and Modeling. ACS Publications, 2019. \ No newline at end of file diff --git a/parse/train/LyjH88yV7F/LyjH88yV7F_content_list.json b/parse/train/LyjH88yV7F/LyjH88yV7F_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..12a86b15c8e12b399048ebd3f6b76dfe757fc135 --- /dev/null +++ b/parse/train/LyjH88yV7F/LyjH88yV7F_content_list.json @@ -0,0 +1,1040 @@ +[ + { + "type": "text", + "text": "Learning Graph Models for Retrosynthesis Prediction ", + "text_level": 1, + "bbox": [ + 330, + 122, + 666, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Vignesh Ram Somnath1 ", + "bbox": [ + 228, + 226, + 392, + 241 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Charlotte Bunne1 ", + "bbox": [ + 455, + 224, + 578, + 241 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Connor W. Coley2 ", + "bbox": [ + 642, + 224, + 769, + 241 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Andreas Krause1 Regina Barzilay3 ", + "bbox": [ + 369, + 261, + 629, + 276 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1Department of Computer Science, ETH 2Department of Chemical Engineering, MIT 3Computer Science and Artificial Intelligence Lab, MIT 1{vsomnath, bunnec, krausea}@ethz.ch, 2ccoley@mit.edu, 3regina@csail.mit.edu ", + "bbox": [ + 186, + 289, + 815, + 347 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 382, + 535, + 400 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Retrosynthesis prediction is a fundamental problem in organic synthesis, where the task is to identify precursor molecules that can be used to synthesize a target molecule. A key consideration in building neural models for this task is aligning model design with strategies adopted by chemists. Building on this viewpoint, this paper introduces a graph-based approach that capitalizes on the idea that the graph topology of precursor molecules is largely unaltered during a chemical reaction. The model first predicts the set of graph edits transforming the target into incomplete molecules called synthons. Next, the model learns to expand synthons into complete molecules by attaching relevant leaving groups. This decomposition simplifies the architecture, making its predictions more interpretable, and also amenable to manual correction. Our model achieves a top-1 accuracy of $5 3 . 7 \\%$ , outperforming previous template-free and semi-template-based methods. ", + "bbox": [ + 233, + 419, + 766, + 584 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 618, + 310, + 636 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Retrosynthesis prediction, first formalized by E. J. Corey [Corey, 1991] is a fundamental problem in organic synthesis that attempts to identify a series of chemical transformations for synthesizing a target molecule. In the single-step formulation, the task is to identify a set of reactant molecules given a target. Beyond simple reactions, many practical tasks involving complex organic molecules are difficult even for expert chemists. As a result, substantial experimental exploration is needed to cover for deficiencies of analytical approaches. This has motivated interest in computer-assisted retrosynthesis [Corey and Wipke, 1969], with a recent surge in machine learning methods [Chen et al., 2019, Coley et al., 2017b, Dai et al., 2019, Zheng et al., 2019, Genheden et al., 2020]. ", + "bbox": [ + 174, + 654, + 825, + 765 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Computationally, the main challenge is how to explore the combinatorial space of reactions that can yield the target molecule. Largely, previous methods for retrosynthesis prediction can be divided into template-based [Coley et al., 2017b, Dai et al., 2019, Segler and Waller, 2017] and template-free [Chen et al., 2019, Zheng et al., 2019] approaches. Template-based methods match a target molecule against a large set of templates, which are molecular subgraph patterns that highlight changes during a chemical reaction. Despite their interpretability, these methods fail to generalize to new reactions. Template-free methods bypass templates by learning a direct mapping from the SMILES [Weininger, 1988] representations of the product to reactants. Despite their greater generalization potential, these methods generate reactant SMILES character by character, increasing generation complexity. ", + "bbox": [ + 174, + 771, + 825, + 896 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/e988f2e4a9f13ae6fa41934806006c24c87183942b7fd7f824274aee3a6237f9.jpg", + "image_caption": [ + "Figure 1: Overview of Our Approach. a. Edit Prediction. We train a model to learn a distribution over possible graph edits. In this case, the correct edit corresponds to breaking the bond marked in red. Applying this edit produces two synthons. b. Synthon Completion. Another model is trained to pick candidate leaving groups (blue) for each synthon from a discrete vocabulary, which are then attached to produce the final reactants. " + ], + "image_footnote": [], + "bbox": [ + 174, + 103, + 812, + 292 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Another important consideration in building retrosynthesis models is aligning model design with strategies adopted by expert chemists. These strategies are influenced by fundamental properties of chemical reactions, independent of complexity level: (i.) the product atoms are always a subset of the reactant atoms1, and (ii.) the molecular graph topology is largely unaltered from products to reactants. For example, in the standard retrosynthesis dataset, only $6 . 3 \\%$ of the atoms in the product undergo any change in connectivity. ", + "bbox": [ + 174, + 404, + 825, + 488 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "This consideration has received more attention in recent semi-template-based methods [Shi et al., 2020, Yan et al., 2020], that generate reactants from a product in two stages: (i.) first identify intermediate molecules called synthons, (ii.) and then complete synthons into reactants by sequential generation of atoms or SMILES characters.. Our model GRAPHRETRO also uses a similar workflow. However, we avoid sequential generation for completing synthons by instead selecting subgraphs called leaving groups from a precomputed vocabulary. This vocabulary is constructed during preprocessing by extracting subgraphs that differ between a synthon and the corresponding reactant. The vocabulary has a small size (170 for USPTO-50k) indicating remarkable redundancy, while covering $9 9 . 7 \\%$ of the test set. Operating at the level of these subgraphs greatly reduces the complexity of reactant generation, with improved empirical performance. This formulation also simplifies our architecture, and makes our predictions more transparent, interpretable and amenable to manual correction. ", + "bbox": [ + 174, + 494, + 825, + 646 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The benchmark dataset for evaluating retrosynthesis models is USPTO-50k [Schneider et al., 2016], which consists of 50000 reactions across 10 reaction classes. The dataset contains an unexpected shortcut towards predicting the edit, in that the product atom with atom-mapping 1 is part of the edit in $7 5 \\%$ of the cases, allowing predictions that depend on the position of the atom to overestimate performance. We canonicalize the product SMILES and remap the existing dataset, thereby removing the shortcut. On this remapped dataset, GRAPHRETRO achieves a top-1 accuracy of $5 3 . 7 \\%$ when the reaction class is not known, outperforming both template-free and semi-template-based methods. ", + "bbox": [ + 174, + 652, + 825, + 750 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Related Work ", + "text_level": 1, + "bbox": [ + 174, + 770, + 321, + 786 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Retrosynthesis Prediction Existing machine learning methods for retrosynthesis prediction can be divided into template-based, template-free and recent semi-template-based approaches. ", + "bbox": [ + 173, + 801, + 823, + 830 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Template-Based: Templates are either hand-crafted by experts [Hartenfeller et al., 2011, Szymkuc´ et al., 2016], or extracted algorithmically from large databases Coley et al. [2017a], Law et al. [2009]. Exhaustively applying large template sets is expensive due to the involved subgraph matching procedure. Template-based methods therefore utilize different ways of prioritizing templates, by either learning a conditional distribution over the template set [Segler and Waller, 2017], ranking templates based on molecular similarities to precedent reactions [Coley et al., 2017b] or directly modelling the joint distribution of templates and reactants using logic variables [Dai et al., 2019]. Despite their interpretability, these methods fail to generalize outside their rule set. ", + "bbox": [ + 181, + 843, + 825, + 886 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 197, + 92, + 825, + 174 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Template-Free: Template-free methods [Liu et al., 2017, Zheng et al., 2019, Chen et al., 2019] learn a direct transformation from products to reactants using architectures from neural machine translation and a string based representation of molecules called SMILES [Weininger, 1988]. Linearizing molecules as strings does not utilize the inherently rich chemical structure. In addition, the reactant SMILES are generated from scratch, character by character. Attempts have been made to improve validity by adding a syntax correcter [Zheng et al., 2019] and a mixture model to improve diversity of suggestions [Chen et al., 2019], but the performance remains worse than [Dai et al., 2019] on the standard retrosynthesis dataset. Sun et al. [2021] formulate retrosynthesis using energy-based models, with additional parameterizations and loss terms to enforce the duality between forward (reaction prediction) and backward (retrosynthesis) prediction. ", + "bbox": [ + 183, + 183, + 825, + 321 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Semi-Template-Based: Our work is closely related to recently proposed semi-template-based methods [Shi et al., 2020, Yan et al., 2020], which first identify synthons and then expand synthons into reactants through sequential generation using either a graph generative model [Shi et al., 2020] or a Transformer [Yan et al., 2020]. To reduce the complexity of reactant generation, we instead complete synthons using subgraphs called leaving groups selected from a precomputed vocabulary. This allows us to view synthon completion as a classification problem instead of a generative one. We also utilize the dependency graph between possible edits, and update edit predictions using a message passing network (MPN) [Gilmer et al., 2017] on this graph. Both innovations together yield a $4 . 8 \\%$ and $3 . 3 \\%$ performance improvement respectively over previous semi-template-based methods. ", + "bbox": [ + 183, + 330, + 825, + 468 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Reaction Center Identification The reaction center covers a small number of participating atoms involved in the reaction. Our work is also related to models that predict reaction outcomes by learning to rank atom pairs based on their likelihood to be in the reaction center [Coley et al., 2019, Jin et al., 2017]. The task of identifying the reaction center is related to the step of deriving the synthons in our formulation. Our work departs from [Coley et al., 2019, Jin et al., 2017] as we utilize the property that new bond formations occur rarely $( \\sim 0 . 1 \\% )$ from products to synthons, allowing us to predict a score only for existing bonds and atoms and reduce prediction complexity from $O ( N ^ { \\bar { 2 } } )$ to $O ( N )$ . We also utilize the dependency graph between possible edits, and update edit predictions using a MPN on this graph. ", + "bbox": [ + 174, + 487, + 825, + 613 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Utilizing Substructures Substructures have been utilized in various tasks from sentence generation by fusing phrases to molecule generation and optimization [Jin et al., 2018, 2020]. Our work is closely related to [Jin et al., 2020] which uses precomputed substructures as building blocks for property-conditioned molecule generation. However, instead of precomputing, synthons —analogous building blocks for reactants— are indirectly learnt during training. ", + "bbox": [ + 174, + 631, + 825, + 702 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 Model Design ", + "text_level": 1, + "bbox": [ + 174, + 724, + 320, + 742 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our approach leverages the property that graph topology is largely unaltered from products to reactants. To achieve this, we first derive suitable building blocks from the product called synthons, and then complete them into valid reactants by adding specific functionalities called leaving groups. These derivations, called edits, are characterized by modifications to bonds or hydrogen counts on atoms. We first train a neural network to predict a score for possible edits (Section 3.1). The edit with the highest score is then applied to the product to obtain synthons. Since the number of unique leaving groups are small, we model leaving group selection as a classification problem over a precomputed vocabulary (Section 3.2). To produce candidate reactants, we attach the predicted leaving group to the corresponding synthon through chemically constrained rules. The overall process is outlined in Figure 1. Before describing the two modules, we introduce relevant preliminaries that set the background for the remainder of the paper. ", + "bbox": [ + 174, + 758, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Retrosynthesis Prediction A retrosynthesis pair $R$ is described by a pair of molecular graphs $( \\mathcal { G } _ { p } , \\mathcal { G } _ { r } )$ , where $\\mathcal { G } _ { p }$ are the products and $\\mathcal { G } _ { r }$ the reactants. A molecular graph is described as $\\mathcal { G } = \\mathbf { \\bar { \\rho } } ( \\mathcal { V } , \\mathcal { E } )$ with atoms $\\nu$ as nodes and bonds $\\mathcal { E }$ as edges. Prior work has focused on the single product case, while reactants can have multiple connected components, i.e. $\\mathcal { G } _ { r } = \\{ \\mathcal { G } _ { r _ { c } } \\} _ { c = 1 } ^ { C }$ . Retrosynthesis pairs are atom-mapped so that each product atom has a unique corresponding reactant atom. The retrosynthesis task then, is to infer $\\{ \\mathcal { G } _ { r _ { c } } \\} _ { c = 1 } ^ { C }$ given $\\mathcal { G } _ { p }$ . ", + "bbox": [ + 173, + 90, + 825, + 176 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Edits Edits consist of (i.) atom pairs $\\left\\{ \\left( a _ { i } , a _ { j } \\right) \\right\\}$ where the bond type changes from products to reactants, and (ii.) atoms $\\left\\{ { a } _ { i } \\right\\}$ where the number of hydrogens attached to the atom change from products to reactants. We denote the set of edits by $E$ . Since retrosynthesis pairs in the training set are atom-mapped, edits can be automatically identified by comparing the atoms and atom pairs in the product to their corresponding reactant counterparts. ", + "bbox": [ + 173, + 189, + 825, + 260 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Synthons and Leaving Groups Applying edits $E$ to the product $\\mathcal { G } _ { p }$ results in incomplete molecules called synthons. Synthons are analogous to rationales or building blocks, which are expanded into valid reactants by adding specific functionalities called leaving groups that are responsible for its reactivity. We denote synthons by $\\mathcal { G } _ { s }$ and leaving groups by $\\mathcal { G } _ { l }$ . We further assume that synthons and leaving groups have the same number of connected components as the reactants, i.e $\\mathcal { G } _ { s } \\doteq \\{ \\mathcal { G } _ { s _ { c } } \\} _ { c = 1 } ^ { C }$ and $\\mathcal { G } _ { l } = \\{ \\mathcal { G } _ { l _ { c } } ^ { \\star } \\} _ { c = 1 } ^ { C }$ . This assumption holds for $9 9 . 9 7 \\%$ reactions in the training set. ", + "bbox": [ + 173, + 275, + 825, + 359 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Formally, our model generates reactants by first predicting the set of edits $E$ that transform $\\mathcal { G } _ { p }$ into $\\mathcal { G } _ { s }$ , followed by predicting a leaving group $\\mathcal { G } _ { l _ { c } }$ to attach to each synthon $\\mathcal { G } _ { s _ { c } }$ . The model is defined as ", + "bbox": [ + 171, + 364, + 823, + 393 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/e5989ffe48f40113cab8f00a06662537b577c1fdbf3a8f9d8153b9ad86f85418.jpg", + "text": "$$\nP ( \\mathcal G _ { r } | \\mathcal G _ { p } ) = \\sum _ { E , \\mathcal G _ { l } } P ( E | \\mathcal G _ { p } ) P ( \\mathcal G _ { l } | \\mathcal G _ { p } , \\mathcal G _ { s } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 364, + 407, + 632, + 443 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\mathcal { G } _ { s } , \\mathcal { G } _ { r }$ are deterministic given $E , { \\mathcal { G } } _ { l }$ , and $\\mathcal { G } _ { p }$ . ", + "bbox": [ + 173, + 449, + 509, + 465 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 Edit Prediction ", + "text_level": 1, + "bbox": [ + 174, + 481, + 318, + 496 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For a given retrosynthesis pair $R = ( \\mathcal G _ { p } , \\mathcal G _ { r } )$ , we predict an edit score only for existing bonds and atoms, instead of every atom pair as in [Coley et al., 2019, Jin et al., 2017]. This choice is motivated by the low frequency $( \\sim 0 . 1 \\% )$ of new bond formations in the training set examples. Coupled with the sparsity of molecular graphs, this reduces the prediction complexity from $O ( N ^ { 2 } )$ to $O ( N )$ for a product with $N$ atoms. Our edit prediction model has variants tailored to single and multiple edit prediction. Since $9 5 \\%$ of the training set consists of single edit examples, the remainder of this section describes the setup for single edit prediction. A detailed description of our multiple edit prediction model can be found in Appendix ??. ", + "bbox": [ + 173, + 506, + 825, + 619 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Each bond $( u , v )$ in $\\mathcal { G } _ { p }$ is associated with a label $y _ { u v k } \\in \\{ 0 , 1 \\}$ indicating whether its bond type $k$ has changed from the products to reactants. Each atom $u$ is associated with a label $y _ { u } \\in \\{ 0 , \\bar { 1 } \\}$ indicating a change in hydrogen count. We predict edit scores using representations that are learnt using a graph encoder. ", + "bbox": [ + 173, + 625, + 825, + 681 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Graph Encoder To obtain atom representations, we use a variant of the message passing network (MPN) described in [Gilmer et al., 2017]. Each atom $u$ has a feature vector $\\mathbf { x } _ { u }$ indicating its atom type, degree and other properties. Each bond $( u , v )$ has a feature vector $\\mathbf { x } _ { u v }$ indicating its aromaticity, bond type and ring membership. For simplicity, we denote the encoding process by $\\mathrm { M P N } ( \\cdot )$ and describe architectural details in Appendix ??. The MPN computes atom representations $\\{ \\mathbf { c } _ { u } | u \\in \\mathcal { G } \\}$ via ", + "bbox": [ + 173, + 695, + 825, + 779 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/b517ef315352815d369d32f3af958c21a2ae42d976e66bdc837220e4bb39c388.jpg", + "text": "$$\n\\{ \\mathbf { c } _ { u } \\} = \\mathrm { M P N } ( \\mathcal { G } , \\{ \\mathbf { x } _ { u } \\} , \\{ \\mathbf { x } _ { u v } \\} _ { v \\in \\mathcal { N } ( u ) } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 367, + 794, + 629, + 813 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\mathcal { N } ( u )$ denotes the neighbors of atom $u$ . The graph representation $\\mathbf { c } _ { \\mathcal { G } }$ is an aggregation of atom representations, i.e. $\\mathbf { c } _ { \\mathcal { G } } = \\dot { \\sum _ { { u } \\in \\mathcal { V } } } \\mathbf { c } _ { u }$ . When $\\mathcal { G }$ has connected components $\\left\\{ { \\mathcal { G } } _ { i } \\right\\}$ , we get a set of graph representations $\\left\\{ \\mathbf { c } _ { \\mathcal { G } _ { i } } \\right\\}$ . For a bond $( u , v )$ , we define its representation $\\mathbf { c } _ { u v } = ( \\operatorname { A B S } ( \\mathbf { c } _ { u } , \\mathbf { c } _ { v } ) | | \\mathbf { c } _ { u } + \\mathbf { c } _ { v } )$ , where ABS denotes absolute difference and $| |$ refers to concatenation. This ensures our representations are permutation invariant. These representations are then used to predict atom and bond edit scores using corresponding neural networks, ", + "bbox": [ + 174, + 825, + 825, + 911 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/43a6f38c88896496f6b8c1e00b965fa739609189983ee4e111bdef371168642c.jpg", + "text": "$$\n\\begin{array} { r } { \\boldsymbol { s } _ { u } = \\mathbf { u _ { a } } ^ { T } \\boldsymbol { \\tau } ( \\mathbf { W _ { a } } \\mathbf { c } _ { u } + b ) \\quad } \\\\ { \\boldsymbol { s } _ { u v k } = \\mathbf { u _ { k } } ^ { T } \\boldsymbol { \\tau } ( \\mathbf { W _ { k } } \\mathbf { c } _ { u v } + b _ { k } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 400, + 87, + 596, + 131 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\tau ( \\cdot )$ is the ReLU activation function. ", + "bbox": [ + 176, + 142, + 457, + 157 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Updating Bond Edit Scores Unlike a typical classification problem where the labels are independent, edits can have possible dependencies between each other. For example, bonds part of a stable system such as an aromatic ring have a greater tendency to remain unchanged (label 0). We attempt to leverage such dependencies to update initial edit scores. To this end, we build a graph with bonds $( u , v )$ as nodes, and introduce an edge between bonds sharing an atom. We use another $\\mathrm { M P N } ( \\cdot )$ on this graph to learn aggregated neighborhood messages $\\mathbf { m } _ { u v }$ , and update the edit scores $s _ { u v k }$ in a manner similar to how LSTMs update representations, ", + "bbox": [ + 173, + 172, + 825, + 270 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/190ca6c12e9d0551c6c507ca3bb0a358e64c6db98378f96a20d19207a9bfc801.jpg", + "text": "$$\n\\begin{array} { r l } & { f _ { u v k } = \\sigma ( \\mathbf { W _ { k x } ^ { f } } \\mathbf { x } _ { u v } + \\mathbf { W _ { k m } ^ { f } } \\mathbf { m } _ { u v } ) } \\\\ & { i _ { u v k } = \\sigma ( \\mathbf { W _ { k x } ^ { i } } \\mathbf { x } _ { u v } + \\mathbf { W _ { k m } ^ { i } } \\mathbf { m } _ { u v } ) } \\\\ & { \\tilde { m } _ { u v k } = \\mathbf { u _ { m } } \\tau ( \\mathbf { W _ { k x } ^ { m } } \\mathbf { x } _ { u v } + \\mathbf { W _ { k m } ^ { m } } \\mathbf { m } _ { u v } ) } \\\\ & { \\tilde { s } _ { u v k } = f _ { u v k } \\cdot s _ { u v k } + i _ { u v k } \\cdot \\tilde { m } _ { u v k } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 369, + 284, + 629, + 362 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Training We train by minimizing the cross-entropy loss over possible bond and atom edits ", + "bbox": [ + 169, + 375, + 781, + 390 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/d39b464af053c1ddc554cc80cf10fc8a8cf6434c8681b4503820a67f7447deed.jpg", + "text": "$$\n\\mathcal { L } _ { e } = - \\sum _ { ( \\mathcal { G } _ { p } , E ) } \\left( \\sum _ { ( ( u , v ) , k ) \\in E } y _ { u v k } \\mathrm { l o g } ( \\widetilde s _ { u v k } ) + \\sum _ { u \\in E } y _ { u } \\mathrm { l o g } ( s _ { u } ) \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 287, + 402, + 710, + 454 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The cross-entropy loss enforces the model to learn a distribution over possible edits instead of reasoning about each edit independently, as with the binary cross entropy loss used in [Jin et al., 2017, Coley et al., 2019]. ", + "bbox": [ + 174, + 467, + 825, + 510 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 Synthon Completion ", + "text_level": 1, + "bbox": [ + 174, + 526, + 356, + 542 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Synthons are completed into valid reactants by adding specific functionalities called leaving groups. This involves two complementary tasks: (i.) selecting the appropriate leaving group, and (ii.) attaching the leaving group to the synthon. As ground truth leaving groups are not directly provided, we extract the leaving groups and construct a vocabulary $\\mathcal { X }$ of unique leaving groups during preprocessing. ", + "bbox": [ + 173, + 553, + 825, + 609 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The vocabulary has a limited size ( $| \\mathcal { X } | = 1 7 0$ for a standard dataset with 50, 000 examples, and 72000 synthons) indicating the redundancy of leaving groups used in accomplishing retrosynthetic transformations. This redundancy also allows us to formulate leaving group selection as a classification problem over $\\mathcal { X }$ , while retaining the ability to generate diverse reactants using different combinations of leaving groups. ", + "bbox": [ + 174, + 614, + 825, + 685 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Vocabulary Construction Before constructing the vocabulary, we align connected components of synthon and reactant graphs by comparing atom mapping overlaps. Using aligned pairs $\\mathcal { G } _ { s _ { c } } =$ $( \\gamma _ { s _ { c } } , \\mathcal { E } _ { s _ { c } } )$ and $\\mathcal { G } _ { r _ { c } } = ( \\nu _ { r _ { c } } , \\mathcal { E } _ { r _ { c } } )$ as input, the leaving group vocabulary $\\mathcal { X }$ is constructed by extracting subgraphs $\\mathcal { G } _ { l _ { c } } = ( \\nu _ { l _ { c } } , \\mathcal { E } _ { l _ { c } } )$ such that $\\smash { \\gamma _ { l _ { c } } = \\gamma _ { r _ { c } } \\setminus \\gamma _ { s _ { c } } }$ . Atoms $\\left\\{ { a } _ { i } \\right\\}$ in the leaving groups that attach to synthons are marked with a special symbol. We also add three tokens to $\\mathcal { X }$ namely START, which indicates the start of synthon completion, END, which indicates that there is no leaving group to add and PAD, which is used to handle variable numbers of synthon components in a minibatch. ", + "bbox": [ + 173, + 700, + 825, + 799 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Leaving Group Selection For synthon component $c \\leq C$ , where $C$ is the number of connected components in the synthon graph, we use three inputs for leaving group selection – the product representation $\\mathbf { c } _ { \\mathcal { G } _ { p } }$ , the synthon component representation $\\mathbf { c } _ { \\mathcal { G } _ { s _ { c } } }$ , and the leaving group representation for the previous synthon component, ${ \\bf e } _ { l _ { c - 1 } }$ . The product and synthon representations are learnt using the $\\mathrm { M P N } ( \\cdot )$ . For each $x _ { i } \\in { \\mathcal { X } }$ , representations can be learnt by either training independent embedding vectors (ind) or by treating each $x _ { i }$ as a subgraph and using the $\\mathrm { M P N } ( \\cdot )$ (shared). In the shared setting, we use the same $\\mathrm { M P N } ( \\cdot )$ as the product and synthons. ", + "bbox": [ + 173, + 814, + 825, + 911 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The leaving group probabilities are then computed by combining $\\mathbf { c } _ { \\mathcal { G } _ { p } } , \\mathbf { c } _ { \\mathcal { G } _ { s _ { c } } }$ and $\\mathbf { e } _ { l _ { c - 1 } }$ via a single layer neural network and softmax function ", + "bbox": [ + 173, + 90, + 823, + 119 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/d2fa89696df25b480905fd11b89872899b1206510d6da72f11724cb4e151b106.jpg", + "text": "$$\n\\hat { q } _ { l _ { c } } = \\mathrm { s o f t m a x } \\left( \\mathbf { U } \\tau \\left( \\mathbf { W } _ { 1 } \\mathbf { c } _ { \\mathcal { G } _ { p } } + \\mathbf { W } _ { 2 } \\mathbf { c } _ { \\mathcal { G } _ { s _ { c } } } + \\mathbf { W } _ { 3 } \\mathbf { e } _ { l _ { \\left( c - 1 \\right) } } \\right) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 305, + 131, + 691, + 151 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\hat { q } _ { l _ { c } }$ is distribution learnt over $\\mathcal { X }$ . Using the representation of the previous leaving group ${ \\bf e } _ { l _ { c - 1 } }$ allows the model to understand combinations of leaving groups that generate the desired product from the reactants. We also include the product representation $\\mathbf { c } _ { \\mathcal { G } _ { p } }$ as the synthon graphs are derived from the product graph. ", + "bbox": [ + 174, + 164, + 825, + 219 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Training For step $c$ , given the one hot encoding of the true leaving group $q _ { l _ { c } }$ , we minimize the cross-entropy loss ", + "bbox": [ + 173, + 233, + 823, + 263 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/c5878d96a04183a5e264e043e021073accc4e3106f900490a5939023894394b7.jpg", + "text": "$$\n\\mathcal { L } _ { s } = \\sum _ { c = 1 } ^ { C } \\mathcal { L } ( \\hat { q } _ { l _ { c } } , q _ { l _ { c } } ) .\n$$", + "text_format": "latex", + "bbox": [ + 428, + 275, + 570, + 319 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Training utilizes teacher-forcing [Williams and Zipser, 1989] so that the model makes predictions given correct histories. During inference, at every step, we use the representation of leaving group from the previous step with the highest predicted probability. ", + "bbox": [ + 174, + 330, + 823, + 372 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Leaving Group Attachment Attaching leaving groups to synthons is a deterministic process and not learnt during training. The task involves identification of the type of bonds to add between attaching atoms in the leaving group (marked during vocabulary construction), and the atom(s) participating in the edit. These bonds can be inferred by applying the valency constraint, which determines the maximum number of neighbors for each atom. The attachment process does not modify any stereochemistry. Given synthons and leaving groups, the attachment process has a $100 \\%$ accuracy. The detailed procedure is described in Appendix ??. ", + "bbox": [ + 173, + 387, + 825, + 484 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.3 Inference ", + "text_level": 1, + "bbox": [ + 174, + 501, + 279, + 515 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Inference is performed using beam search with a log-likelihood scoring function. For a beam width $n$ , we select $n$ edits with highest scores and apply them to the product to obtain $n$ synthons, where each synthon can consist of multiple connected components. The synthons form the nodes for beam search. Each node maintains a cumulative score by aggregating the log-likelihoods of the edit and predicted leaving groups. Leaving group inference starts with a connected component for each synthon, and selects $n$ leaving groups with highest log-likelihoods. From the $n ^ { 2 }$ possibilities, we select $n$ nodes with the highest cumulative scores. This process is repeated until all nodes have a leaving group predicted for each synthon component. ", + "bbox": [ + 174, + 525, + 826, + 637 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 Evaluation ", + "text_level": 1, + "bbox": [ + 174, + 656, + 295, + 672 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Evaluating retrosynthesis models is challenging as multiple sets of reactants can be generated from the same product. To deal with this, previous works [Coley et al., 2017b, Dai et al., 2019] evaluate the ability of the model to recover retrosynthetic strategies recorded in the dataset. ", + "bbox": [ + 174, + 686, + 825, + 728 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Data We use the benchmark dataset USPTO-50k [Schneider et al., 2016] for all our experiments. The dataset contains 50, 000 atom-mapped reactions across 10 reaction classes. We use the same dataset version and splits as provided by [Dai et al., 2019]. The USPTO-50k dataset contains a shortcut in that the product atom with atom-mapping 1 is part of the edit in $\\sim 7 5 \\%$ of the cases. If the product SMILES is not canonicalized, predictions utilizing operations that depend on the position of the atom or bond will be able to use the shortcut, and overestimate performance. We canonicalize the product SMILES, and reassign atom-mappings to the reactant atoms based on the canonical ordering, which removes the shortcut. Details on the remapping procedure can be found in Appendix ??. ", + "bbox": [ + 174, + 742, + 825, + 854 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Evaluation We use the top- $\\mathbf { \\nabla } \\cdot n$ accuracy $( n = 1 , 3 , 5 , 1 0 )$ ) as our evaluation metric, defined as the fraction of examples where the recorded reactants are suggested by the model with rank $\\leq n$ . Following prior work [Coley et al., 2017b, Zheng et al., 2019, Dai et al., 2019], we compute the accuracy by comparing the canonical SMILES of predicted reactants to the ground truth. Atom-mapping is excluded from this comparison, but stereochemistry, which describes the relative orientation of atoms in the molecule, is retained. The evaluation is carried out for two settings, with the reaction class being known or unknown. ", + "bbox": [ + 176, + 869, + 825, + 911 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/b50491b8ada14574c6239d28c43d14386b5d359906cdb5f2c186422249a03113.jpg", + "table_caption": [ + "Table 1: Top- $n$ exact match accuracy. Best values within each section are highlighted in bold. " + ], + "table_footnote": [], + "table_body": "
ModelTop-n Accuracy (%)
Reaction class knownReaction class unknown
1351013510
Template-Based
RETROSIM [Coley et al.,2017b]52.973.881.288.137.354.763.374.1
NEURALSYM [Segler and Waller,2017]55.376.081.485.144.465.372.478.9
GLN [Dai et ai., 2019]64.279.185.290.052.569.075.683.7
DUALTB [Sun et al.,2021]67.784.888.992.055.274.680.586.9
Template-Free
SCROP [Zheng et al.,2019]59.074.878.181.143.760.065.268.7
LV-TRANSFORMER [Chen et al.,2019]----40.565.172.879.4
DUALTF [Sun et al., 2021]65.781.984.785.953.670.774.677.0
Semi-Template-Based
G2Gs [Shi et al.,2020]61.081.386.088.748.967.672.575.5
RETROXPERT [Yan et al.,2020]62.175.878.580.950.461.162.363.4
GRAPHRETRO (ours)63.981.585.288.153.768.372.275.5
", + "bbox": [ + 171, + 116, + 826, + 366 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 392, + 825, + 448 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Baselines For evaluating overall performance, we compare GRAPHRETRO to nine baselines — four template-based, three template-free, and two semi-template-based methods. These include: ", + "bbox": [ + 174, + 465, + 825, + 494 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Template-Based: RETROSIM Coley et al. [2017b] ranks templates for a given target molecule by computing molecular similarities to precedent reactions. NEURALSYM [Segler and Waller, 2017] trains a model to rank templates given a target molecule. GLN [Dai et al., 2019] models the joint distribution of templates and reactants in a hierarchical fashion using logic variables. DUALTB [Sun et al., 2021] uses an energy-based model formulation for retrosynthesis, with additional parameterizations and loss terms to enforce the duality between forward (reaction prediction) and backward (retrosynthesis prediction). Inference is carried out using reactant candidates obtained by applying an extracted template set to the products. ", + "bbox": [ + 181, + 507, + 825, + 619 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Template-Free: SCROP [Zheng et al., 2019], LV-TRANSFORMER [Chen et al., 2019] and DUALTF [Sun et al., 2021] use the Transformer architecture [Vaswani et al., 2017] to output reactant SMILES given a product SMILES. To improve the validity of their suggestions, SCROP include a second Transformer that functions as a syntax correcter. LV-TRANSFORMER uses a latent variable mixture model to improve diversity of suggestions. DUALTF utilizes additional parameterizations and loss terms to enforce the duality between forward (reaction prediction) and backward (retrosynthesis prediction). ", + "bbox": [ + 183, + 626, + 825, + 722 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Semi-Template-Based: G2GS [Shi et al., 2020] and RETROXPERT [Yan et al., 2020] first identify synthons, and then expand the synthons into reactants by either sequential generation of atoms and bonds (G2Gs), or using the Transformer architecture (RETROXPERT). The training dataset for the Transformer in [Yan et al., 2020] is augmented with incorrectly predicted synthons with the goal of learning a correction mechanism. ", + "bbox": [ + 183, + 729, + 825, + 799 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Results for NEURALSYM are taken from [Dai et al., 2019]. The authors in [Yan et al., 2020] report their performance being affected by the dataset leakage2. Thus, we use the most recent results from their website on the canonicalized dataset. For remaining baselines, we directly use the values reported in their paper. For the synthon completion module, we use the ind configuration given its better empirical performance. ", + "bbox": [ + 176, + 813, + 823, + 882 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1 Overall Performance ", + "text_level": 1, + "bbox": [ + 174, + 90, + 357, + 106 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Reaction class unknown As shown in Table 1, when the reaction class is unknown, GRAPHRETRO outperforms G2GS by $4 . 8 \\%$ and and RETROXPERT by $3 . 3 \\%$ in top-1 accuracy. Performance improvements are also seen for larger $n$ , except for $n = 5$ . Barring DUALTB, the top-1 accuracy is also better than other template-free and template-based methods. For larger $n$ , one reason for lower top- $n$ accuracies than most template-based methods is that templates already contain combinations of leaving group patterns. In contrast, our model learns to discover these during training. A second hypothesis to this end is that simply adding log-likelihood scores from edit prediction and synthon completion models may be suboptimal and bias the beam search in the direction of the more dominating term. We leave it to future work to investigate scoring functions that rank the attachment. ", + "bbox": [ + 173, + 121, + 825, + 246 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Reaction class known When the reaction class is known, GRAPHRETRO outperforms G2GS and RETROXPERT by a margin of $3 \\%$ and $2 \\%$ respectively in top-1 accuracy. GRAPHRETRO also outperforms all the template-free methods in top- $^ n$ accuracy. for GRAPHRETRO are also better than most template-based and template-free methods. When the reaction class is known, RETROSIM and GLN restrict template sets corresponding to the reaction class, thus improving performance. The increased edit prediction performance (Section 4.2) for GRAPHRETRO helps outweigh this factor, achieving comparable or better performance till $n = 5$ . ", + "bbox": [ + 174, + 271, + 825, + 368 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2 Individual Module Performance ", + "text_level": 1, + "bbox": [ + 174, + 396, + 436, + 411 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To gain more insight into the working of GRAPHRETRO, we evaluate the top- $\\boldsymbol { n }$ accuracy $\\mathbf { \\nabla } _ { n } =$ $1 , 2 , 3 , 5 )$ of edit prediction and synthon completion modules, along with corresponding ablation studies, with results shown in Table 2. ", + "bbox": [ + 174, + 426, + 825, + 468 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Edit Prediction For the edit prediction module, we compare the true edit(s) to top- $\\boldsymbol { n }$ edits predicted by the model. We also consider two ablation studies, one where we directly use the initial edit scores without updating them, and the other where we predict edits using atom-pairs instead of existing bonds and atoms. Both design choices lead to improvements in performance, as shown in Table 2. We hypothesize that the larger improvement compared to edit prediction using atom-pairs is due to the easier optimization procedure, with lesser imbalance between labels 1 and 0. ", + "bbox": [ + 173, + 493, + 825, + 577 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Synthon Completion For evaluating the synthon completion module, we first apply the true edits to obtain synthons, and compare the true leaving groups to top- $\\mathbf { \\nabla } \\cdot n$ leaving groups predicted by the model. We test the performance of both the ind and shared configurations. Both configurations perform similarly, and are able to identify $\\sim 9 7 \\%$ (close to its upper bound of $9 9 . 7 \\%$ ) of the true leaving groups in its top-5 choices, when the reaction class is known. The top-1, 3 and 5 accuracies of the synthon completion for unknown reaction classes for G2Gs are $6 1 . 1 \\%$ , $8 1 . 5 \\%$ and $8 6 . 7 \\%$ respectively, while ours are $7 5 . 6 \\%$ , $9 2 . 5 \\%$ and $9 6 . 1 \\%$ , indicating a $10 \\%$ performance improvement using a classification formulation over the generative one adopted by G2Gs. ", + "bbox": [ + 173, + 603, + 825, + 714 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/a7464e4972dde4bef7b44f291b1851eda3f6ff349da317bc359b2a73e891edfc.jpg", + "table_caption": [ + "Table 2: Performance Study of edit prediction and synthon completion modules " + ], + "table_footnote": [], + "table_body": "
SettingTop-n Accuracy (%)
Reaction class knownReaction class unknown
12351235
Edit Prediction84.692.293.794.570.885.189.592.7
- without edit score updates84.392.193.794.570.184.889.492.6
- predicting on atom pairs81.989.590.992.168.683.288.391.8
Synthon Completion (ind)77.489.594.297.675.687.492.596.1
Synthon Completion (shared)76.989.693.997.474.987.792.996.3
", + "bbox": [ + 173, + 767, + 826, + 909 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.3 Example Predictions ", + "text_level": 1, + "bbox": [ + 174, + 92, + 356, + 106 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In Figure 2, we visualize the model predictions and the ground truth for three cases. Figure 2a shows an example where the model identifies both the edits and leaving groups correctly. In Figure 2b, the correct edit is identified but the predicted leaving groups are incorrect. We hypothesize this is due to the fact that in the training set, leaving groups attaching to the carbonyl carbon $\\scriptstyle ( \\mathbf { C } = \\mathbf { O } )$ are small (e.g. -OH, - $\\mathrm { - N H _ { 2 } }$ , halides). The true leaving group in this example, however, is large. The model is unable to reason about this and predicts the small leaving group -I. In Figure 2c, the model identifies the edit and consequently the leaving group incorrectly. This highlights a limitation of our model. If the edit is predicted incorrectly, the model cannot suggest the true precursors. ", + "bbox": [ + 173, + 116, + 825, + 228 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.4 Limitations ", + "text_level": 1, + "bbox": [ + 174, + 243, + 294, + 258 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The simplified and interpretable construction of GRAPHRETRO comes with certain limitations. First, the overall performance of the model is limited by the performance of the edit prediction step. If the predicted edit is incorrect, the true reactants cannot be salvaged. This limitation is partly remedied by our model design, that allows for user intervention to correct the edit. Second, our method is reliant on atom-mapping for extracting edits and leaving groups. Extracting edits directly based on substructure matching currently suffer from false positives, and heuristics to correct for these result in correct edits in only ${ \\sim } 9 0 \\%$ of the cases. Third, our formulation assumes that we have as many synthons as reactants, which is violated in some reactions. We leave it to future work to extend the model to realize a single reactant from multiple synthons, and introduce more chemically meaningful edit correction mechanisms. ", + "bbox": [ + 173, + 268, + 825, + 407 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/9df65c90e84b75ac55b8dc08692726ee93cf492df4b05d29c79c8c792f98fccd.jpg", + "image_caption": [ + "Figure 2: Example Predictions. The true edit and incorrect edit (if any) are highlighted in green and red respectively. The true and predicted leaving groups are highlighted in blue. a. Correctly predicted example by the model. b. Correctly predicted edit but incorrectly predicted leaving groups. c. Incorrectly predicted edit and leaving group. " + ], + "image_footnote": [], + "bbox": [ + 267, + 417, + 710, + 743 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 Conclusion ", + "text_level": 1, + "bbox": [ + 174, + 838, + 299, + 856 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Previous methods for single-step retrosynthesis either restrict prediction to a template set, are insensitive to molecular graph structure or generate molecules from scratch. We address these shortcomings by introducing a graph-based semi-template-based model inspired by a chemist’s workflow, enhancing the interpretability of retrosynthesis models. Given a target molecule, we first identify synthetic building blocks (synthons) which are then realized into valid reactants, thus avoiding molecule generation from scratch. Our model outperforms previous semi-template-methods by significant margins on the benchmark dataset. Future work aims to extend the model to realize a single reactant from multiple synthons, and introduce more chemically meaningful components to improve the synergy between such tools for retrosynthesis prediction and a practitioner’s expertise. ", + "bbox": [ + 174, + 869, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 92, + 825, + 174 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgements ", + "text_level": 1, + "bbox": [ + 176, + 194, + 338, + 210 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This research was supported by the Machine Learning for Pharmaceutical Discovery and Synthesis Consortium at MIT. V.R.S. was also supported by the Zeno Karl Schindler Foundation. C.B. was supported by the Swiss National Science Foundation under the National Center of Competence in Research (NCCR) Catalysis under grant agreement 51NF40 180544. We thank the Leonhard scientific computing cluster at ETH Zürich for providing computational resources. ", + "bbox": [ + 174, + 224, + 825, + 295 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 174, + 314, + 266, + 330 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "B. Chen, T. Shen, T. S. Jaakkola, and R. Barzilay. Learning to Make Generalizable and Diverse Predictions for Retrosynthesis. In Submission, 2019. \nC. W. Coley, R. Barzilay, T. S. Jaakkola, W. H. Green, and K. F. Jensen. Prediction of Organic Reaction Outcomes Using Machine Learning. In ACS Central Science. ACS Publications, 2017a. \nC. W. Coley, L. Rogers, W. H. Green, and K. F. Jensen. Computer-Assisted Retrosynthesis Based on Molecular Similarity. ACS Central Science, 3, 2017b. \nC. W. Coley, W. Jin, L. Rogers, T. F. Jamison, T. S. Jaakkola, W. H. Green, R. Barzilay, and K. F. Jensen. 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Our model achieves a top-1 accuracy of", + "type": "text" + }, + { + "bbox": [ + 439, + 441, + 466, + 452 + ], + "score": 0.85, + "content": "5 3 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 441, + 470, + 453 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 452, + 435, + 464 + ], + "spans": [ + { + "bbox": [ + 141, + 452, + 435, + 464 + ], + "score": 1.0, + "content": "outperforming previous template-free and semi-template-based methods.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16.5, + "bbox_fs": [ + 141, + 333, + 470, + 464 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 490, + 190, + 504 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 192, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 192, + 506 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 518, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "score": 1.0, + "content": "Retrosynthesis prediction, first formalized by E. J. Corey [Corey, 1991] is a fundamental problem", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 528, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 543 + ], + "score": 1.0, + "content": "in organic synthesis that attempts to identify a series of chemical transformations for synthesizing", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 541, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 553 + ], + "score": 1.0, + "content": "a target molecule. In the single-step formulation, the task is to identify a set of reactant molecules", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "given a target. Beyond simple reactions, many practical tasks involving complex organic molecules", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 562, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 574 + ], + "score": 1.0, + "content": "are difficult even for expert chemists. As a result, substantial experimental exploration is needed", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 573, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 585 + ], + "score": 1.0, + "content": "to cover for deficiencies of analytical approaches. This has motivated interest in computer-assisted", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "score": 1.0, + "content": "retrosynthesis [Corey and Wipke, 1969], with a recent surge in machine learning methods [Chen", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 595, + 475, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 475, + 607 + ], + "score": 1.0, + "content": "et al., 2019, Coley et al., 2017b, Dai et al., 2019, Zheng et al., 2019, Genheden et al., 2020].", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 518, + 506, + 607 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 611, + 505, + 710 + ], + "lines": [ + { + "bbox": [ + 106, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "Computationally, the main challenge is how to explore the combinatorial space of reactions that can", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "score": 1.0, + "content": "yield the target molecule. Largely, previous methods for retrosynthesis prediction can be divided into", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "template-based [Coley et al., 2017b, Dai et al., 2019, Segler and Waller, 2017] and template-free", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "[Chen et al., 2019, Zheng et al., 2019] approaches. Template-based methods match a target molecule", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 654, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 669 + ], + "score": 1.0, + "content": "against a large set of templates, which are molecular subgraph patterns that highlight changes during", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "a chemical reaction. Despite their interpretability, these methods fail to generalize to new reactions.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 675, + 507, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 507, + 690 + ], + "score": 1.0, + "content": "Template-free methods bypass templates by learning a direct mapping from the SMILES [Weininger,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "score": 1.0, + "content": "1988] representations of the product to reactants. Despite their greater generalization potential, these", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 698, + 480, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 480, + 712 + ], + "score": 1.0, + "content": "methods generate reactant SMILES character by character, increasing generation complexity.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 610, + 507, + 712 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 82, + 497, + 232 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 82, + 497, + 232 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 82, + 497, + 232 + ], + "spans": [ + { + "bbox": [ + 107, + 82, + 497, + 232 + ], + "score": 0.971, + "type": "image", + "image_path": "e988f2e4a9f13ae6fa41934806006c24c87183942b7fd7f824274aee3a6237f9.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 82, + 497, + 132.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 132.0, + 497, + 182.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 182.0, + 497, + 232.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 243, + 505, + 298 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 244, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 505, + 255 + ], + "score": 1.0, + "content": "Figure 1: Overview of Our Approach. a. Edit Prediction. We train a model to learn a distribution", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 254, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 266 + ], + "score": 1.0, + "content": "over possible graph edits. In this case, the correct edit corresponds to breaking the bond marked in", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "score": 1.0, + "content": "red. Applying this edit produces two synthons. b. Synthon Completion. Another model is trained", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "score": 1.0, + "content": "to pick candidate leaving groups (blue) for each synthon from a discrete vocabulary, which are then", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 288, + 262, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 262, + 299 + ], + "score": 1.0, + "content": "attached to produce the final reactants.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 320, + 505, + 387 + ], + "lines": [ + { + "bbox": [ + 106, + 321, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 505, + 333 + ], + "score": 1.0, + "content": "Another important consideration in building retrosynthesis models is aligning model design with", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "strategies adopted by expert chemists. These strategies are influenced by fundamental properties of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 343, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 505, + 355 + ], + "score": 1.0, + "content": "chemical reactions, independent of complexity level: (i.) the product atoms are always a subset of the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 353, + 507, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 507, + 366 + ], + "score": 1.0, + "content": "reactant atoms1, and (ii.) the molecular graph topology is largely unaltered from products to reactants.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 363, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 335, + 378 + ], + "score": 1.0, + "content": "For example, in the standard retrosynthesis dataset, only", + "type": "text" + }, + { + "bbox": [ + 335, + 365, + 358, + 375 + ], + "score": 0.86, + "content": "6 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 363, + 505, + 378 + ], + "score": 1.0, + "content": "of the atoms in the product undergo", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 376, + 217, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 217, + 389 + ], + "score": 1.0, + "content": "any change in connectivity.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 392, + 505, + 512 + ], + "lines": [ + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "score": 1.0, + "content": "This consideration has received more attention in recent semi-template-based methods [Shi et al., 2020,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "Yan et al., 2020], that generate reactants from a product in two stages: (i.) first identify intermediate", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 414, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 506, + 426 + ], + "score": 1.0, + "content": "molecules called synthons, (ii.) and then complete synthons into reactants by sequential generation of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "score": 1.0, + "content": "atoms or SMILES characters.. Our model GRAPHRETRO also uses a similar workflow. However, we", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 434, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 449 + ], + "score": 1.0, + "content": "avoid sequential generation for completing synthons by instead selecting subgraphs called leaving", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "groups from a precomputed vocabulary. This vocabulary is constructed during preprocessing by", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "score": 1.0, + "content": "extracting subgraphs that differ between a synthon and the corresponding reactant. The vocabulary", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 468, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 465, + 481 + ], + "score": 1.0, + "content": "has a small size (170 for USPTO-50k) indicating remarkable redundancy, while covering", + "type": "text" + }, + { + "bbox": [ + 466, + 468, + 493, + 479 + ], + "score": 0.86, + "content": "9 9 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 468, + 506, + 481 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 479, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 492 + ], + "score": 1.0, + "content": "the test set. Operating at the level of these subgraphs greatly reduces the complexity of reactant", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 490, + 507, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 507, + 504 + ], + "score": 1.0, + "content": "generation, with improved empirical performance. This formulation also simplifies our architecture,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 502, + 483, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 483, + 513 + ], + "score": 1.0, + "content": "and makes our predictions more transparent, interpretable and amenable to manual correction.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 517, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "score": 1.0, + "content": "The benchmark dataset for evaluating retrosynthesis models is USPTO-50k [Schneider et al., 2016],", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "which consists of 50000 reactions across 10 reaction classes. The dataset contains an unexpected", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 539, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 551 + ], + "score": 1.0, + "content": "shortcut towards predicting the edit, in that the product atom with atom-mapping 1 is part of the edit", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 117, + 563 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 550, + 137, + 561 + ], + "score": 0.88, + "content": "7 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "of the cases, allowing predictions that depend on the position of the atom to overestimate", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 560, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 575 + ], + "score": 1.0, + "content": "performance. We canonicalize the product SMILES and remap the existing dataset, thereby removing", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 438, + 584 + ], + "score": 1.0, + "content": "the shortcut. On this remapped dataset, GRAPHRETRO achieves a top-1 accuracy of", + "type": "text" + }, + { + "bbox": [ + 439, + 572, + 466, + 582 + ], + "score": 0.87, + "content": "5 3 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "when the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 583, + 496, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 496, + 595 + ], + "score": 1.0, + "content": "reaction class is not known, outperforming both template-free and semi-template-based methods.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 107, + 610, + 197, + 623 + ], + "lines": [ + { + "bbox": [ + 105, + 609, + 198, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 198, + 625 + ], + "score": 1.0, + "content": "2 Related Work", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 635, + 504, + 658 + ], + "lines": [ + { + "bbox": [ + 105, + 634, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 505, + 649 + ], + "score": 1.0, + "content": "Retrosynthesis Prediction Existing machine learning methods for retrosynthesis prediction can", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 647, + 467, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 647, + 467, + 659 + ], + "score": 1.0, + "content": "be divided into template-based, template-free and recent semi-template-based approaches.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 111, + 668, + 505, + 702 + ], + "lines": [ + { + "bbox": [ + 111, + 668, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 111, + 668, + 506, + 681 + ], + "score": 1.0, + "content": "Template-Based: Templates are either hand-crafted by experts [Hartenfeller et al., 2011, Szymkuc´", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 120, + 679, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 120, + 679, + 506, + 691 + ], + "score": 1.0, + "content": "et al., 2016], or extracted algorithmically from large databases Coley et al. 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Edit Prediction. We train a model to learn a distribution", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 254, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 266 + ], + "score": 1.0, + "content": "over possible graph edits. In this case, the correct edit corresponds to breaking the bond marked in", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "score": 1.0, + "content": "red. Applying this edit produces two synthons. b. Synthon Completion. Another model is trained", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "score": 1.0, + "content": "to pick candidate leaving groups (blue) for each synthon from a discrete vocabulary, which are then", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 288, + 262, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 262, + 299 + ], + "score": 1.0, + "content": "attached to produce the final reactants.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 320, + 505, + 387 + ], + "lines": [ + { + "bbox": [ + 106, + 321, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 505, + 333 + ], + "score": 1.0, + "content": "Another important consideration in building retrosynthesis models is aligning model design with", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "strategies adopted by expert chemists. These strategies are influenced by fundamental properties of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 343, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 505, + 355 + ], + "score": 1.0, + "content": "chemical reactions, independent of complexity level: (i.) the product atoms are always a subset of the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 353, + 507, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 507, + 366 + ], + "score": 1.0, + "content": "reactant atoms1, and (ii.) the molecular graph topology is largely unaltered from products to reactants.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 363, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 335, + 378 + ], + "score": 1.0, + "content": "For example, in the standard retrosynthesis dataset, only", + "type": "text" + }, + { + "bbox": [ + 335, + 365, + 358, + 375 + ], + "score": 0.86, + "content": "6 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 363, + 505, + 378 + ], + "score": 1.0, + "content": "of the atoms in the product undergo", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 376, + 217, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 217, + 389 + ], + "score": 1.0, + "content": "any change in connectivity.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 321, + 507, + 389 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 392, + 505, + 512 + ], + "lines": [ + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "score": 1.0, + "content": "This consideration has received more attention in recent semi-template-based methods [Shi et al., 2020,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "Yan et al., 2020], that generate reactants from a product in two stages: (i.) first identify intermediate", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 414, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 506, + 426 + ], + "score": 1.0, + "content": "molecules called synthons, (ii.) and then complete synthons into reactants by sequential generation of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "score": 1.0, + "content": "atoms or SMILES characters.. Our model GRAPHRETRO also uses a similar workflow. However, we", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 434, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 449 + ], + "score": 1.0, + "content": "avoid sequential generation for completing synthons by instead selecting subgraphs called leaving", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "groups from a precomputed vocabulary. This vocabulary is constructed during preprocessing by", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "score": 1.0, + "content": "extracting subgraphs that differ between a synthon and the corresponding reactant. The vocabulary", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 468, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 465, + 481 + ], + "score": 1.0, + "content": "has a small size (170 for USPTO-50k) indicating remarkable redundancy, while covering", + "type": "text" + }, + { + "bbox": [ + 466, + 468, + 493, + 479 + ], + "score": 0.86, + "content": "9 9 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 468, + 506, + 481 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 479, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 492 + ], + "score": 1.0, + "content": "the test set. Operating at the level of these subgraphs greatly reduces the complexity of reactant", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 490, + 507, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 507, + 504 + ], + "score": 1.0, + "content": "generation, with improved empirical performance. This formulation also simplifies our architecture,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 502, + 483, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 483, + 513 + ], + "score": 1.0, + "content": "and makes our predictions more transparent, interpretable and amenable to manual correction.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 391, + 507, + 513 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 517, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "score": 1.0, + "content": "The benchmark dataset for evaluating retrosynthesis models is USPTO-50k [Schneider et al., 2016],", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "which consists of 50000 reactions across 10 reaction classes. The dataset contains an unexpected", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 539, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 551 + ], + "score": 1.0, + "content": "shortcut towards predicting the edit, in that the product atom with atom-mapping 1 is part of the edit", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 117, + 563 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 550, + 137, + 561 + ], + "score": 0.88, + "content": "7 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "of the cases, allowing predictions that depend on the position of the atom to overestimate", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 560, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 575 + ], + "score": 1.0, + "content": "performance. We canonicalize the product SMILES and remap the existing dataset, thereby removing", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 438, + 584 + ], + "score": 1.0, + "content": "the shortcut. 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[2017a], Law et al.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 120, + 689, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 120, + 689, + 505, + 703 + ], + "score": 1.0, + "content": "[2009]. Exhaustively applying large template sets is expensive due to the involved subgraph", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 119, + 71, + 506, + 87 + ], + "spans": [ + { + "bbox": [ + 119, + 71, + 506, + 87 + ], + "score": 1.0, + "content": "matching procedure. Template-based methods therefore utilize different ways of prioritizing", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 120, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 120, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "templates, by either learning a conditional distribution over the template set [Segler and Waller,", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 119, + 93, + 507, + 108 + ], + "spans": [ + { + "bbox": [ + 119, + 93, + 507, + 108 + ], + "score": 1.0, + "content": "2017], ranking templates based on molecular similarities to precedent reactions [Coley et al.,", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 120, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 120, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "2017b] or directly modelling the joint distribution of templates and reactants using logic variables", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 120, + 115, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 120, + 115, + 506, + 129 + ], + "score": 1.0, + "content": "[Dai et al., 2019]. Despite their interpretability, these methods fail to generalize outside their rule", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 120, + 128, + 138, + 140 + ], + "spans": [ + { + "bbox": [ + 120, + 128, + 138, + 140 + ], + "score": 1.0, + "content": "set.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 36, + "bbox_fs": [ + 111, + 668, + 506, + 703 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 121, + 73, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 119, + 71, + 506, + 87 + ], + "spans": [ + { + "bbox": [ + 119, + 71, + 506, + 87 + ], + "score": 1.0, + "content": "matching procedure. Template-based methods therefore utilize different ways of prioritizing", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 120, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 120, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "templates, by either learning a conditional distribution over the template set [Segler and Waller,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 119, + 93, + 507, + 108 + ], + "spans": [ + { + "bbox": [ + 119, + 93, + 507, + 108 + ], + "score": 1.0, + "content": "2017], ranking templates based on molecular similarities to precedent reactions [Coley et al.,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 120, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 120, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "2017b] or directly modelling the joint distribution of templates and reactants using logic variables", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 120, + 115, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 120, + 115, + 506, + 129 + ], + "score": 1.0, + "content": "[Dai et al., 2019]. Despite their interpretability, these methods fail to generalize outside their rule", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 120, + 128, + 138, + 140 + ], + "spans": [ + { + "bbox": [ + 120, + 128, + 138, + 140 + ], + "score": 1.0, + "content": "set.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 112, + 145, + 505, + 255 + ], + "lines": [ + { + "bbox": [ + 111, + 145, + 506, + 158 + ], + "spans": [ + { + "bbox": [ + 111, + 145, + 506, + 158 + ], + "score": 1.0, + "content": "Template-Free: Template-free methods [Liu et al., 2017, Zheng et al., 2019, Chen et al., 2019]", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 120, + 156, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 120, + 156, + 506, + 169 + ], + "score": 1.0, + "content": "learn a direct transformation from products to reactants using architectures from neural machine", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 120, + 167, + 507, + 180 + ], + "spans": [ + { + "bbox": [ + 120, + 167, + 507, + 180 + ], + "score": 1.0, + "content": "translation and a string based representation of molecules called SMILES [Weininger, 1988].", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 120, + 178, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 120, + 178, + 506, + 190 + ], + "score": 1.0, + "content": "Linearizing molecules as strings does not utilize the inherently rich chemical structure. In addition,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 120, + 189, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 120, + 189, + 506, + 202 + ], + "score": 1.0, + "content": "the reactant SMILES are generated from scratch, character by character. Attempts have been", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 120, + 200, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 120, + 200, + 506, + 213 + ], + "score": 1.0, + "content": "made to improve validity by adding a syntax correcter [Zheng et al., 2019] and a mixture model", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 120, + 211, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 120, + 211, + 505, + 223 + ], + "score": 1.0, + "content": "to improve diversity of suggestions [Chen et al., 2019], but the performance remains worse than", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 120, + 222, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 120, + 222, + 505, + 235 + ], + "score": 1.0, + "content": "[Dai et al., 2019] on the standard retrosynthesis dataset. Sun et al. [2021] formulate retrosynthesis", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 120, + 233, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 120, + 233, + 506, + 245 + ], + "score": 1.0, + "content": "using energy-based models, with additional parameterizations and loss terms to enforce the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 120, + 242, + 473, + 257 + ], + "spans": [ + { + "bbox": [ + 120, + 242, + 473, + 257 + ], + "score": 1.0, + "content": "duality between forward (reaction prediction) and backward (retrosynthesis) prediction.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 112, + 262, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 110, + 261, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 110, + 261, + 506, + 273 + ], + "score": 1.0, + "content": "Semi-Template-Based: Our work is closely related to recently proposed semi-template-based", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 121, + 273, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 121, + 273, + 505, + 284 + ], + "score": 1.0, + "content": "methods [Shi et al., 2020, Yan et al., 2020], which first identify synthons and then expand", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 120, + 284, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 120, + 284, + 505, + 296 + ], + "score": 1.0, + "content": "synthons into reactants through sequential generation using either a graph generative model [Shi", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 120, + 295, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 120, + 295, + 506, + 307 + ], + "score": 1.0, + "content": "et al., 2020] or a Transformer [Yan et al., 2020]. To reduce the complexity of reactant generation,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 119, + 305, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 119, + 305, + 506, + 318 + ], + "score": 1.0, + "content": "we instead complete synthons using subgraphs called leaving groups selected from a precomputed", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 120, + 316, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 120, + 316, + 506, + 329 + ], + "score": 1.0, + "content": "vocabulary. This allows us to view synthon completion as a classification problem instead of a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 119, + 326, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 119, + 326, + 505, + 340 + ], + "score": 1.0, + "content": "generative one. We also utilize the dependency graph between possible edits, and update edit", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 120, + 338, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 120, + 338, + 506, + 352 + ], + "score": 1.0, + "content": "predictions using a message passing network (MPN) [Gilmer et al., 2017] on this graph. Both", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 120, + 348, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 120, + 348, + 232, + 362 + ], + "score": 1.0, + "content": "innovations together yield a", + "type": "text" + }, + { + "bbox": [ + 232, + 349, + 254, + 360 + ], + "score": 0.88, + "content": "4 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 348, + 271, + 362 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 271, + 349, + 293, + 360 + ], + "score": 0.85, + "content": "3 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 348, + 505, + 362 + ], + "score": 1.0, + "content": "performance improvement respectively over previous", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 120, + 360, + 245, + 372 + ], + "spans": [ + { + "bbox": [ + 120, + 360, + 245, + 372 + ], + "score": 1.0, + "content": "semi-template-based methods.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 386, + 505, + 486 + ], + "lines": [ + { + "bbox": [ + 106, + 385, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 506, + 400 + ], + "score": 1.0, + "content": "Reaction Center Identification The reaction center covers a small number of participating atoms", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 396, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 411 + ], + "score": 1.0, + "content": "involved in the reaction. Our work is also related to models that predict reaction outcomes by learning", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 407, + 507, + 422 + ], + "spans": [ + { + "bbox": [ + 104, + 407, + 507, + 422 + ], + "score": 1.0, + "content": "to rank atom pairs based on their likelihood to be in the reaction center [Coley et al., 2019, Jin et al.,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "2017]. The task of identifying the reaction center is related to the step of deriving the synthons in our", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 429, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 444 + ], + "score": 1.0, + "content": "formulation. Our work departs from [Coley et al., 2019, Jin et al., 2017] as we utilize the property", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 442, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 263, + 453 + ], + "score": 1.0, + "content": "that new bond formations occur rarely", + "type": "text" + }, + { + "bbox": [ + 263, + 442, + 297, + 452 + ], + "score": 0.89, + "content": "( \\sim 0 . 1 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 442, + 505, + 453 + ], + "score": 1.0, + "content": "from products to synthons, allowing us to predict a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 418, + 465 + ], + "score": 1.0, + "content": "score only for existing bonds and atoms and reduce prediction complexity from", + "type": "text" + }, + { + "bbox": [ + 418, + 452, + 448, + 464 + ], + "score": 0.92, + "content": "O ( N ^ { \\bar { 2 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 451, + 460, + 465 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 460, + 452, + 485, + 464 + ], + "score": 0.91, + "content": "O ( N )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 451, + 506, + 465 + ], + "score": 1.0, + "content": ". We", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 462, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 477 + ], + "score": 1.0, + "content": "also utilize the dependency graph between possible edits, and update edit predictions using a MPN", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 474, + 163, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 163, + 486 + ], + "score": 1.0, + "content": "on this graph.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "Utilizing Substructures Substructures have been utilized in various tasks from sentence generation", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 511, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 525 + ], + "score": 1.0, + "content": "by fusing phrases to molecule generation and optimization [Jin et al., 2018, 2020]. Our work is", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "closely related to [Jin et al., 2020] which uses precomputed substructures as building blocks for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 533, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 547 + ], + "score": 1.0, + "content": "property-conditioned molecule generation. However, instead of precomputing, synthons —analogous", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 543, + 377, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 377, + 558 + ], + "score": 1.0, + "content": "building blocks for reactants— are indirectly learnt during training.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37 + }, + { + "type": "title", + "bbox": [ + 107, + 574, + 196, + 588 + ], + "lines": [ + { + "bbox": [ + 104, + 571, + 198, + 592 + ], + "spans": [ + { + "bbox": [ + 104, + 571, + 198, + 592 + ], + "score": 1.0, + "content": "3 Model Design", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 601, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 506, + 615 + ], + "score": 1.0, + "content": "Our approach leverages the property that graph topology is largely unaltered from products to", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "score": 1.0, + "content": "reactants. To achieve this, we first derive suitable building blocks from the product called synthons,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 622, + 507, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 507, + 637 + ], + "score": 1.0, + "content": "and then complete them into valid reactants by adding specific functionalities called leaving groups.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 634, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 506, + 647 + ], + "score": 1.0, + "content": "These derivations, called edits, are characterized by modifications to bonds or hydrogen counts on", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "atoms. We first train a neural network to predict a score for possible edits (Section 3.1). The edit with", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 655, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 670 + ], + "score": 1.0, + "content": "the highest score is then applied to the product to obtain synthons. Since the number of unique leaving", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "groups are small, we model leaving group selection as a classification problem over a precomputed", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "vocabulary (Section 3.2). To produce candidate reactants, we attach the predicted leaving group to", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "the corresponding synthon through chemically constrained rules. The overall process is outlined", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "in Figure 1. Before describing the two modules, we introduce relevant preliminaries that set the", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 710, + 280, + 724 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 280, + 724 + ], + "score": 1.0, + "content": "background for the remainder of the paper.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 46 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 740, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 740, + 309, + 752 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 121, + 73, + 505, + 138 + ], + "lines": [], + "index": 2.5, + "bbox_fs": [ + 119, + 71, + 507, + 140 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 112, + 145, + 505, + 255 + ], + "lines": [ + { + "bbox": [ + 111, + 145, + 506, + 158 + ], + "spans": [ + { + "bbox": [ + 111, + 145, + 506, + 158 + ], + "score": 1.0, + "content": "Template-Free: Template-free methods [Liu et al., 2017, Zheng et al., 2019, Chen et al., 2019]", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 120, + 156, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 120, + 156, + 506, + 169 + ], + "score": 1.0, + "content": "learn a direct transformation from products to reactants using architectures from neural machine", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 120, + 167, + 507, + 180 + ], + "spans": [ + { + "bbox": [ + 120, + 167, + 507, + 180 + ], + "score": 1.0, + "content": "translation and a string based representation of molecules called SMILES [Weininger, 1988].", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 120, + 178, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 120, + 178, + 506, + 190 + ], + "score": 1.0, + "content": "Linearizing molecules as strings does not utilize the inherently rich chemical structure. In addition,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 120, + 189, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 120, + 189, + 506, + 202 + ], + "score": 1.0, + "content": "the reactant SMILES are generated from scratch, character by character. Attempts have been", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 120, + 200, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 120, + 200, + 506, + 213 + ], + "score": 1.0, + "content": "made to improve validity by adding a syntax correcter [Zheng et al., 2019] and a mixture model", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 120, + 211, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 120, + 211, + 505, + 223 + ], + "score": 1.0, + "content": "to improve diversity of suggestions [Chen et al., 2019], but the performance remains worse than", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 120, + 222, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 120, + 222, + 505, + 235 + ], + "score": 1.0, + "content": "[Dai et al., 2019] on the standard retrosynthesis dataset. Sun et al. [2021] formulate retrosynthesis", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 120, + 233, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 120, + 233, + 506, + 245 + ], + "score": 1.0, + "content": "using energy-based models, with additional parameterizations and loss terms to enforce the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 120, + 242, + 473, + 257 + ], + "spans": [ + { + "bbox": [ + 120, + 242, + 473, + 257 + ], + "score": 1.0, + "content": "duality between forward (reaction prediction) and backward (retrosynthesis) prediction.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10.5, + "bbox_fs": [ + 111, + 145, + 507, + 257 + ] + }, + { + "type": "text", + "bbox": [ + 112, + 262, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 110, + 261, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 110, + 261, + 506, + 273 + ], + "score": 1.0, + "content": "Semi-Template-Based: Our work is closely related to recently proposed semi-template-based", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 121, + 273, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 121, + 273, + 505, + 284 + ], + "score": 1.0, + "content": "methods [Shi et al., 2020, Yan et al., 2020], which first identify synthons and then expand", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 120, + 284, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 120, + 284, + 505, + 296 + ], + "score": 1.0, + "content": "synthons into reactants through sequential generation using either a graph generative model [Shi", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 120, + 295, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 120, + 295, + 506, + 307 + ], + "score": 1.0, + "content": "et al., 2020] or a Transformer [Yan et al., 2020]. To reduce the complexity of reactant generation,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 119, + 305, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 119, + 305, + 506, + 318 + ], + "score": 1.0, + "content": "we instead complete synthons using subgraphs called leaving groups selected from a precomputed", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 120, + 316, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 120, + 316, + 506, + 329 + ], + "score": 1.0, + "content": "vocabulary. This allows us to view synthon completion as a classification problem instead of a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 119, + 326, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 119, + 326, + 505, + 340 + ], + "score": 1.0, + "content": "generative one. We also utilize the dependency graph between possible edits, and update edit", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 120, + 338, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 120, + 338, + 506, + 352 + ], + "score": 1.0, + "content": "predictions using a message passing network (MPN) [Gilmer et al., 2017] on this graph. Both", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 120, + 348, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 120, + 348, + 232, + 362 + ], + "score": 1.0, + "content": "innovations together yield a", + "type": "text" + }, + { + "bbox": [ + 232, + 349, + 254, + 360 + ], + "score": 0.88, + "content": "4 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 348, + 271, + 362 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 271, + 349, + 293, + 360 + ], + "score": 0.85, + "content": "3 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 348, + 505, + 362 + ], + "score": 1.0, + "content": "performance improvement respectively over previous", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 120, + 360, + 245, + 372 + ], + "spans": [ + { + "bbox": [ + 120, + 360, + 245, + 372 + ], + "score": 1.0, + "content": "semi-template-based methods.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20.5, + "bbox_fs": [ + 110, + 261, + 506, + 372 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 386, + 505, + 486 + ], + "lines": [ + { + "bbox": [ + 106, + 385, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 506, + 400 + ], + "score": 1.0, + "content": "Reaction Center Identification The reaction center covers a small number of participating atoms", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 396, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 411 + ], + "score": 1.0, + "content": "involved in the reaction. Our work is also related to models that predict reaction outcomes by learning", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 407, + 507, + 422 + ], + "spans": [ + { + "bbox": [ + 104, + 407, + 507, + 422 + ], + "score": 1.0, + "content": "to rank atom pairs based on their likelihood to be in the reaction center [Coley et al., 2019, Jin et al.,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "2017]. The task of identifying the reaction center is related to the step of deriving the synthons in our", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 429, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 444 + ], + "score": 1.0, + "content": "formulation. Our work departs from [Coley et al., 2019, Jin et al., 2017] as we utilize the property", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 442, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 263, + 453 + ], + "score": 1.0, + "content": "that new bond formations occur rarely", + "type": "text" + }, + { + "bbox": [ + 263, + 442, + 297, + 452 + ], + "score": 0.89, + "content": "( \\sim 0 . 1 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 442, + 505, + 453 + ], + "score": 1.0, + "content": "from products to synthons, allowing us to predict a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 418, + 465 + ], + "score": 1.0, + "content": "score only for existing bonds and atoms and reduce prediction complexity from", + "type": "text" + }, + { + "bbox": [ + 418, + 452, + 448, + 464 + ], + "score": 0.92, + "content": "O ( N ^ { \\bar { 2 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 451, + 460, + 465 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 460, + 452, + 485, + 464 + ], + "score": 0.91, + "content": "O ( N )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 451, + 506, + 465 + ], + "score": 1.0, + "content": ". We", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 462, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 477 + ], + "score": 1.0, + "content": "also utilize the dependency graph between possible edits, and update edit predictions using a MPN", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 474, + 163, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 163, + 486 + ], + "score": 1.0, + "content": "on this graph.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30, + "bbox_fs": [ + 104, + 385, + 507, + 486 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "Utilizing Substructures Substructures have been utilized in various tasks from sentence generation", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 511, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 525 + ], + "score": 1.0, + "content": "by fusing phrases to molecule generation and optimization [Jin et al., 2018, 2020]. Our work is", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "closely related to [Jin et al., 2020] which uses precomputed substructures as building blocks for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 533, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 547 + ], + "score": 1.0, + "content": "property-conditioned molecule generation. However, instead of precomputing, synthons —analogous", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 543, + 377, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 377, + 558 + ], + "score": 1.0, + "content": "building blocks for reactants— are indirectly learnt during training.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 500, + 506, + 558 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 574, + 196, + 588 + ], + "lines": [ + { + "bbox": [ + 104, + 571, + 198, + 592 + ], + "spans": [ + { + "bbox": [ + 104, + 571, + 198, + 592 + ], + "score": 1.0, + "content": "3 Model Design", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 601, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 506, + 615 + ], + "score": 1.0, + "content": "Our approach leverages the property that graph topology is largely unaltered from products to", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "score": 1.0, + "content": "reactants. To achieve this, we first derive suitable building blocks from the product called synthons,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 622, + 507, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 507, + 637 + ], + "score": 1.0, + "content": "and then complete them into valid reactants by adding specific functionalities called leaving groups.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 634, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 506, + 647 + ], + "score": 1.0, + "content": "These derivations, called edits, are characterized by modifications to bonds or hydrogen counts on", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "atoms. We first train a neural network to predict a score for possible edits (Section 3.1). The edit with", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 655, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 670 + ], + "score": 1.0, + "content": "the highest score is then applied to the product to obtain synthons. Since the number of unique leaving", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "groups are small, we model leaving group selection as a classification problem over a precomputed", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "vocabulary (Section 3.2). To produce candidate reactants, we attach the predicted leaving group to", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "the corresponding synthon through chemically constrained rules. The overall process is outlined", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "in Figure 1. Before describing the two modules, we introduce relevant preliminaries that set the", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 710, + 280, + 724 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 280, + 724 + ], + "score": 1.0, + "content": "background for the remainder of the paper.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 601, + 507, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 140 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 316, + 85 + ], + "score": 1.0, + "content": "Retrosynthesis Prediction A retrosynthesis pair", + "type": "text" + }, + { + "bbox": [ + 316, + 73, + 325, + 83 + ], + "score": 0.81, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "is described by a pair of molecular graphs", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 139, + 96 + ], + "score": 0.92, + "content": "( \\mathcal { G } _ { p } , \\mathcal { G } _ { r } )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 83, + 168, + 97 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 169, + 84, + 180, + 96 + ], + "score": 0.89, + "content": "\\mathcal { G } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 83, + 260, + 97 + ], + "score": 1.0, + "content": "are the products and", + "type": "text" + }, + { + "bbox": [ + 261, + 84, + 272, + 95 + ], + "score": 0.88, + "content": "\\mathcal { G } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 83, + 459, + 97 + ], + "score": 1.0, + "content": "the reactants. A molecular graph is described as", + "type": "text" + }, + { + "bbox": [ + 459, + 83, + 505, + 96 + ], + "score": 0.93, + "content": "\\mathcal { G } = \\mathbf { \\bar { \\rho } } ( \\mathcal { V } , \\mathcal { E } )", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 151, + 106 + ], + "score": 1.0, + "content": "with atoms", + "type": "text" + }, + { + "bbox": [ + 151, + 95, + 160, + 104 + ], + "score": 0.79, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 95, + 237, + 106 + ], + "score": 1.0, + "content": "as nodes and bonds", + "type": "text" + }, + { + "bbox": [ + 238, + 95, + 245, + 104 + ], + "score": 0.82, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "as edges. Prior work has focused on the single product case, while", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 104, + 506, + 121 + ], + "spans": [ + { + "bbox": [ + 104, + 104, + 335, + 121 + ], + "score": 1.0, + "content": "reactants can have multiple connected components, i.e.", + "type": "text" + }, + { + "bbox": [ + 336, + 105, + 399, + 118 + ], + "score": 0.93, + "content": "\\mathcal { G } _ { r } = \\{ \\mathcal { G } _ { r _ { c } } \\} _ { c = 1 } ^ { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 104, + 506, + 121 + ], + "score": 1.0, + "content": ". Retrosynthesis pairs are", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "atom-mapped so that each product atom has a unique corresponding reactant atom. The retrosynthesis", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 123, + 269, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 188, + 142 + ], + "score": 1.0, + "content": "task then, is to infer", + "type": "text" + }, + { + "bbox": [ + 188, + 127, + 227, + 140 + ], + "score": 0.93, + "content": "\\{ \\mathcal { G } _ { r _ { c } } \\} _ { c = 1 } ^ { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 123, + 252, + 142 + ], + "score": 1.0, + "content": "given", + "type": "text" + }, + { + "bbox": [ + 253, + 127, + 264, + 140 + ], + "score": 0.89, + "content": "\\mathcal { G } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 123, + 269, + 142 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 150, + 505, + 206 + ], + "lines": [ + { + "bbox": [ + 105, + 149, + 506, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 267, + 164 + ], + "score": 1.0, + "content": "Edits Edits consist of (i.) atom pairs", + "type": "text" + }, + { + "bbox": [ + 267, + 151, + 308, + 163 + ], + "score": 0.94, + "content": "\\left\\{ \\left( a _ { i } , a _ { j } \\right) \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 149, + 506, + 164 + ], + "score": 1.0, + "content": "where the bond type changes from products to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 162, + 504, + 174 + ], + "spans": [ + { + "bbox": [ + 106, + 162, + 210, + 174 + ], + "score": 1.0, + "content": "reactants, and (ii.) atoms", + "type": "text" + }, + { + "bbox": [ + 211, + 162, + 230, + 174 + ], + "score": 0.93, + "content": "\\left\\{ { a } _ { i } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 162, + 504, + 174 + ], + "score": 1.0, + "content": "where the number of hydrogens attached to the atom change from", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 172, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 313, + 186 + ], + "score": 1.0, + "content": "products to reactants. We denote the set of edits by", + "type": "text" + }, + { + "bbox": [ + 314, + 173, + 323, + 183 + ], + "score": 0.8, + "content": "E", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 172, + 506, + 186 + ], + "score": 1.0, + "content": ". Since retrosynthesis pairs in the training set", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 183, + 505, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 505, + 197 + ], + "score": 1.0, + "content": "are atom-mapped, edits can be automatically identified by comparing the atoms and atom pairs in the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 194, + 318, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 318, + 207 + ], + "score": 1.0, + "content": "product to their corresponding reactant counterparts.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 218, + 505, + 285 + ], + "lines": [ + { + "bbox": [ + 106, + 218, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 303, + 231 + ], + "score": 1.0, + "content": "Synthons and Leaving Groups Applying edits", + "type": "text" + }, + { + "bbox": [ + 303, + 219, + 312, + 228 + ], + "score": 0.84, + "content": "E", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 218, + 368, + 231 + ], + "score": 1.0, + "content": "to the product", + "type": "text" + }, + { + "bbox": [ + 368, + 219, + 380, + 231 + ], + "score": 0.89, + "content": "\\mathcal { G } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 218, + 505, + 231 + ], + "score": 1.0, + "content": "results in incomplete molecules", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 229, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 505, + 242 + ], + "score": 1.0, + "content": "called synthons. Synthons are analogous to rationales or building blocks, which are expanded into", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 239, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 505, + 254 + ], + "score": 1.0, + "content": "valid reactants by adding specific functionalities called leaving groups that are responsible for its", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 251, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 241, + 264 + ], + "score": 1.0, + "content": "reactivity. We denote synthons by", + "type": "text" + }, + { + "bbox": [ + 242, + 252, + 253, + 262 + ], + "score": 0.89, + "content": "\\mathcal { G } _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 251, + 343, + 264 + ], + "score": 1.0, + "content": "and leaving groups by", + "type": "text" + }, + { + "bbox": [ + 344, + 252, + 353, + 262 + ], + "score": 0.86, + "content": "\\mathcal { G } _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 251, + 505, + 264 + ], + "score": 1.0, + "content": ". We further assume that synthons and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 259, + 504, + 278 + ], + "spans": [ + { + "bbox": [ + 104, + 259, + 441, + 278 + ], + "score": 1.0, + "content": "leaving groups have the same number of connected components as the reactants, i.e", + "type": "text" + }, + { + "bbox": [ + 441, + 261, + 504, + 275 + ], + "score": 0.92, + "content": "\\mathcal { G } _ { s } \\doteq \\{ \\mathcal { G } _ { s _ { c } } \\} _ { c = 1 } ^ { C }", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 269, + 442, + 288 + ], + "spans": [ + { + "bbox": [ + 104, + 269, + 123, + 288 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 272, + 183, + 285 + ], + "score": 0.93, + "content": "\\mathcal { G } _ { l } = \\{ \\mathcal { G } _ { l _ { c } } ^ { \\star } \\} _ { c = 1 } ^ { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 269, + 294, + 288 + ], + "score": 1.0, + "content": ". This assumption holds for", + "type": "text" + }, + { + "bbox": [ + 294, + 273, + 326, + 284 + ], + "score": 0.86, + "content": "9 9 . 9 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 269, + 442, + 288 + ], + "score": 1.0, + "content": "reactions in the training set.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 105, + 289, + 504, + 312 + ], + "lines": [ + { + "bbox": [ + 105, + 288, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 404, + 303 + ], + "score": 1.0, + "content": "Formally, our model generates reactants by first predicting the set of edits", + "type": "text" + }, + { + "bbox": [ + 405, + 290, + 414, + 299 + ], + "score": 0.83, + "content": "E", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 288, + 474, + 303 + ], + "score": 1.0, + "content": "that transform", + "type": "text" + }, + { + "bbox": [ + 474, + 290, + 486, + 302 + ], + "score": 0.89, + "content": "\\mathcal { G } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 288, + 506, + 303 + ], + "score": 1.0, + "content": "into", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 107, + 301, + 118, + 311 + ], + "score": 0.86, + "content": "\\mathcal { G } _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 300, + 275, + 313 + ], + "score": 1.0, + "content": ", followed by predicting a leaving group", + "type": "text" + }, + { + "bbox": [ + 276, + 301, + 289, + 312 + ], + "score": 0.9, + "content": "\\mathcal { G } _ { l _ { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 300, + 390, + 313 + ], + "score": 1.0, + "content": "to attach to each synthon", + "type": "text" + }, + { + "bbox": [ + 390, + 301, + 405, + 312 + ], + "score": 0.9, + "content": "\\mathcal { G } _ { s _ { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 300, + 506, + 313 + ], + "score": 1.0, + "content": ". 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This choice is motivated", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 195, + 437 + ], + "score": 1.0, + "content": "by the low frequency", + "type": "text" + }, + { + "bbox": [ + 195, + 425, + 229, + 435 + ], + "score": 0.93, + "content": "( \\sim 0 . 1 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "of new bond formations in the training set examples. 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For a bond", + "type": "text" + }, + { + "bbox": [ + 241, + 678, + 265, + 690 + ], + "score": 0.9, + "content": "( u , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 677, + 377, + 691 + ], + "score": 1.0, + "content": ", we define its representation", + "type": "text" + }, + { + "bbox": [ + 378, + 678, + 502, + 690 + ], + "score": 0.9, + "content": "\\mathbf { c } _ { u v } = ( \\operatorname { A B S } ( \\mathbf { c } _ { u } , \\mathbf { c } _ { v } ) | | \\mathbf { c } _ { u } + \\mathbf { c } _ { v } )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 677, + 506, + 691 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 688, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 277, + 702 + ], + "score": 1.0, + "content": "where ABS denotes absolute difference and", + "type": "text" + }, + { + "bbox": [ + 278, + 690, + 284, + 701 + ], + "score": 0.85, + "content": "| |", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 688, + 505, + 702 + ], + "score": 1.0, + "content": "refers to concatenation. This ensures our representations", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "are permutation invariant. These representations are then used to predict atom and bond edit scores", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 711, + 259, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 259, + 723 + ], + "score": 1.0, + "content": "using corresponding neural networks,", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 11, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 140 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 316, + 85 + ], + "score": 1.0, + "content": "Retrosynthesis Prediction A retrosynthesis pair", + "type": "text" + }, + { + "bbox": [ + 316, + 73, + 325, + 83 + ], + "score": 0.81, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "is described by a pair of molecular graphs", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 139, + 96 + ], + "score": 0.92, + "content": "( \\mathcal { G } _ { p } , \\mathcal { G } _ { r } )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 83, + 168, + 97 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 169, + 84, + 180, + 96 + ], + "score": 0.89, + "content": "\\mathcal { G } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 83, + 260, + 97 + ], + "score": 1.0, + "content": "are the products and", + "type": "text" + }, + { + "bbox": [ + 261, + 84, + 272, + 95 + ], + "score": 0.88, + "content": "\\mathcal { G } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 83, + 459, + 97 + ], + "score": 1.0, + "content": "the reactants. A molecular graph is described as", + "type": "text" + }, + { + "bbox": [ + 459, + 83, + 505, + 96 + ], + "score": 0.93, + "content": "\\mathcal { G } = \\mathbf { \\bar { \\rho } } ( \\mathcal { V } , \\mathcal { E } )", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 151, + 106 + ], + "score": 1.0, + "content": "with atoms", + "type": "text" + }, + { + "bbox": [ + 151, + 95, + 160, + 104 + ], + "score": 0.79, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 95, + 237, + 106 + ], + "score": 1.0, + "content": "as nodes and bonds", + "type": "text" + }, + { + "bbox": [ + 238, + 95, + 245, + 104 + ], + "score": 0.82, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "as edges. Prior work has focused on the single product case, while", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 104, + 506, + 121 + ], + "spans": [ + { + "bbox": [ + 104, + 104, + 335, + 121 + ], + "score": 1.0, + "content": "reactants can have multiple connected components, i.e.", + "type": "text" + }, + { + "bbox": [ + 336, + 105, + 399, + 118 + ], + "score": 0.93, + "content": "\\mathcal { G } _ { r } = \\{ \\mathcal { G } _ { r _ { c } } \\} _ { c = 1 } ^ { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 104, + 506, + 121 + ], + "score": 1.0, + "content": ". Retrosynthesis pairs are", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "atom-mapped so that each product atom has a unique corresponding reactant atom. The retrosynthesis", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 123, + 269, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 188, + 142 + ], + "score": 1.0, + "content": "task then, is to infer", + "type": "text" + }, + { + "bbox": [ + 188, + 127, + 227, + 140 + ], + "score": 0.93, + "content": "\\{ \\mathcal { G } _ { r _ { c } } \\} _ { c = 1 } ^ { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 123, + 252, + 142 + ], + "score": 1.0, + "content": "given", + "type": "text" + }, + { + "bbox": [ + 253, + 127, + 264, + 140 + ], + "score": 0.89, + "content": "\\mathcal { G } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 123, + 269, + 142 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 104, + 72, + 506, + 142 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 150, + 505, + 206 + ], + "lines": [ + { + "bbox": [ + 105, + 149, + 506, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 267, + 164 + ], + "score": 1.0, + "content": "Edits Edits consist of (i.) atom pairs", + "type": "text" + }, + { + "bbox": [ + 267, + 151, + 308, + 163 + ], + "score": 0.94, + "content": "\\left\\{ \\left( a _ { i } , a _ { j } \\right) \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 149, + 506, + 164 + ], + "score": 1.0, + "content": "where the bond type changes from products to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 162, + 504, + 174 + ], + "spans": [ + { + "bbox": [ + 106, + 162, + 210, + 174 + ], + "score": 1.0, + "content": "reactants, and (ii.) atoms", + "type": "text" + }, + { + "bbox": [ + 211, + 162, + 230, + 174 + ], + "score": 0.93, + "content": "\\left\\{ { a } _ { i } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 162, + 504, + 174 + ], + "score": 1.0, + "content": "where the number of hydrogens attached to the atom change from", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 172, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 313, + 186 + ], + "score": 1.0, + "content": "products to reactants. We denote the set of edits by", + "type": "text" + }, + { + "bbox": [ + 314, + 173, + 323, + 183 + ], + "score": 0.8, + "content": "E", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 172, + 506, + 186 + ], + "score": 1.0, + "content": ". Since retrosynthesis pairs in the training set", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 183, + 505, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 505, + 197 + ], + "score": 1.0, + "content": "are atom-mapped, edits can be automatically identified by comparing the atoms and atom pairs in the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 194, + 318, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 318, + 207 + ], + "score": 1.0, + "content": "product to their corresponding reactant counterparts.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 149, + 506, + 207 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 218, + 505, + 285 + ], + "lines": [ + { + "bbox": [ + 106, + 218, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 303, + 231 + ], + "score": 1.0, + "content": "Synthons and Leaving Groups Applying edits", + "type": "text" + }, + { + "bbox": [ + 303, + 219, + 312, + 228 + ], + "score": 0.84, + "content": "E", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 218, + 368, + 231 + ], + "score": 1.0, + "content": "to the product", + "type": "text" + }, + { + "bbox": [ + 368, + 219, + 380, + 231 + ], + "score": 0.89, + "content": "\\mathcal { G } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 218, + 505, + 231 + ], + "score": 1.0, + "content": "results in incomplete molecules", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 229, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 505, + 242 + ], + "score": 1.0, + "content": "called synthons. Synthons are analogous to rationales or building blocks, which are expanded into", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 239, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 505, + 254 + ], + "score": 1.0, + "content": "valid reactants by adding specific functionalities called leaving groups that are responsible for its", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 251, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 241, + 264 + ], + "score": 1.0, + "content": "reactivity. We denote synthons by", + "type": "text" + }, + { + "bbox": [ + 242, + 252, + 253, + 262 + ], + "score": 0.89, + "content": "\\mathcal { G } _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 251, + 343, + 264 + ], + "score": 1.0, + "content": "and leaving groups by", + "type": "text" + }, + { + "bbox": [ + 344, + 252, + 353, + 262 + ], + "score": 0.86, + "content": "\\mathcal { G } _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 251, + 505, + 264 + ], + "score": 1.0, + "content": ". We further assume that synthons and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 259, + 504, + 278 + ], + "spans": [ + { + "bbox": [ + 104, + 259, + 441, + 278 + ], + "score": 1.0, + "content": "leaving groups have the same number of connected components as the reactants, i.e", + "type": "text" + }, + { + "bbox": [ + 441, + 261, + 504, + 275 + ], + "score": 0.92, + "content": "\\mathcal { G } _ { s } \\doteq \\{ \\mathcal { G } _ { s _ { c } } \\} _ { c = 1 } ^ { C }", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 269, + 442, + 288 + ], + "spans": [ + { + "bbox": [ + 104, + 269, + 123, + 288 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 272, + 183, + 285 + ], + "score": 0.93, + "content": "\\mathcal { G } _ { l } = \\{ \\mathcal { G } _ { l _ { c } } ^ { \\star } \\} _ { c = 1 } ^ { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 269, + 294, + 288 + ], + "score": 1.0, + "content": ". This assumption holds for", + "type": "text" + }, + { + "bbox": [ + 294, + 273, + 326, + 284 + ], + "score": 0.86, + "content": "9 9 . 9 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 269, + 442, + 288 + ], + "score": 1.0, + "content": "reactions in the training set.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5, + "bbox_fs": [ + 104, + 218, + 505, + 288 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 289, + 504, + 312 + ], + "lines": [ + { + "bbox": [ + 105, + 288, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 404, + 303 + ], + "score": 1.0, + "content": "Formally, our model generates reactants by first predicting the set of edits", + "type": "text" + }, + { + "bbox": [ + 405, + 290, + 414, + 299 + ], + "score": 0.83, + "content": "E", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 288, + 474, + 303 + ], + "score": 1.0, + "content": "that transform", + "type": "text" + }, + { + "bbox": [ + 474, + 290, + 486, + 302 + ], + "score": 0.89, + "content": "\\mathcal { G } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 288, + 506, + 303 + ], + "score": 1.0, + "content": "into", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 107, + 301, + 118, + 311 + ], + "score": 0.86, + "content": "\\mathcal { G } _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 300, + 275, + 313 + ], + "score": 1.0, + "content": ", followed by predicting a leaving group", + "type": "text" + }, + { + "bbox": [ + 276, + 301, + 289, + 312 + ], + "score": 0.9, + "content": "\\mathcal { G } _ { l _ { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 300, + 390, + 313 + ], + "score": 1.0, + "content": "to attach to each synthon", + "type": "text" + }, + { + "bbox": [ + 390, + 301, + 405, + 312 + ], + "score": 0.9, + "content": "\\mathcal { G } _ { s _ { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 300, + 506, + 313 + ], + "score": 1.0, + "content": ". The model is defined as", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 288, + 506, + 313 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 323, + 387, + 351 + ], + "lines": [ + { + "bbox": [ + 223, + 323, + 387, + 351 + ], + "spans": [ + { + "bbox": [ + 223, + 323, + 387, + 351 + ], + "score": 0.94, + "content": "P ( \\mathcal G _ { r } | \\mathcal G _ { p } ) = \\sum _ { E , \\mathcal G _ { l } } P ( E | \\mathcal G _ { p } ) P ( \\mathcal G _ { l } | \\mathcal G _ { p } , \\mathcal G _ { s } ) ,", + "type": "interline_equation", + "image_path": "e5989ffe48f40113cab8f00a06662537b577c1fdbf3a8f9d8153b9ad86f85418.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 223, + 323, + 387, + 351 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 356, + 312, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 312, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 133, + 371 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 357, + 159, + 368 + ], + "score": 0.92, + "content": "\\mathcal { G } _ { s } , \\mathcal { G } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 354, + 253, + 371 + ], + "score": 1.0, + "content": "are deterministic given", + "type": "text" + }, + { + "bbox": [ + 254, + 357, + 276, + 368 + ], + "score": 0.91, + "content": "E , { \\mathcal { G } } _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 354, + 297, + 371 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 297, + 357, + 308, + 369 + ], + "score": 0.89, + "content": "\\mathcal { G } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 354, + 312, + 371 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 354, + 312, + 371 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 381, + 195, + 393 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 197, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 197, + 395 + ], + "score": 1.0, + "content": "3.1 Edit Prediction", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 401, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 232, + 416 + ], + "score": 1.0, + "content": "For a given retrosynthesis pair", + "type": "text" + }, + { + "bbox": [ + 233, + 402, + 288, + 414 + ], + "score": 0.93, + "content": "R = ( \\mathcal G _ { p } , \\mathcal G _ { r } )", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 402, + 506, + 416 + ], + "score": 1.0, + "content": ", we predict an edit score only for existing bonds and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "atoms, instead of every atom pair as in [Coley et al., 2019, Jin et al., 2017]. This choice is motivated", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 195, + 437 + ], + "score": 1.0, + "content": "by the low frequency", + "type": "text" + }, + { + "bbox": [ + 195, + 425, + 229, + 435 + ], + "score": 0.93, + "content": "( \\sim 0 . 1 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "of new bond formations in the training set examples. Coupled with", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 434, + 507, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 415, + 448 + ], + "score": 1.0, + "content": "the sparsity of molecular graphs, this reduces the prediction complexity from", + "type": "text" + }, + { + "bbox": [ + 415, + 435, + 445, + 447 + ], + "score": 0.92, + "content": "O ( N ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 434, + 457, + 448 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 457, + 435, + 482, + 447 + ], + "score": 0.92, + "content": "O ( N )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 434, + 507, + 448 + ], + "score": 1.0, + "content": "for a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 160, + 459 + ], + "score": 1.0, + "content": "product with", + "type": "text" + }, + { + "bbox": [ + 161, + 446, + 171, + 456 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "atoms. Our edit prediction model has variants tailored to single and multiple edit", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 181, + 469 + ], + "score": 1.0, + "content": "prediction. Since", + "type": "text" + }, + { + "bbox": [ + 181, + 457, + 201, + 468 + ], + "score": 0.87, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "of the training set consists of single edit examples, the remainder of this", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 467, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 481 + ], + "score": 1.0, + "content": "section describes the setup for single edit prediction. A detailed description of our multiple edit", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 479, + 297, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 297, + 491 + ], + "score": 1.0, + "content": "prediction model can be found in Appendix ??.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 402, + 507, + 491 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 495, + 505, + 540 + ], + "lines": [ + { + "bbox": [ + 105, + 494, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 153, + 508 + ], + "score": 1.0, + "content": "Each bond", + "type": "text" + }, + { + "bbox": [ + 154, + 495, + 177, + 507 + ], + "score": 0.91, + "content": "( u , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 494, + 190, + 508 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 190, + 495, + 202, + 507 + ], + "score": 0.88, + "content": "\\mathcal { G } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 494, + 309, + 508 + ], + "score": 1.0, + "content": "is associated with a label", + "type": "text" + }, + { + "bbox": [ + 309, + 495, + 368, + 507 + ], + "score": 0.93, + "content": "y _ { u v k } \\in \\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 494, + 505, + 508 + ], + "score": 1.0, + "content": "indicating whether its bond type", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 506, + 504, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 114, + 516 + ], + "score": 0.76, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 506, + 342, + 518 + ], + "score": 1.0, + "content": "has changed from the products to reactants. Each atom", + "type": "text" + }, + { + "bbox": [ + 342, + 509, + 349, + 516 + ], + "score": 0.75, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 506, + 455, + 518 + ], + "score": 1.0, + "content": "is associated with a label", + "type": "text" + }, + { + "bbox": [ + 455, + 506, + 504, + 518 + ], + "score": 0.93, + "content": "y _ { u } \\in \\{ 0 , \\bar { 1 } \\}", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "indicating a change in hydrogen count. We predict edit scores using representations that are learnt", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 528, + 199, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 199, + 540 + ], + "score": 1.0, + "content": "using a graph encoder.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 494, + 506, + 540 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 551, + 505, + 617 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "score": 1.0, + "content": "Graph Encoder To obtain atom representations, we use a variant of the message passing network", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 562, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 323, + 575 + ], + "score": 1.0, + "content": "(MPN) described in [Gilmer et al., 2017]. 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This ensures our representations", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "are permutation invariant. These representations are then used to predict atom and bond edit scores", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 711, + 259, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 259, + 723 + ], + "score": 1.0, + "content": "using corresponding neural networks,", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43.5, + "bbox_fs": [ + 104, + 653, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "interline_equation", + "bbox": [ + 245, + 69, + 365, + 104 + ], + "lines": [ + { + "bbox": [ + 245, + 69, + 365, + 104 + ], + "spans": [ + { + "bbox": [ + 245, + 69, + 365, + 104 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\boldsymbol { s } _ { u } = \\mathbf { u _ { a } } ^ { T } \\boldsymbol { \\tau } ( \\mathbf { W _ { a } } \\mathbf { c } _ { u } + b ) \\quad } \\\\ { \\boldsymbol { s } _ { u v k } = \\mathbf { u _ { k } } ^ { T } \\boldsymbol { \\tau } ( \\mathbf { W _ { k } } \\mathbf { c } _ { u v } + b _ { k } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "43a6f38c88896496f6b8c1e00b965fa739609189983ee4e111bdef371168642c.jpg" + } + ] + } + ], + "index": 0.5, + "virtual_lines": [ + { + "bbox": [ + 245, + 69, + 365, + 86.5 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 245, + 86.5, + 365, + 104.0 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 113, + 280, + 125 + ], + "lines": [ + { + "bbox": [ + 106, + 112, + 281, + 126 + ], + "spans": [ + { + "bbox": [ + 106, + 112, + 133, + 126 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 113, + 150, + 125 + ], + "score": 0.91, + "content": "\\tau ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 112, + 281, + 126 + ], + "score": 1.0, + "content": "is the ReLU activation function.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 137, + 505, + 214 + ], + "lines": [ + { + "bbox": [ + 106, + 136, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 506, + 151 + ], + "score": 1.0, + "content": "Updating Bond Edit Scores Unlike a typical classification problem where the labels are indepen-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "dent, edits can have possible dependencies between each other. For example, bonds part of a stable", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "system such as an aromatic ring have a greater tendency to remain unchanged (label 0). We attempt", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 170, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 182 + ], + "score": 1.0, + "content": "to leverage such dependencies to update initial edit scores. To this end, we build a graph with bonds", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 129, + 193 + ], + "score": 0.91, + "content": "( u , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 181, + 456, + 194 + ], + "score": 1.0, + "content": "as nodes, and introduce an edge between bonds sharing an atom. We use another", + "type": "text" + }, + { + "bbox": [ + 456, + 181, + 491, + 193 + ], + "score": 0.43, + "content": "\\mathrm { M P N } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "on", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 104, + 191, + 331, + 205 + ], + "score": 1.0, + "content": "this graph to learn aggregated neighborhood messages", + "type": "text" + }, + { + "bbox": [ + 332, + 193, + 352, + 203 + ], + "score": 0.88, + "content": "\\mathbf { m } _ { u v }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 191, + 465, + 205 + ], + "score": 1.0, + "content": ", and update the edit scores", + "type": "text" + }, + { + "bbox": [ + 466, + 194, + 486, + 203 + ], + "score": 0.9, + "content": "s _ { u v k }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "in a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 202, + 326, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 326, + 216 + ], + "score": 1.0, + "content": "manner similar to how LSTMs update representations,", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 225, + 385, + 287 + ], + "lines": [ + { + "bbox": [ + 226, + 225, + 385, + 287 + ], + "spans": [ + { + "bbox": [ + 226, + 225, + 385, + 287 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { f _ { u v k } = \\sigma ( \\mathbf { W _ { k x } ^ { f } } \\mathbf { x } _ { u v } + \\mathbf { W _ { k m } ^ { f } } \\mathbf { m } _ { u v } ) } \\\\ & { i _ { u v k } = \\sigma ( \\mathbf { W _ { k x } ^ { i } } \\mathbf { x } _ { u v } + \\mathbf { W _ { k m } ^ { i } } \\mathbf { m } _ { u v } ) } \\\\ & { \\tilde { m } _ { u v k } = \\mathbf { u _ { m } } \\tau ( \\mathbf { W _ { k x } ^ { m } } \\mathbf { x } _ { u v } + \\mathbf { W _ { k m } ^ { m } } \\mathbf { m } _ { u v } ) } \\\\ & { \\tilde { s } _ { u v k } = f _ { u v k } \\cdot s _ { u v k } + i _ { u v k } \\cdot \\tilde { m } _ { u v k } . } \\end{array}", + "type": "interline_equation", + "image_path": "190ca6c12e9d0551c6c507ca3bb0a358e64c6db98378f96a20d19207a9bfc801.jpg" + } + ] + } + ], + "index": 11.5, + "virtual_lines": [ + { + "bbox": [ + 226, + 225, + 385, + 240.5 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 226, + 240.5, + 385, + 256.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 226, + 256.0, + 385, + 271.5 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 226, + 271.5, + 385, + 287.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 297, + 478, + 309 + ], + "lines": [ + { + "bbox": [ + 106, + 296, + 479, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 479, + 310 + ], + "score": 1.0, + "content": "Training We train by minimizing the cross-entropy loss over possible bond and atom edits", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 176, + 319, + 435, + 360 + ], + "lines": [ + { + "bbox": [ + 176, + 319, + 435, + 360 + ], + "spans": [ + { + "bbox": [ + 176, + 319, + 435, + 360 + ], + "score": 0.94, + "content": "\\mathcal { L } _ { e } = - \\sum _ { ( \\mathcal { G } _ { p } , E ) } \\left( \\sum _ { ( ( u , v ) , k ) \\in E } y _ { u v k } \\mathrm { l o g } ( \\widetilde s _ { u v k } ) + \\sum _ { u \\in E } y _ { u } \\mathrm { l o g } ( s _ { u } ) \\right) .", + "type": "interline_equation", + "image_path": "d39b464af053c1ddc554cc80cf10fc8a8cf6434c8681b4503820a67f7447deed.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 176, + 319, + 435, + 332.6666666666667 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 176, + 332.6666666666667, + 435, + 346.33333333333337 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 176, + 346.33333333333337, + 435, + 360.00000000000006 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 370, + 505, + 404 + ], + "lines": [ + { + "bbox": [ + 106, + 370, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 506, + 383 + ], + "score": 1.0, + "content": "The cross-entropy loss enforces the model to learn a distribution over possible edits instead of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "score": 1.0, + "content": "reasoning about each edit independently, as with the binary cross entropy loss used in [Jin et al., 2017,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 393, + 185, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 185, + 405 + ], + "score": 1.0, + "content": "Coley et al., 2019].", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "title", + "bbox": [ + 107, + 417, + 218, + 430 + ], + "lines": [ + { + "bbox": [ + 105, + 416, + 219, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 219, + 432 + ], + "score": 1.0, + "content": "3.2 Synthon Completion", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 438, + 505, + 483 + ], + "lines": [ + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "score": 1.0, + "content": "Synthons are completed into valid reactants by adding specific functionalities called leaving groups.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "score": 1.0, + "content": "This involves two complementary tasks: (i.) selecting the appropriate leaving group, and (ii.) attaching", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 460, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 474 + ], + "score": 1.0, + "content": "the leaving group to the synthon. As ground truth leaving groups are not directly provided, we extract", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 471, + 493, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 291, + 485 + ], + "score": 1.0, + "content": "the leaving groups and construct a vocabulary", + "type": "text" + }, + { + "bbox": [ + 292, + 472, + 302, + 481 + ], + "score": 0.84, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 471, + 493, + 485 + ], + "score": 1.0, + "content": "of unique leaving groups during preprocessing.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 487, + 505, + 543 + ], + "lines": [ + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 252, + 500 + ], + "score": 1.0, + "content": "The vocabulary has a limited size (", + "type": "text" + }, + { + "bbox": [ + 252, + 487, + 297, + 500 + ], + "score": 0.89, + "content": "| \\mathcal { X } | = 1 7 0", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "for a standard dataset with 50, 000 examples, and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 498, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 512 + ], + "score": 1.0, + "content": "72000 synthons) indicating the redundancy of leaving groups used in accomplishing retrosynthetic", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 509, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 523 + ], + "score": 1.0, + "content": "transformations. This redundancy also allows us to formulate leaving group selection as a classi-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 520, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 198, + 534 + ], + "score": 1.0, + "content": "fication problem over", + "type": "text" + }, + { + "bbox": [ + 198, + 521, + 208, + 530 + ], + "score": 0.8, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 520, + 506, + 534 + ], + "score": 1.0, + "content": ", while retaining the ability to generate diverse reactants using different", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 531, + 237, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 237, + 544 + ], + "score": 1.0, + "content": "combinations of leaving groups.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 555, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 569 + ], + "score": 1.0, + "content": "Vocabulary Construction Before constructing the vocabulary, we align connected components", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 565, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 477, + 580 + ], + "score": 1.0, + "content": "of synthon and reactant graphs by comparing atom mapping overlaps. Using aligned pairs", + "type": "text" + }, + { + "bbox": [ + 478, + 567, + 505, + 578 + ], + "score": 0.9, + "content": "\\mathcal { G } _ { s _ { c } } =", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 576, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 146, + 590 + ], + "score": 0.91, + "content": "( \\gamma _ { s _ { c } } , \\mathcal { E } _ { s _ { c } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 576, + 164, + 592 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 164, + 577, + 231, + 589 + ], + "score": 0.91, + "content": "\\mathcal { G } _ { r _ { c } } = ( \\nu _ { r _ { c } } , \\mathcal { E } _ { r _ { c } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 576, + 384, + 592 + ], + "score": 1.0, + "content": "as input, the leaving group vocabulary", + "type": "text" + }, + { + "bbox": [ + 384, + 578, + 394, + 587 + ], + "score": 0.83, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 576, + 505, + 592 + ], + "score": 1.0, + "content": "is constructed by extracting", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 588, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 150, + 602 + ], + "score": 1.0, + "content": "subgraphs", + "type": "text" + }, + { + "bbox": [ + 150, + 589, + 214, + 600 + ], + "score": 0.91, + "content": "\\mathcal { G } _ { l _ { c } } = ( \\nu _ { l _ { c } } , \\mathcal { E } _ { l _ { c } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 588, + 254, + 602 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 254, + 588, + 318, + 600 + ], + "score": 0.92, + "content": "\\smash { \\gamma _ { l _ { c } } = \\gamma _ { r _ { c } } \\setminus \\gamma _ { s _ { c } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 588, + 353, + 602 + ], + "score": 1.0, + "content": ". Atoms", + "type": "text" + }, + { + "bbox": [ + 353, + 588, + 372, + 600 + ], + "score": 0.92, + "content": "\\left\\{ { a } _ { i } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 588, + 505, + 602 + ], + "score": 1.0, + "content": "in the leaving groups that attach", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 599, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 401, + 611 + ], + "score": 1.0, + "content": "to synthons are marked with a special symbol. 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For example, bonds part of a stable", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "system such as an aromatic ring have a greater tendency to remain unchanged (label 0). We attempt", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 170, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 182 + ], + "score": 1.0, + "content": "to leverage such dependencies to update initial edit scores. To this end, we build a graph with bonds", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 129, + 193 + ], + "score": 0.91, + "content": "( u , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 181, + 456, + 194 + ], + "score": 1.0, + "content": "as nodes, and introduce an edge between bonds sharing an atom. 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As ground truth leaving groups are not directly provided, we extract", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 471, + 493, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 291, + 485 + ], + "score": 1.0, + "content": "the leaving groups and construct a vocabulary", + "type": "text" + }, + { + "bbox": [ + 292, + 472, + 302, + 481 + ], + "score": 0.84, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 471, + 493, + 485 + ], + "score": 1.0, + "content": "of unique leaving groups during preprocessing.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 438, + 506, + 485 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 487, + 505, + 543 + ], + "lines": [ + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 252, + 500 + ], + "score": 1.0, + "content": "The vocabulary has a limited size (", + "type": "text" + }, + { + "bbox": [ + 252, + 487, + 297, + 500 + ], + "score": 0.89, + "content": "| \\mathcal { X } | = 1 7 0", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "for a standard dataset with 50, 000 examples, and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 498, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 512 + ], + "score": 1.0, + "content": "72000 synthons) indicating the redundancy of leaving groups used in accomplishing retrosynthetic", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 509, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 523 + ], + "score": 1.0, + "content": "transformations. This redundancy also allows us to formulate leaving group selection as a classi-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 520, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 198, + 534 + ], + "score": 1.0, + "content": "fication problem over", + "type": "text" + }, + { + "bbox": [ + 198, + 521, + 208, + 530 + ], + "score": 0.8, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 520, + 506, + 534 + ], + "score": 1.0, + "content": ", while retaining the ability to generate diverse reactants using different", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 531, + 237, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 237, + 544 + ], + "score": 1.0, + "content": "combinations of leaving groups.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 487, + 506, + 544 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 555, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 569 + ], + "score": 1.0, + "content": "Vocabulary Construction Before constructing the vocabulary, we align connected components", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 565, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 477, + 580 + ], + "score": 1.0, + "content": "of synthon and reactant graphs by comparing atom mapping overlaps. Using aligned pairs", + "type": "text" + }, + { + "bbox": [ + 478, + 567, + 505, + 578 + ], + "score": 0.9, + "content": "\\mathcal { G } _ { s _ { c } } =", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 576, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 146, + 590 + ], + "score": 0.91, + "content": "( \\gamma _ { s _ { c } } , \\mathcal { E } _ { s _ { c } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 576, + 164, + 592 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 164, + 577, + 231, + 589 + ], + "score": 0.91, + "content": "\\mathcal { G } _ { r _ { c } } = ( \\nu _ { r _ { c } } , \\mathcal { E } _ { r _ { c } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 576, + 384, + 592 + ], + "score": 1.0, + "content": "as input, the leaving group vocabulary", + "type": "text" + }, + { + "bbox": [ + 384, + 578, + 394, + 587 + ], + "score": 0.83, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 576, + 505, + 592 + ], + "score": 1.0, + "content": "is constructed by extracting", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 588, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 150, + 602 + ], + "score": 1.0, + "content": "subgraphs", + "type": "text" + }, + { + "bbox": [ + 150, + 589, + 214, + 600 + ], + "score": 0.91, + "content": "\\mathcal { G } _ { l _ { c } } = ( \\nu _ { l _ { c } } , \\mathcal { E } _ { l _ { c } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 588, + 254, + 602 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 254, + 588, + 318, + 600 + ], + "score": 0.92, + "content": "\\smash { \\gamma _ { l _ { c } } = \\gamma _ { r _ { c } } \\setminus \\gamma _ { s _ { c } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 588, + 353, + 602 + ], + "score": 1.0, + "content": ". Atoms", + "type": "text" + }, + { + "bbox": [ + 353, + 588, + 372, + 600 + ], + "score": 0.92, + "content": "\\left\\{ { a } _ { i } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 588, + 505, + 602 + ], + "score": 1.0, + "content": "in the leaving groups that attach", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 599, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 401, + 611 + ], + "score": 1.0, + "content": "to synthons are marked with a special symbol. 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The attachment process does not", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 480, + 375 + ], + "score": 1.0, + "content": "modify any stereochemistry. Given synthons and leaving groups, the attachment process has a", + "type": "text" + }, + { + "bbox": [ + 480, + 362, + 505, + 372 + ], + "score": 0.85, + "content": "100 \\%", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 372, + 358, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 358, + 385 + ], + "score": 1.0, + "content": "accuracy. The detailed procedure is described in Appendix ??.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 107, + 397, + 171, + 408 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 173, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 173, + 410 + ], + "score": 1.0, + "content": "3.3 Inference", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 416, + 506, + 505 + ], + "lines": [ + { + "bbox": [ + 106, + 417, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 495, + 429 + ], + "score": 1.0, + "content": "Inference is performed using beam search with a log-likelihood scoring function. 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The synthons form the nodes for beam search.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 449, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 506, + 462 + ], + "score": 1.0, + "content": "Each node maintains a cumulative score by aggregating the log-likelihoods of the edit and predicted", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 461, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 506, + 473 + ], + "score": 1.0, + "content": "leaving groups. Leaving group inference starts with a connected component for each synthon, and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 136, + 484 + ], + "score": 1.0, + "content": "selects", + "type": "text" + }, + { + "bbox": [ + 136, + 474, + 143, + 482 + ], + "score": 0.7, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 471, + 364, + 484 + ], + "score": 1.0, + "content": "leaving groups with highest log-likelihoods. From the", + "type": "text" + }, + { + "bbox": [ + 365, + 471, + 376, + 482 + ], + "score": 0.86, + "content": "n ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 471, + 470, + 484 + ], + "score": 1.0, + "content": "possibilities, we select", + "type": "text" + }, + { + "bbox": [ + 470, + 474, + 478, + 482 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 471, + 506, + 484 + ], + "score": 1.0, + "content": "nodes", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 481, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 506, + 496 + ], + "score": 1.0, + "content": "with the highest cumulative scores. This process is repeated until all nodes have a leaving group", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 494, + 263, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 263, + 506 + ], + "score": 1.0, + "content": "predicted for each synthon component.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25.5 + }, + { + "type": "title", + "bbox": [ + 107, + 520, + 181, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 182, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 182, + 535 + ], + "score": 1.0, + "content": "4 Evaluation", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 544, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "Evaluating retrosynthesis models is challenging as multiple sets of reactants can be generated from", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 556, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 505, + 568 + ], + "score": 1.0, + "content": "the same product. To deal with this, previous works [Coley et al., 2017b, Dai et al., 2019] evaluate", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 566, + 438, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 438, + 579 + ], + "score": 1.0, + "content": "the ability of the model to recover retrosynthetic strategies recorded in the dataset.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 588, + 505, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 602 + ], + "score": 1.0, + "content": "Data We use the benchmark dataset USPTO-50k [Schneider et al., 2016] for all our experiments.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "The dataset contains 50, 000 atom-mapped reactions across 10 reaction classes. We use the same", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 612, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 506, + 623 + ], + "score": 1.0, + "content": "dataset version and splits as provided by [Dai et al., 2019]. The USPTO-50k dataset contains a", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 403, + 635 + ], + "score": 1.0, + "content": "shortcut in that the product atom with atom-mapping 1 is part of the edit in", + "type": "text" + }, + { + "bbox": [ + 404, + 622, + 429, + 633 + ], + "score": 0.89, + "content": "\\sim 7 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 622, + 506, + 635 + ], + "score": 1.0, + "content": "of the cases. If the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "score": 1.0, + "content": "product SMILES is not canonicalized, predictions utilizing operations that depend on the position of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "the atom or bond will be able to use the shortcut, and overestimate performance. We canonicalize the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 655, + 507, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 507, + 668 + ], + "score": 1.0, + "content": "product SMILES, and reassign atom-mappings to the reactant atoms based on the canonical ordering,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 665, + 487, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 487, + 678 + ], + "score": 1.0, + "content": "which removes the shortcut. 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Using the representation of the previous leaving group", + "type": "text" + }, + { + "bbox": [ + 483, + 133, + 504, + 143 + ], + "score": 0.88, + "content": "{ \\bf e } _ { l _ { c - 1 } }", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 140, + 506, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 506, + 154 + ], + "score": 1.0, + "content": "allows the model to understand combinations of leaving groups that generate the desired product", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 151, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 354, + 165 + ], + "score": 1.0, + "content": "from the reactants. 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The task involves identification of the type of bonds to add between", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "attaching atoms in the leaving group (marked during vocabulary construction), and the atom(s)", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "score": 1.0, + "content": "participating in the edit. These bonds can be inferred by applying the valency constraint, which", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 351, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 506, + 363 + ], + "score": 1.0, + "content": "determines the maximum number of neighbors for each atom. The attachment process does not", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 480, + 375 + ], + "score": 1.0, + "content": "modify any stereochemistry. Given synthons and leaving groups, the attachment process has a", + "type": "text" + }, + { + "bbox": [ + 480, + 362, + 505, + 372 + ], + "score": 0.85, + "content": "100 \\%", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 372, + 358, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 358, + 385 + ], + "score": 1.0, + "content": "accuracy. The detailed procedure is described in Appendix ??.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 307, + 506, + 385 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 397, + 171, + 408 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 173, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 173, + 410 + ], + "score": 1.0, + "content": "3.3 Inference", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 416, + 506, + 505 + ], + "lines": [ + { + "bbox": [ + 106, + 417, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 495, + 429 + ], + "score": 1.0, + "content": "Inference is performed using beam search with a log-likelihood scoring function. For a beam width", + "type": "text" + }, + { + "bbox": [ + 496, + 419, + 502, + 427 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 417, + 506, + 429 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 428, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 145, + 440 + ], + "score": 1.0, + "content": "we select", + "type": "text" + }, + { + "bbox": [ + 145, + 430, + 153, + 438 + ], + "score": 0.73, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 428, + 409, + 440 + ], + "score": 1.0, + "content": "edits with highest scores and apply them to the product to obtain", + "type": "text" + }, + { + "bbox": [ + 410, + 430, + 417, + 438 + ], + "score": 0.64, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 428, + 506, + 440 + ], + "score": 1.0, + "content": "synthons, where each", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 439, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 451 + ], + "score": 1.0, + "content": "synthon can consist of multiple connected components. The synthons form the nodes for beam search.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 449, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 506, + 462 + ], + "score": 1.0, + "content": "Each node maintains a cumulative score by aggregating the log-likelihoods of the edit and predicted", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 461, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 506, + 473 + ], + "score": 1.0, + "content": "leaving groups. Leaving group inference starts with a connected component for each synthon, and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 136, + 484 + ], + "score": 1.0, + "content": "selects", + "type": "text" + }, + { + "bbox": [ + 136, + 474, + 143, + 482 + ], + "score": 0.7, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 471, + 364, + 484 + ], + "score": 1.0, + "content": "leaving groups with highest log-likelihoods. From the", + "type": "text" + }, + { + "bbox": [ + 365, + 471, + 376, + 482 + ], + "score": 0.86, + "content": "n ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 471, + 470, + 484 + ], + "score": 1.0, + "content": "possibilities, we select", + "type": "text" + }, + { + "bbox": [ + 470, + 474, + 478, + 482 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 471, + 506, + 484 + ], + "score": 1.0, + "content": "nodes", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 481, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 506, + 496 + ], + "score": 1.0, + "content": "with the highest cumulative scores. This process is repeated until all nodes have a leaving group", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 494, + 263, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 263, + 506 + ], + "score": 1.0, + "content": "predicted for each synthon component.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 417, + 506, + 506 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 520, + 181, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 182, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 182, + 535 + ], + "score": 1.0, + "content": "4 Evaluation", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 544, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "Evaluating retrosynthesis models is challenging as multiple sets of reactants can be generated from", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 556, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 505, + 568 + ], + "score": 1.0, + "content": "the same product. To deal with this, previous works [Coley et al., 2017b, Dai et al., 2019] evaluate", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 566, + 438, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 438, + 579 + ], + "score": 1.0, + "content": "the ability of the model to recover retrosynthetic strategies recorded in the dataset.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 545, + 505, + 579 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 588, + 505, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 602 + ], + "score": 1.0, + "content": "Data We use the benchmark dataset USPTO-50k [Schneider et al., 2016] for all our experiments.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "The dataset contains 50, 000 atom-mapped reactions across 10 reaction classes. We use the same", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 612, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 506, + 623 + ], + "score": 1.0, + "content": "dataset version and splits as provided by [Dai et al., 2019]. The USPTO-50k dataset contains a", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 403, + 635 + ], + "score": 1.0, + "content": "shortcut in that the product atom with atom-mapping 1 is part of the edit in", + "type": "text" + }, + { + "bbox": [ + 404, + 622, + 429, + 633 + ], + "score": 0.89, + "content": "\\sim 7 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 622, + 506, + 635 + ], + "score": 1.0, + "content": "of the cases. If the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "score": 1.0, + "content": "product SMILES is not canonicalized, predictions utilizing operations that depend on the position of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "the atom or bond will be able to use the shortcut, and overestimate performance. We canonicalize the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 655, + 507, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 507, + 668 + ], + "score": 1.0, + "content": "product SMILES, and reassign atom-mappings to the reactant atoms based on the canonical ordering,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 665, + 487, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 487, + 678 + ], + "score": 1.0, + "content": "which removes the shortcut. Details on the remapping procedure can be found in Appendix ??.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 588, + 507, + 678 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 689, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 227, + 702 + ], + "score": 1.0, + "content": "Evaluation We use the top-", + "type": "text" + }, + { + "bbox": [ + 227, + 691, + 234, + 699 + ], + "score": 0.75, + "content": "\\mathbf { \\nabla } \\cdot n", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 688, + 276, + 702 + ], + "score": 1.0, + "content": "accuracy", + "type": "text" + }, + { + "bbox": [ + 277, + 689, + 338, + 700 + ], + "score": 0.76, + "content": "( n = 1 , 3 , 5 , 1 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 688, + 506, + 702 + ], + "score": 1.0, + "content": ") as our evaluation metric, defined as the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 700, + 507, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 451, + 713 + ], + "score": 1.0, + "content": "fraction of examples where the recorded reactants are suggested by the model with rank", + "type": "text" + }, + { + "bbox": [ + 451, + 700, + 469, + 711 + ], + "score": 0.87, + "content": "\\leq n", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 700, + 507, + 713 + ], + "score": 1.0, + "content": ". Follow-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 710, + 506, + 725 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 725 + ], + "score": 1.0, + "content": "ing prior work [Coley et al., 2017b, Zheng et al., 2019, Dai et al., 2019], we compute the accuracy", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 311, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 324 + ], + "score": 1.0, + "content": "by comparing the canonical SMILES of predicted reactants to the ground truth. Atom-mapping is", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 505, + 334 + ], + "score": 1.0, + "content": "excluded from this comparison, but stereochemistry, which describes the relative orientation of atoms", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 334, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 345 + ], + "score": 1.0, + "content": "in the molecule, is retained. The evaluation is carried out for two settings, with the reaction class", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 345, + 213, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 213, + 357 + ], + "score": 1.0, + "content": "being known or unknown.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 688, + 507, + 725 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 105, + 92, + 506, + 290 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 110, + 77, + 495, + 89 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 76, + 497, + 90 + ], + "spans": [ + { + "bbox": [ + 113, + 76, + 169, + 90 + ], + "score": 1.0, + "content": "Table 1: Top-", + "type": "text" + }, + { + "bbox": [ + 169, + 79, + 176, + 88 + ], + "score": 0.5, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 76, + 497, + 90 + ], + "score": 1.0, + "content": "exact match accuracy. Best values within each section are highlighted in bold.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 105, + 92, + 506, + 290 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 92, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 290 + ], + "score": 0.983, + "html": "
ModelTop-n Accuracy (%)
Reaction class knownReaction class unknown
1351013510
Template-Based
RETROSIM [Coley et al.,2017b]52.973.881.288.137.354.763.374.1
NEURALSYM [Segler and Waller,2017]55.376.081.485.144.465.372.478.9
GLN [Dai et ai., 2019]64.279.185.290.052.569.075.683.7
DUALTB [Sun et al.,2021]67.784.888.992.055.274.680.586.9
Template-Free
SCROP [Zheng et al.,2019]59.074.878.181.143.760.065.268.7
LV-TRANSFORMER [Chen et al.,2019]----40.565.172.879.4
DUALTF [Sun et al., 2021]65.781.984.785.953.670.774.677.0
Semi-Template-Based
G2Gs [Shi et al.,2020]61.081.386.088.748.967.672.575.5
RETROXPERT [Yan et al.,2020]62.175.878.580.950.461.162.363.4
GRAPHRETRO (ours)63.981.585.288.153.768.372.275.5
", + "type": "table", + "image_path": "b50491b8ada14574c6239d28c43d14386b5d359906cdb5f2c186422249a03113.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 105, + 92, + 506, + 158.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 105, + 158.0, + 506, + 224.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 105, + 224.0, + 506, + 290.0 + ], + "spans": [], + "index": 3 + } + ] + } + ], + "index": 1.0 + }, + { + "type": "text", + "bbox": [ + 106, + 311, + 505, + 355 + ], + "lines": [ + { + "bbox": [ + 105, + 311, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 324 + ], + "score": 1.0, + "content": "by comparing the canonical SMILES of predicted reactants to the ground truth. Atom-mapping is", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 505, + 334 + ], + "score": 1.0, + "content": "excluded from this comparison, but stereochemistry, which describes the relative orientation of atoms", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 334, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 345 + ], + "score": 1.0, + "content": "in the molecule, is retained. The evaluation is carried out for two settings, with the reaction class", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 345, + 213, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 213, + 357 + ], + "score": 1.0, + "content": "being known or unknown.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 369, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 106, + 368, + 499, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 499, + 382 + ], + "score": 1.0, + "content": "Baselines For evaluating overall performance, we compare GRAPHRETRO to nine baselines —", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 379, + 491, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 491, + 392 + ], + "score": 1.0, + "content": "four template-based, three template-free, and two semi-template-based methods. These include:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 111, + 402, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 110, + 402, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 110, + 402, + 506, + 416 + ], + "score": 1.0, + "content": "Template-Based: RETROSIM Coley et al. [2017b] ranks templates for a given target molecule by", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 120, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 120, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "computing molecular similarities to precedent reactions. NEURALSYM [Segler and Waller, 2017]", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 120, + 424, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 120, + 424, + 506, + 438 + ], + "score": 1.0, + "content": "trains a model to rank templates given a target molecule. GLN [Dai et al., 2019] models the joint", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 121, + 436, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 121, + 436, + 505, + 448 + ], + "score": 1.0, + "content": "distribution of templates and reactants in a hierarchical fashion using logic variables. DUALTB", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 120, + 447, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 120, + 447, + 506, + 459 + ], + "score": 1.0, + "content": "[Sun et al., 2021] uses an energy-based model formulation for retrosynthesis, with additional", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 120, + 458, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 120, + 458, + 506, + 470 + ], + "score": 1.0, + "content": "parameterizations and loss terms to enforce the duality between forward (reaction prediction) and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 120, + 469, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 120, + 469, + 506, + 480 + ], + "score": 1.0, + "content": "backward (retrosynthesis prediction). Inference is carried out using reactant candidates obtained", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 120, + 479, + 336, + 493 + ], + "spans": [ + { + "bbox": [ + 120, + 479, + 336, + 493 + ], + "score": 1.0, + "content": "by applying an extracted template set to the products.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 112, + 496, + 505, + 572 + ], + "lines": [ + { + "bbox": [ + 111, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 111, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "Template-Free: SCROP [Zheng et al., 2019], LV-TRANSFORMER [Chen et al., 2019] and DUALTF", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 121, + 507, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 121, + 507, + 506, + 519 + ], + "score": 1.0, + "content": "[Sun et al., 2021] use the Transformer architecture [Vaswani et al., 2017] to output reactant", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 120, + 517, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 120, + 517, + 506, + 531 + ], + "score": 1.0, + "content": "SMILES given a product SMILES. To improve the validity of their suggestions, SCROP in-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 120, + 529, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 120, + 529, + 506, + 541 + ], + "score": 1.0, + "content": "clude a second Transformer that functions as a syntax correcter. LV-TRANSFORMER uses a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 120, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 120, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "latent variable mixture model to improve diversity of suggestions. DUALTF utilizes additional", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 120, + 550, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 120, + 550, + 506, + 564 + ], + "score": 1.0, + "content": "parameterizations and loss terms to enforce the duality between forward (reaction prediction) and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 121, + 562, + 272, + 574 + ], + "spans": [ + { + "bbox": [ + 121, + 562, + 272, + 574 + ], + "score": 1.0, + "content": "backward (retrosynthesis prediction).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 112, + 578, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 111, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 111, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "Semi-Template-Based: G2GS [Shi et al., 2020] and RETROXPERT [Yan et al., 2020] first identify", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 120, + 589, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 120, + 589, + 506, + 602 + ], + "score": 1.0, + "content": "synthons, and then expand the synthons into reactants by either sequential generation of atoms", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 120, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 120, + 599, + 506, + 613 + ], + "score": 1.0, + "content": "and bonds (G2Gs), or using the Transformer architecture (RETROXPERT). The training dataset", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 120, + 611, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 120, + 611, + 505, + 624 + ], + "score": 1.0, + "content": "for the Transformer in [Yan et al., 2020] is augmented with incorrectly predicted synthons with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 121, + 622, + 301, + 634 + ], + "spans": [ + { + "bbox": [ + 121, + 622, + 301, + 634 + ], + "score": 1.0, + "content": "the goal of learning a correction mechanism.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 108, + 644, + 504, + 699 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "Results for NEURALSYM are taken from [Dai et al., 2019]. The authors in [Yan et al., 2020] report", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "their performance being affected by the dataset leakage2. Thus, we use the most recent results", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 679 + ], + "score": 1.0, + "content": "from their website on the canonicalized dataset. For remaining baselines, we directly use the values", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "reported in their paper. For the synthon completion module, we use the ind configuration given its", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 688, + 227, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 227, + 701 + ], + "score": 1.0, + "content": "better empirical performance.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 120, + 712, + 268, + 722 + ], + "lines": [ + { + "bbox": [ + 119, + 710, + 270, + 724 + ], + "spans": [ + { + "bbox": [ + 119, + 710, + 270, + 724 + ], + "score": 1.0, + "content": "2https://github.com/uta-smile/RetroXpert", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 105, + 92, + 506, + 290 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 110, + 77, + 495, + 89 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 76, + 497, + 90 + ], + "spans": [ + { + "bbox": [ + 113, + 76, + 169, + 90 + ], + "score": 1.0, + "content": "Table 1: Top-", + "type": "text" + }, + { + "bbox": [ + 169, + 79, + 176, + 88 + ], + "score": 0.5, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 76, + 497, + 90 + ], + "score": 1.0, + "content": "exact match accuracy. Best values within each section are highlighted in bold.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 105, + 92, + 506, + 290 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 92, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 290 + ], + "score": 0.983, + "html": "
ModelTop-n Accuracy (%)
Reaction class knownReaction class unknown
1351013510
Template-Based
RETROSIM [Coley et al.,2017b]52.973.881.288.137.354.763.374.1
NEURALSYM [Segler and Waller,2017]55.376.081.485.144.465.372.478.9
GLN [Dai et ai., 2019]64.279.185.290.052.569.075.683.7
DUALTB [Sun et al.,2021]67.784.888.992.055.274.680.586.9
Template-Free
SCROP [Zheng et al.,2019]59.074.878.181.143.760.065.268.7
LV-TRANSFORMER [Chen et al.,2019]----40.565.172.879.4
DUALTF [Sun et al., 2021]65.781.984.785.953.670.774.677.0
Semi-Template-Based
G2Gs [Shi et al.,2020]61.081.386.088.748.967.672.575.5
RETROXPERT [Yan et al.,2020]62.175.878.580.950.461.162.363.4
GRAPHRETRO (ours)63.981.585.288.153.768.372.275.5
", + "type": "table", + "image_path": "b50491b8ada14574c6239d28c43d14386b5d359906cdb5f2c186422249a03113.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 105, + 92, + 506, + 158.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 105, + 158.0, + 506, + 224.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 105, + 224.0, + 506, + 290.0 + ], + "spans": [], + "index": 3 + } + ] + } + ], + "index": 1.0 + }, + { + "type": "text", + "bbox": [ + 106, + 311, + 505, + 355 + ], + "lines": [], + "index": 5.5, + "bbox_fs": [ + 105, + 311, + 506, + 357 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 369, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 106, + 368, + 499, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 499, + 382 + ], + "score": 1.0, + "content": "Baselines For evaluating overall performance, we compare GRAPHRETRO to nine baselines —", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 379, + 491, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 491, + 392 + ], + "score": 1.0, + "content": "four template-based, three template-free, and two semi-template-based methods. These include:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 368, + 499, + 392 + ] + }, + { + "type": "text", + "bbox": [ + 111, + 402, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 110, + 402, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 110, + 402, + 506, + 416 + ], + "score": 1.0, + "content": "Template-Based: RETROSIM Coley et al. [2017b] ranks templates for a given target molecule by", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 120, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 120, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "computing molecular similarities to precedent reactions. NEURALSYM [Segler and Waller, 2017]", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 120, + 424, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 120, + 424, + 506, + 438 + ], + "score": 1.0, + "content": "trains a model to rank templates given a target molecule. GLN [Dai et al., 2019] models the joint", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 121, + 436, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 121, + 436, + 505, + 448 + ], + "score": 1.0, + "content": "distribution of templates and reactants in a hierarchical fashion using logic variables. DUALTB", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 120, + 447, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 120, + 447, + 506, + 459 + ], + "score": 1.0, + "content": "[Sun et al., 2021] uses an energy-based model formulation for retrosynthesis, with additional", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 120, + 458, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 120, + 458, + 506, + 470 + ], + "score": 1.0, + "content": "parameterizations and loss terms to enforce the duality between forward (reaction prediction) and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 120, + 469, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 120, + 469, + 506, + 480 + ], + "score": 1.0, + "content": "backward (retrosynthesis prediction). Inference is carried out using reactant candidates obtained", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 120, + 479, + 336, + 493 + ], + "spans": [ + { + "bbox": [ + 120, + 479, + 336, + 493 + ], + "score": 1.0, + "content": "by applying an extracted template set to the products.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13.5, + "bbox_fs": [ + 110, + 402, + 506, + 493 + ] + }, + { + "type": "text", + "bbox": [ + 112, + 496, + 505, + 572 + ], + "lines": [ + { + "bbox": [ + 111, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 111, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "Template-Free: SCROP [Zheng et al., 2019], LV-TRANSFORMER [Chen et al., 2019] and DUALTF", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 121, + 507, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 121, + 507, + 506, + 519 + ], + "score": 1.0, + "content": "[Sun et al., 2021] use the Transformer architecture [Vaswani et al., 2017] to output reactant", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 120, + 517, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 120, + 517, + 506, + 531 + ], + "score": 1.0, + "content": "SMILES given a product SMILES. To improve the validity of their suggestions, SCROP in-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 120, + 529, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 120, + 529, + 506, + 541 + ], + "score": 1.0, + "content": "clude a second Transformer that functions as a syntax correcter. LV-TRANSFORMER uses a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 120, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 120, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "latent variable mixture model to improve diversity of suggestions. DUALTF utilizes additional", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 120, + 550, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 120, + 550, + 506, + 564 + ], + "score": 1.0, + "content": "parameterizations and loss terms to enforce the duality between forward (reaction prediction) and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 121, + 562, + 272, + 574 + ], + "spans": [ + { + "bbox": [ + 121, + 562, + 272, + 574 + ], + "score": 1.0, + "content": "backward (retrosynthesis prediction).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21, + "bbox_fs": [ + 111, + 496, + 506, + 574 + ] + }, + { + "type": "text", + "bbox": [ + 112, + 578, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 111, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 111, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "Semi-Template-Based: G2GS [Shi et al., 2020] and RETROXPERT [Yan et al., 2020] first identify", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 120, + 589, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 120, + 589, + 506, + 602 + ], + "score": 1.0, + "content": "synthons, and then expand the synthons into reactants by either sequential generation of atoms", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 120, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 120, + 599, + 506, + 613 + ], + "score": 1.0, + "content": "and bonds (G2Gs), or using the Transformer architecture (RETROXPERT). The training dataset", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 120, + 611, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 120, + 611, + 505, + 624 + ], + "score": 1.0, + "content": "for the Transformer in [Yan et al., 2020] is augmented with incorrectly predicted synthons with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 121, + 622, + 301, + 634 + ], + "spans": [ + { + "bbox": [ + 121, + 622, + 301, + 634 + ], + "score": 1.0, + "content": "the goal of learning a correction mechanism.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27, + "bbox_fs": [ + 111, + 578, + 506, + 634 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 644, + 504, + 699 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "Results for NEURALSYM are taken from [Dai et al., 2019]. The authors in [Yan et al., 2020] report", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "their performance being affected by the dataset leakage2. Thus, we use the most recent results", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 679 + ], + "score": 1.0, + "content": "from their website on the canonicalized dataset. For remaining baselines, we directly use the values", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "reported in their paper. For the synthon completion module, we use the ind configuration given its", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 688, + 227, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 227, + 701 + ], + "score": 1.0, + "content": "better empirical performance.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 644, + 506, + 701 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 72, + 219, + 84 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 221, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 221, + 86 + ], + "score": 1.0, + "content": "4.1 Overall Performance", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 96, + 505, + 195 + ], + "lines": [ + { + "bbox": [ + 106, + 96, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 96, + 505, + 108 + ], + "score": 1.0, + "content": "Reaction class unknown As shown in Table 1, when the reaction class is unknown, GRAPHRETRO", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 107, + 506, + 120 + ], + "spans": [ + { + "bbox": [ + 106, + 107, + 204, + 120 + ], + "score": 1.0, + "content": "outperforms G2GS by", + "type": "text" + }, + { + "bbox": [ + 204, + 107, + 227, + 118 + ], + "score": 0.88, + "content": "4 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 107, + 343, + 120 + ], + "score": 1.0, + "content": "and and RETROXPERT by", + "type": "text" + }, + { + "bbox": [ + 343, + 107, + 366, + 118 + ], + "score": 0.87, + "content": "3 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 107, + 506, + 120 + ], + "score": 1.0, + "content": "in top-1 accuracy. Performance", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 118, + 506, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 259, + 131 + ], + "score": 1.0, + "content": "improvements are also seen for larger", + "type": "text" + }, + { + "bbox": [ + 259, + 120, + 267, + 128 + ], + "score": 0.59, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 118, + 313, + 131 + ], + "score": 1.0, + "content": ", except for", + "type": "text" + }, + { + "bbox": [ + 313, + 119, + 339, + 128 + ], + "score": 0.89, + "content": "n = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 118, + 506, + 131 + ], + "score": 1.0, + "content": ". Barring DUALTB, the top-1 accuracy is", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 128, + 506, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 409, + 142 + ], + "score": 1.0, + "content": "also better than other template-free and template-based methods. For larger", + "type": "text" + }, + { + "bbox": [ + 410, + 131, + 417, + 139 + ], + "score": 0.67, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 128, + 506, + 142 + ], + "score": 1.0, + "content": ", one reason for lower", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 140, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 123, + 152 + ], + "score": 1.0, + "content": "top-", + "type": "text" + }, + { + "bbox": [ + 123, + 142, + 130, + 150 + ], + "score": 0.66, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 140, + 505, + 152 + ], + "score": 1.0, + "content": "accuracies than most template-based methods is that templates already contain combinations", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 150, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 505, + 165 + ], + "score": 1.0, + "content": "of leaving group patterns. In contrast, our model learns to discover these during training. A second", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 162, + 505, + 175 + ], + "spans": [ + { + "bbox": [ + 106, + 162, + 505, + 175 + ], + "score": 1.0, + "content": "hypothesis to this end is that simply adding log-likelihood scores from edit prediction and synthon", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 506, + 186 + ], + "score": 1.0, + "content": "completion models may be suboptimal and bias the beam search in the direction of the more", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 183, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 506, + 195 + ], + "score": 1.0, + "content": "dominating term. We leave it to future work to investigate scoring functions that rank the attachment.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 215, + 505, + 292 + ], + "lines": [ + { + "bbox": [ + 106, + 216, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 505, + 227 + ], + "score": 1.0, + "content": "Reaction class known When the reaction class is known, GRAPHRETRO outperforms G2GS", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 246, + 239 + ], + "score": 1.0, + "content": "and RETROXPERT by a margin of", + "type": "text" + }, + { + "bbox": [ + 247, + 226, + 262, + 237 + ], + "score": 0.88, + "content": "3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 226, + 280, + 239 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 280, + 226, + 294, + 237 + ], + "score": 0.87, + "content": "2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 226, + 505, + 239 + ], + "score": 1.0, + "content": "respectively in top-1 accuracy. GRAPHRETRO also", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 237, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 302, + 250 + ], + "score": 1.0, + "content": "outperforms all the template-free methods in top-", + "type": "text" + }, + { + "bbox": [ + 302, + 239, + 309, + 247 + ], + "score": 0.72, + "content": "^ n", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 237, + 505, + 250 + ], + "score": 1.0, + "content": "accuracy. for GRAPHRETRO are also better than", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 248, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 505, + 261 + ], + "score": 1.0, + "content": "most template-based and template-free methods. When the reaction class is known, RETROSIM and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 259, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 505, + 272 + ], + "score": 1.0, + "content": "GLN restrict template sets corresponding to the reaction class, thus improving performance. The", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 269, + 507, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 507, + 284 + ], + "score": 1.0, + "content": "increased edit prediction performance (Section 4.2) for GRAPHRETRO helps outweigh this factor,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 281, + 329, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 298, + 294 + ], + "score": 1.0, + "content": "achieving comparable or better performance till", + "type": "text" + }, + { + "bbox": [ + 299, + 281, + 324, + 291 + ], + "score": 0.9, + "content": "n = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 281, + 329, + 294 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 107, + 314, + 267, + 326 + ], + "lines": [ + { + "bbox": [ + 106, + 314, + 268, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 268, + 327 + ], + "score": 1.0, + "content": "4.2 Individual Module Performance", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 338, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 336, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 432, + 351 + ], + "score": 1.0, + "content": "To gain more insight into the working of GRAPHRETRO, we evaluate the top-", + "type": "text" + }, + { + "bbox": [ + 433, + 340, + 440, + 348 + ], + "score": 0.7, + "content": "\\boldsymbol { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 336, + 483, + 351 + ], + "score": 1.0, + "content": "accuracy", + "type": "text" + }, + { + "bbox": [ + 484, + 339, + 505, + 349 + ], + "score": 0.79, + "content": "\\mathbf { \\nabla } _ { n } =", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 348, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 143, + 360 + ], + "score": 0.59, + "content": "1 , 2 , 3 , 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 348, + 505, + 361 + ], + "score": 1.0, + "content": "of edit prediction and synthon completion modules, along with corresponding ablation", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 359, + 261, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 261, + 371 + ], + "score": 1.0, + "content": "studies, with results shown in Table 2.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 391, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 437, + 404 + ], + "score": 1.0, + "content": "Edit Prediction For the edit prediction module, we compare the true edit(s) to top-", + "type": "text" + }, + { + "bbox": [ + 438, + 394, + 445, + 402 + ], + "score": 0.75, + "content": "\\boldsymbol { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "edits predicted", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "by the model. We also consider two ablation studies, one where we directly use the initial edit scores", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 412, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 104, + 412, + 506, + 427 + ], + "score": 1.0, + "content": "without updating them, and the other where we predict edits using atom-pairs instead of existing", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "score": 1.0, + "content": "bonds and atoms. Both design choices lead to improvements in performance, as shown in Table 2.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "We hypothesize that the larger improvement compared to edit prediction using atom-pairs is due to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 446, + 429, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 429, + 458 + ], + "score": 1.0, + "content": "the easier optimization procedure, with lesser imbalance between labels 1 and 0.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 478, + 505, + 566 + ], + "lines": [ + { + "bbox": [ + 106, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "Synthon Completion For evaluating the synthon completion module, we first apply the true edits", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 104, + 489, + 365, + 502 + ], + "score": 1.0, + "content": "to obtain synthons, and compare the true leaving groups to top-", + "type": "text" + }, + { + "bbox": [ + 365, + 491, + 372, + 500 + ], + "score": 0.73, + "content": "\\mathbf { \\nabla } \\cdot n", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "leaving groups predicted by the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "model. We test the performance of both the ind and shared configurations. Both configurations", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 281, + 524 + ], + "score": 1.0, + "content": "perform similarly, and are able to identify", + "type": "text" + }, + { + "bbox": [ + 281, + 511, + 310, + 522 + ], + "score": 0.83, + "content": "\\sim 9 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 511, + 427, + 524 + ], + "score": 1.0, + "content": "(close to its upper bound of", + "type": "text" + }, + { + "bbox": [ + 428, + 511, + 455, + 522 + ], + "score": 0.86, + "content": "9 9 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 511, + 506, + 524 + ], + "score": 1.0, + "content": ") of the true", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "leaving groups in its top-5 choices, when the reaction class is known. The top-1, 3 and 5 accuracies", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 398, + 545 + ], + "score": 1.0, + "content": "of the synthon completion for unknown reaction classes for G2Gs are", + "type": "text" + }, + { + "bbox": [ + 398, + 533, + 425, + 543 + ], + "score": 0.85, + "content": "6 1 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 533, + 430, + 545 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 430, + 533, + 458, + 544 + ], + "score": 0.85, + "content": "8 1 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 533, + 477, + 545 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 477, + 533, + 505, + 544 + ], + "score": 0.86, + "content": "8 6 . 7 \\%", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 544, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 214, + 556 + ], + "score": 1.0, + "content": "respectively, while ours are", + "type": "text" + }, + { + "bbox": [ + 214, + 544, + 241, + 555 + ], + "score": 0.85, + "content": "7 5 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 544, + 244, + 556 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 244, + 544, + 271, + 554 + ], + "score": 0.85, + "content": "9 2 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 544, + 288, + 556 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 288, + 544, + 315, + 554 + ], + "score": 0.88, + "content": "9 6 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 544, + 366, + 556 + ], + "score": 1.0, + "content": ", indicating a", + "type": "text" + }, + { + "bbox": [ + 366, + 544, + 398, + 554 + ], + "score": 0.87, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 544, + 506, + 556 + ], + "score": 1.0, + "content": "performance improvement", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 555, + 411, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 411, + 567 + ], + "score": 1.0, + "content": "using a classification formulation over the generative one adopted by G2Gs.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5 + }, + { + "type": "table", + "bbox": [ + 106, + 608, + 506, + 720 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 142, + 592, + 467, + 604 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 141, + 591, + 468, + 606 + ], + "spans": [ + { + "bbox": [ + 141, + 591, + 468, + 606 + ], + "score": 1.0, + "content": "Table 2: Performance Study of edit prediction and synthon completion modules", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "table_body", + "bbox": [ + 106, + 608, + 506, + 720 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 608, + 506, + 720 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 506, + 720 + ], + "score": 0.984, + "html": "
SettingTop-n Accuracy (%)
Reaction class knownReaction class unknown
12351235
Edit Prediction84.692.293.794.570.885.189.592.7
- without edit score updates84.392.193.794.570.184.889.492.6
- predicting on atom pairs81.989.590.992.168.683.288.391.8
Synthon Completion (ind)77.489.594.297.675.687.492.596.1
Synthon Completion (shared)76.989.693.997.474.987.792.996.3
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Performance", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 118, + 506, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 259, + 131 + ], + "score": 1.0, + "content": "improvements are also seen for larger", + "type": "text" + }, + { + "bbox": [ + 259, + 120, + 267, + 128 + ], + "score": 0.59, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 118, + 313, + 131 + ], + "score": 1.0, + "content": ", except for", + "type": "text" + }, + { + "bbox": [ + 313, + 119, + 339, + 128 + ], + "score": 0.89, + "content": "n = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 118, + 506, + 131 + ], + "score": 1.0, + "content": ". Barring DUALTB, the top-1 accuracy is", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 128, + 506, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 409, + 142 + ], + "score": 1.0, + "content": "also better than other template-free and template-based methods. For larger", + "type": "text" + }, + { + "bbox": [ + 410, + 131, + 417, + 139 + ], + "score": 0.67, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 128, + 506, + 142 + ], + "score": 1.0, + "content": ", one reason for lower", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 140, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 123, + 152 + ], + "score": 1.0, + "content": "top-", + "type": "text" + }, + { + "bbox": [ + 123, + 142, + 130, + 150 + ], + "score": 0.66, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 140, + 505, + 152 + ], + "score": 1.0, + "content": "accuracies than most template-based methods is that templates already contain combinations", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 150, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 505, + 165 + ], + "score": 1.0, + "content": "of leaving group patterns. In contrast, our model learns to discover these during training. A second", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 162, + 505, + 175 + ], + "spans": [ + { + "bbox": [ + 106, + 162, + 505, + 175 + ], + "score": 1.0, + "content": "hypothesis to this end is that simply adding log-likelihood scores from edit prediction and synthon", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 506, + 186 + ], + "score": 1.0, + "content": "completion models may be suboptimal and bias the beam search in the direction of the more", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 183, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 506, + 195 + ], + "score": 1.0, + "content": "dominating term. We leave it to future work to investigate scoring functions that rank the attachment.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 96, + 506, + 195 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 215, + 505, + 292 + ], + "lines": [ + { + "bbox": [ + 106, + 216, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 505, + 227 + ], + "score": 1.0, + "content": "Reaction class known When the reaction class is known, GRAPHRETRO outperforms G2GS", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 246, + 239 + ], + "score": 1.0, + "content": "and RETROXPERT by a margin of", + "type": "text" + }, + { + "bbox": [ + 247, + 226, + 262, + 237 + ], + "score": 0.88, + "content": "3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 226, + 280, + 239 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 280, + 226, + 294, + 237 + ], + "score": 0.87, + "content": "2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 226, + 505, + 239 + ], + "score": 1.0, + "content": "respectively in top-1 accuracy. GRAPHRETRO also", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 237, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 302, + 250 + ], + "score": 1.0, + "content": "outperforms all the template-free methods in top-", + "type": "text" + }, + { + "bbox": [ + 302, + 239, + 309, + 247 + ], + "score": 0.72, + "content": "^ n", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 237, + 505, + 250 + ], + "score": 1.0, + "content": "accuracy. for GRAPHRETRO are also better than", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 248, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 505, + 261 + ], + "score": 1.0, + "content": "most template-based and template-free methods. When the reaction class is known, RETROSIM and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 259, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 505, + 272 + ], + "score": 1.0, + "content": "GLN restrict template sets corresponding to the reaction class, thus improving performance. The", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 269, + 507, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 507, + 284 + ], + "score": 1.0, + "content": "increased edit prediction performance (Section 4.2) for GRAPHRETRO helps outweigh this factor,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 281, + 329, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 298, + 294 + ], + "score": 1.0, + "content": "achieving comparable or better performance till", + "type": "text" + }, + { + "bbox": [ + 299, + 281, + 324, + 291 + ], + "score": 0.9, + "content": "n = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 281, + 329, + 294 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 216, + 507, + 294 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 314, + 267, + 326 + ], + "lines": [ + { + "bbox": [ + 106, + 314, + 268, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 268, + 327 + ], + "score": 1.0, + "content": "4.2 Individual Module Performance", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 338, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 336, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 432, + 351 + ], + "score": 1.0, + "content": "To gain more insight into the working of GRAPHRETRO, we evaluate the top-", + "type": "text" + }, + { + "bbox": [ + 433, + 340, + 440, + 348 + ], + "score": 0.7, + "content": "\\boldsymbol { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 336, + 483, + 351 + ], + "score": 1.0, + "content": "accuracy", + "type": "text" + }, + { + "bbox": [ + 484, + 339, + 505, + 349 + ], + "score": 0.79, + "content": "\\mathbf { \\nabla } _ { n } =", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 348, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 143, + 360 + ], + "score": 0.59, + "content": "1 , 2 , 3 , 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 348, + 505, + 361 + ], + "score": 1.0, + "content": "of edit prediction and synthon completion modules, along with corresponding ablation", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 359, + 261, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 261, + 371 + ], + "score": 1.0, + "content": "studies, with results shown in Table 2.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 336, + 505, + 371 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 391, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 437, + 404 + ], + "score": 1.0, + "content": "Edit Prediction For the edit prediction module, we compare the true edit(s) to top-", + "type": "text" + }, + { + "bbox": [ + 438, + 394, + 445, + 402 + ], + "score": 0.75, + "content": "\\boldsymbol { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "edits predicted", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "by the model. We also consider two ablation studies, one where we directly use the initial edit scores", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 412, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 104, + 412, + 506, + 427 + ], + "score": 1.0, + "content": "without updating them, and the other where we predict edits using atom-pairs instead of existing", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "score": 1.0, + "content": "bonds and atoms. Both design choices lead to improvements in performance, as shown in Table 2.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "We hypothesize that the larger improvement compared to edit prediction using atom-pairs is due to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 446, + 429, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 429, + 458 + ], + "score": 1.0, + "content": "the easier optimization procedure, with lesser imbalance between labels 1 and 0.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5, + "bbox_fs": [ + 104, + 390, + 506, + 458 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 478, + 505, + 566 + ], + "lines": [ + { + "bbox": [ + 106, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "Synthon Completion For evaluating the synthon completion module, we first apply the true edits", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 104, + 489, + 365, + 502 + ], + "score": 1.0, + "content": "to obtain synthons, and compare the true leaving groups to top-", + "type": "text" + }, + { + "bbox": [ + 365, + 491, + 372, + 500 + ], + "score": 0.73, + "content": "\\mathbf { \\nabla } \\cdot n", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "leaving groups predicted by the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "model. We test the performance of both the ind and shared configurations. Both configurations", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 281, + 524 + ], + "score": 1.0, + "content": "perform similarly, and are able to identify", + "type": "text" + }, + { + "bbox": [ + 281, + 511, + 310, + 522 + ], + "score": 0.83, + "content": "\\sim 9 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 511, + 427, + 524 + ], + "score": 1.0, + "content": "(close to its upper bound of", + "type": "text" + }, + { + "bbox": [ + 428, + 511, + 455, + 522 + ], + "score": 0.86, + "content": "9 9 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 511, + 506, + 524 + ], + "score": 1.0, + "content": ") of the true", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "leaving groups in its top-5 choices, when the reaction class is known. The top-1, 3 and 5 accuracies", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 398, + 545 + ], + "score": 1.0, + "content": "of the synthon completion for unknown reaction classes for G2Gs are", + "type": "text" + }, + { + "bbox": [ + 398, + 533, + 425, + 543 + ], + "score": 0.85, + "content": "6 1 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 533, + 430, + 545 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 430, + 533, + 458, + 544 + ], + "score": 0.85, + "content": "8 1 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 533, + 477, + 545 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 477, + 533, + 505, + 544 + ], + "score": 0.86, + "content": "8 6 . 7 \\%", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 544, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 214, + 556 + ], + "score": 1.0, + "content": "respectively, while ours are", + "type": "text" + }, + { + "bbox": [ + 214, + 544, + 241, + 555 + ], + "score": 0.85, + "content": "7 5 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 544, + 244, + 556 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 244, + 544, + 271, + 554 + ], + "score": 0.85, + "content": "9 2 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 544, + 288, + 556 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 288, + 544, + 315, + 554 + ], + "score": 0.88, + "content": "9 6 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 544, + 366, + 556 + ], + "score": 1.0, + "content": ", indicating a", + "type": "text" + }, + { + "bbox": [ + 366, + 544, + 398, + 554 + ], + "score": 0.87, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 544, + 506, + 556 + ], + "score": 1.0, + "content": "performance improvement", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 555, + 411, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 411, + 567 + ], + "score": 1.0, + "content": "using a classification formulation over the generative one adopted by G2Gs.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 478, + 506, + 567 + ] + }, + { + "type": "table", + "bbox": [ + 106, + 608, + 506, + 720 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 142, + 592, + 467, + 604 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 141, + 591, + 468, + 606 + ], + "spans": [ + { + "bbox": [ + 141, + 591, + 468, + 606 + ], + "score": 1.0, + "content": "Table 2: Performance Study of edit prediction and synthon completion modules", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "table_body", + "bbox": [ + 106, + 608, + 506, + 720 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 608, + 506, + 720 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 506, + 720 + ], + "score": 0.984, + "html": "
SettingTop-n Accuracy (%)
Reaction class knownReaction class unknown
12351235
Edit Prediction84.692.293.794.570.885.189.592.7
- without edit score updates84.392.193.794.570.184.889.492.6
- predicting on atom pairs81.989.590.992.168.683.288.391.8
Synthon Completion (ind)77.489.594.297.675.687.492.596.1
Synthon Completion (shared)76.989.693.997.474.987.792.996.3
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Figure 2a shows", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 117 + ], + "score": 1.0, + "content": "an example where the model identifies both the edits and leaving groups correctly. In Figure 2b, the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 113, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 104, + 113, + 506, + 128 + ], + "score": 1.0, + "content": "correct edit is identified but the predicted leaving groups are incorrect. We hypothesize this is due to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 124, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 421, + 139 + ], + "score": 1.0, + "content": "the fact that in the training set, leaving groups attaching to the carbonyl carbon", + "type": "text" + }, + { + "bbox": [ + 422, + 126, + 446, + 136 + ], + "score": 0.67, + "content": "\\scriptstyle ( \\mathbf { C } = \\mathbf { O } )", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 124, + 506, + 139 + ], + "score": 1.0, + "content": "are small (e.g.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 136, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 104, + 136, + 129, + 149 + ], + "score": 1.0, + "content": "-OH, -", + "type": "text" + }, + { + "bbox": [ + 129, + 136, + 150, + 147 + ], + "score": 0.87, + "content": "\\mathrm { - N H _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 136, + 506, + 149 + ], + "score": 1.0, + "content": ", halides). The true leaving group in this example, however, is large. The model is unable", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 159 + ], + "score": 1.0, + "content": "to reason about this and predicts the small leaving group -I. In Figure 2c, the model identifies the edit", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 158, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 104, + 158, + 506, + 171 + ], + "score": 1.0, + "content": "and consequently the leaving group incorrectly. This highlights a limitation of our model. If the edit", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 385, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 385, + 182 + ], + "score": 1.0, + "content": "is predicted incorrectly, the model cannot suggest the true precursors.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4.5 + }, + { + "type": "title", + "bbox": [ + 107, + 193, + 180, + 205 + ], + "lines": [ + { + "bbox": [ + 105, + 191, + 182, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 182, + 206 + ], + "score": 1.0, + "content": "4.4 Limitations", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 213, + 505, + 323 + ], + "lines": [ + { + "bbox": [ + 106, + 213, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 506, + 225 + ], + "score": 1.0, + "content": "The simplified and interpretable construction of GRAPHRETRO comes with certain limitations. First,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 224, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 224, + 505, + 235 + ], + "score": 1.0, + "content": "the overall performance of the model is limited by the performance of the edit prediction step. If the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 235, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 505, + 248 + ], + "score": 1.0, + "content": "predicted edit is incorrect, the true reactants cannot be salvaged. This limitation is partly remedied", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 246, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 258 + ], + "score": 1.0, + "content": "by our model design, that allows for user intervention to correct the edit. 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Extracting edits directly based on", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 269, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 505, + 280 + ], + "score": 1.0, + "content": "substructure matching currently suffer from false positives, and heuristics to correct for these result", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 201, + 291 + ], + "score": 1.0, + "content": "in correct edits in only", + "type": "text" + }, + { + "bbox": [ + 202, + 279, + 227, + 289 + ], + "score": 0.89, + "content": "{ \\sim } 9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 278, + 506, + 291 + ], + "score": 1.0, + "content": "of the cases. Third, our formulation assumes that we have as many", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 290, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 302 + ], + "score": 1.0, + "content": "synthons as reactants, which is violated in some reactions. 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The true edit and incorrect edit (if any) are highlighted in green", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "and red respectively. The true and predicted leaving groups are highlighted in blue. a. Correctly", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 622, + 507, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 507, + 635 + ], + "score": 1.0, + "content": "predicted example by the model. b. Correctly predicted edit but incorrectly predicted leaving groups.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 631, + 297, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 297, + 646 + ], + "score": 1.0, + "content": "c. Incorrectly predicted edit and leaving group.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40.5 + } + ], + "index": 34.75 + }, + { + "type": "title", + "bbox": [ + 107, + 664, + 183, + 678 + ], + "lines": [ + { + "bbox": [ + 104, + 662, + 185, + 680 + ], + "spans": [ + { + "bbox": [ + 104, + 662, + 185, + 680 + ], + "score": 1.0, + "content": "5 Conclusion", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 689, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "score": 1.0, + "content": "Previous methods for single-step retrosynthesis either restrict prediction to a template set, are", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "insensitive to molecular graph structure or generate molecules from scratch. 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Figure 2a shows", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 117 + ], + "score": 1.0, + "content": "an example where the model identifies both the edits and leaving groups correctly. In Figure 2b, the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 113, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 104, + 113, + 506, + 128 + ], + "score": 1.0, + "content": "correct edit is identified but the predicted leaving groups are incorrect. We hypothesize this is due to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 124, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 421, + 139 + ], + "score": 1.0, + "content": "the fact that in the training set, leaving groups attaching to the carbonyl carbon", + "type": "text" + }, + { + "bbox": [ + 422, + 126, + 446, + 136 + ], + "score": 0.67, + "content": "\\scriptstyle ( \\mathbf { C } = \\mathbf { O } )", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 124, + 506, + 139 + ], + "score": 1.0, + "content": "are small (e.g.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 136, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 104, + 136, + 129, + 149 + ], + "score": 1.0, + "content": "-OH, -", + "type": "text" + }, + { + "bbox": [ + 129, + 136, + 150, + 147 + ], + "score": 0.87, + "content": "\\mathrm { - N H _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 136, + 506, + 149 + ], + "score": 1.0, + "content": ", halides). The true leaving group in this example, however, is large. The model is unable", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 159 + ], + "score": 1.0, + "content": "to reason about this and predicts the small leaving group -I. In Figure 2c, the model identifies the edit", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 158, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 104, + 158, + 506, + 171 + ], + "score": 1.0, + "content": "and consequently the leaving group incorrectly. This highlights a limitation of our model. If the edit", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 385, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 385, + 182 + ], + "score": 1.0, + "content": "is predicted incorrectly, the model cannot suggest the true precursors.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4.5, + "bbox_fs": [ + 104, + 93, + 506, + 182 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 193, + 180, + 205 + ], + "lines": [ + { + "bbox": [ + 105, + 191, + 182, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 182, + 206 + ], + "score": 1.0, + "content": "4.4 Limitations", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 213, + 505, + 323 + ], + "lines": [ + { + "bbox": [ + 106, + 213, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 506, + 225 + ], + "score": 1.0, + "content": "The simplified and interpretable construction of GRAPHRETRO comes with certain limitations. First,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 224, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 224, + 505, + 235 + ], + "score": 1.0, + "content": "the overall performance of the model is limited by the performance of the edit prediction step. If the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 235, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 505, + 248 + ], + "score": 1.0, + "content": "predicted edit is incorrect, the true reactants cannot be salvaged. This limitation is partly remedied", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 246, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 258 + ], + "score": 1.0, + "content": "by our model design, that allows for user intervention to correct the edit. Second, our method is", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 257, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 270 + ], + "score": 1.0, + "content": "reliant on atom-mapping for extracting edits and leaving groups. Extracting edits directly based on", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 269, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 505, + 280 + ], + "score": 1.0, + "content": "substructure matching currently suffer from false positives, and heuristics to correct for these result", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 201, + 291 + ], + "score": 1.0, + "content": "in correct edits in only", + "type": "text" + }, + { + "bbox": [ + 202, + 279, + 227, + 289 + ], + "score": 0.89, + "content": "{ \\sim } 9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 278, + 506, + 291 + ], + "score": 1.0, + "content": "of the cases. Third, our formulation assumes that we have as many", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 290, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 302 + ], + "score": 1.0, + "content": "synthons as reactants, which is violated in some reactions. We leave it to future work to extend the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 300, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 313 + ], + "score": 1.0, + "content": "model to realize a single reactant from multiple synthons, and introduce more chemically meaningful", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 311, + 221, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 221, + 324 + ], + "score": 1.0, + "content": "edit correction mechanisms.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 213, + 506, + 324 + ] + }, + { + "type": "image", + "bbox": [ + 164, + 331, + 435, + 589 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 164, + 331, + 435, + 589 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 164, + 331, + 435, + 589 + ], + "spans": [ + { + "bbox": [ + 164, + 331, + 435, + 589 + ], + "score": 0.969, + "type": "image", + "image_path": "9df65c90e84b75ac55b8dc08692726ee93cf492df4b05d29c79c8c792f98fccd.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 164, + 331, + 435, + 344.57894736842104 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 164, + 344.57894736842104, + 435, + 358.1578947368421 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 164, + 358.1578947368421, + 435, + 371.7368421052631 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 164, + 371.7368421052631, + 435, + 385.31578947368416 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 164, + 385.31578947368416, + 435, + 398.8947368421052 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 164, + 398.8947368421052, + 435, + 412.47368421052624 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 164, + 412.47368421052624, + 435, + 426.0526315789473 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 164, + 426.0526315789473, + 435, + 439.6315789473683 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 164, + 439.6315789473683, + 435, + 453.21052631578937 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 164, + 453.21052631578937, + 435, + 466.7894736842104 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 164, + 466.7894736842104, + 435, + 480.36842105263145 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 164, + 480.36842105263145, + 435, + 493.9473684210525 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 164, + 493.9473684210525, + 435, + 507.52631578947353 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 164, + 507.52631578947353, + 435, + 521.1052631578946 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 164, + 521.1052631578946, + 435, + 534.6842105263156 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 164, + 534.6842105263156, + 435, + 548.2631578947367 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 164, + 548.2631578947367, + 435, + 561.8421052631577 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 164, + 561.8421052631577, + 435, + 575.4210526315787 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 164, + 575.4210526315787, + 435, + 588.9999999999998 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 599, + 506, + 644 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "Figure 2: Example Predictions. The true edit and incorrect edit (if any) are highlighted in green", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "and red respectively. The true and predicted leaving groups are highlighted in blue. a. Correctly", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 622, + 507, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 507, + 635 + ], + "score": 1.0, + "content": "predicted example by the model. b. Correctly predicted edit but incorrectly predicted leaving groups.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 631, + 297, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 297, + 646 + ], + "score": 1.0, + "content": "c. Incorrectly predicted edit and leaving group.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40.5 + } + ], + "index": 34.75 + }, + { + "type": "title", + "bbox": [ + 107, + 664, + 183, + 678 + ], + "lines": [ + { + "bbox": [ + 104, + 662, + 185, + 680 + ], + "spans": [ + { + "bbox": [ + 104, + 662, + 185, + 680 + ], + "score": 1.0, + "content": "5 Conclusion", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 689, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "score": 1.0, + "content": "Previous methods for single-step retrosynthesis either restrict prediction to a template set, are", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "insensitive to molecular graph structure or generate molecules from scratch. We address these", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 711, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 506, + 724 + ], + "score": 1.0, + "content": "shortcomings by introducing a graph-based semi-template-based model inspired by a chemist’s", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "workflow, enhancing the interpretability of retrosynthesis models. Given a target molecule, we", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 95 + ], + "score": 1.0, + "content": "first identify synthetic building blocks (synthons) which are then realized into valid reactants, thus", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 504, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 504, + 106 + ], + "score": 1.0, + "content": "avoiding molecule generation from scratch. Our model outperforms previous semi-template-methods", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "by significant margins on the benchmark dataset. Future work aims to extend the model to realize a", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 504, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 504, + 128 + ], + "score": 1.0, + "content": "single reactant from multiple synthons, and introduce more chemically meaningful components to", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 128, + 501, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 501, + 139 + ], + "score": 1.0, + "content": "improve the synergy between such tools for retrosynthesis prediction and a practitioner’s expertise.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 688, + 506, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "workflow, enhancing the interpretability of retrosynthesis models. Given a target molecule, we", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 95 + ], + "score": 1.0, + "content": "first identify synthetic building blocks (synthons) which are then realized into valid reactants, thus", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 504, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 504, + 106 + ], + "score": 1.0, + "content": "avoiding molecule generation from scratch. Our model outperforms previous semi-template-methods", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "by significant margins on the benchmark dataset. Future work aims to extend the model to realize a", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 504, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 504, + 128 + ], + "score": 1.0, + "content": "single reactant from multiple synthons, and introduce more chemically meaningful components to", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 128, + 501, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 501, + 139 + ], + "score": 1.0, + "content": "improve the synergy between such tools for retrosynthesis prediction and a practitioner’s expertise.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 108, + 154, + 207, + 167 + ], + "lines": [ + { + "bbox": [ + 105, + 152, + 208, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 152, + 208, + 170 + ], + "score": 1.0, + "content": "Acknowledgements", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 178, + 505, + 234 + ], + "lines": [ + { + "bbox": [ + 106, + 179, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 106, + 179, + 505, + 191 + ], + "score": 1.0, + "content": "This research was supported by the Machine Learning for Pharmaceutical Discovery and Synthesis", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 189, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 505, + 203 + ], + "score": 1.0, + "content": "Consortium at MIT. V.R.S. was also supported by the Zeno Karl Schindler Foundation. C.B. was", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 200, + 506, + 214 + ], + "spans": [ + { + "bbox": [ + 104, + 200, + 506, + 214 + ], + "score": 1.0, + "content": "supported by the Swiss National Science Foundation under the National Center of Competence", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 211, + 506, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 506, + 224 + ], + "score": 1.0, + "content": "in Research (NCCR) Catalysis under grant agreement 51NF40 180544. 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ModelTop-n Accuracy (%)
Reaction class knownReaction class unknown
1351013510
Template-Based
RETROSIM [Coley et al.,2017b]52.973.881.288.137.354.763.374.1
NEURALSYM [Segler and Waller,2017]55.376.081.485.144.465.372.478.9
GLN [Dai et ai., 2019]64.279.185.290.052.569.075.683.7
DUALTB [Sun et al.,2021]67.784.888.992.055.274.680.586.9
Template-Free
SCROP [Zheng et al.,2019]59.074.878.181.143.760.065.268.7
LV-TRANSFORMER [Chen et al.,2019]----40.565.172.879.4
DUALTF [Sun et al., 2021]65.781.984.785.953.670.774.677.0
Semi-Template-Based
G2Gs [Shi et al.,2020]61.081.386.088.748.967.672.575.5
RETROXPERT [Yan et al.,2020]62.175.878.580.950.461.162.363.4
GRAPHRETRO (ours)63.981.585.288.153.768.372.275.5
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SettingTop-n Accuracy (%)
Reaction class knownReaction class unknown
12351235
Edit Prediction84.692.293.794.570.885.189.592.7
- without edit score updates84.392.193.794.570.184.889.492.6
- predicting on atom pairs81.989.590.992.168.683.288.391.8
Synthon Completion (ind)77.489.594.297.675.687.492.596.1
Synthon Completion (shared)76.989.693.997.474.987.792.996.3
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Specifically, individual models are first trained and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 378, + 469, + 390 + ], + "spans": [ + { + "bbox": [ + 141, + 378, + 469, + 390 + ], + "score": 1.0, + "content": "regarded as teachers, and then the multilingual model is trained to fit the training", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 388, + 469, + 401 + ], + "spans": [ + { + "bbox": [ + 141, + 388, + 469, + 401 + ], + "score": 1.0, + "content": "data and match the outputs of individual models simultaneously through knowl-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 400, + 470, + 412 + ], + "spans": [ + { + "bbox": [ + 142, + 400, + 470, + 412 + ], + "score": 1.0, + "content": "edge distillation. Experiments on IWSLT, WMT and Ted talk translation datasets", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 411, + 469, + 422 + ], + "spans": [ + { + "bbox": [ + 142, + 411, + 469, + 422 + ], + "score": 1.0, + "content": "demonstrate the effectiveness of our method. Particularly, we show that one model", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 421, + 469, + 434 + ], + "spans": [ + { + "bbox": [ + 141, + 421, + 469, + 434 + ], + "score": 1.0, + "content": "is enough to handle multiple languages (up to 44 languages in our experiment),", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 433, + 402, + 444 + ], + "spans": [ + { + "bbox": [ + 142, + 433, + 402, + 444 + ], + "score": 1.0, + "content": "with comparable or even better accuracy than individual models.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16, + "bbox_fs": [ + 141, + 300, + 470, + 444 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 466, + 206, + 479 + ], + "lines": [ + { + "bbox": [ + 105, + 465, + 208, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 208, + 482 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 492, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 492, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 505, + 504 + ], + "score": 1.0, + "content": "Neural Machine Translation (NMT) has witnessed rapid development in recent years (Bahdanau", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 503, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 506, + 515 + ], + "score": 1.0, + "content": "et al., 2015; Luong et al., 2015b; Wu et al., 2016; Gehring et al., 2017; Vaswani et al., 2017; Wu", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "et al., 2018; Song et al., 2018; Shen et al., 2018; Guo et al., 2018; He et al., 2018; Gong et al.,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 524, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 538 + ], + "score": 1.0, + "content": "2018), including advanced model structures (Gehring et al., 2017; Vaswani et al., 2017) and human", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "score": 1.0, + "content": "parity achievements (Hassan et al., 2018). While conventional NMT can well handle single pair", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "score": 1.0, + "content": "translation, training a separate model for each language pair is resource consuming, considering", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "score": 1.0, + "content": "there are thousands of languages in the world1. Therefore, multilingual NMT (Johnson et al., 2017;", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 566, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 104, + 566, + 506, + 583 + ], + "score": 1.0, + "content": "Firat et al., 2016; Ha et al., 2016; Lu et al., 2018) is developed which handles multiple language", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 580, + 426, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 426, + 592 + ], + "score": 1.0, + "content": "pairs in one model, greatly reducing the offline training and online serving cost.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28, + "bbox_fs": [ + 104, + 492, + 506, + 592 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 596, + 505, + 674 + ], + "lines": [ + { + "bbox": [ + 105, + 597, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 609 + ], + "score": 1.0, + "content": "Previous works on multilingual NMT mainly focus on model architecture design through parameter", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 607, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 506, + 621 + ], + "score": 1.0, + "content": "sharing, e.g., sharing encoder, decoder or attention module (Firat et al., 2016; Lu et al., 2018) or", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 617, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 505, + 632 + ], + "score": 1.0, + "content": "sharing the entire models (Johnson et al., 2017; Ha et al., 2016). They achieve comparable accuracy", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 629, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 506, + 643 + ], + "score": 1.0, + "content": "with individual models (each language pair with a separate model) when the languages are similar to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 640, + 506, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 506, + 654 + ], + "score": 1.0, + "content": "each other and the number of language pairs is small (e.g., two or three). However, when handling", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 651, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 505, + 664 + ], + "score": 1.0, + "content": "more language pairs (dozens or even hundreds), the translation accuracy of multilingual model is", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 662, + 361, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 361, + 676 + ], + "score": 1.0, + "content": "usually inferior to individual models, due to language diversity.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 597, + 506, + 676 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 679, + 503, + 702 + ], + "lines": [ + { + "bbox": [ + 106, + 679, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 106, + 679, + 505, + 692 + ], + "score": 1.0, + "content": "It is challenging to train a multilingual translation model supporting dozens of language pairs while", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 690, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 505, + 703 + ], + "score": 1.0, + "content": "achieving comparable accuracy as individual models. 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The student model can (nearly)", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 505, + 182 + ], + "score": 1.0, + "content": "match the accuracy of the cumbersome teacher model (or the ensemble of multiple models) with", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 200, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 200, + 193 + ], + "score": 1.0, + "content": "knowledge distillation.", + "type": "text", + "cross_page": true + } + ], + "index": 9 + } + ], + "index": 40.5, + "bbox_fs": [ + 106, + 679, + 505, + 703 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 192 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "of higher accuracy than the multilingual model in conventional model training, we propose to trans-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "fer the knowledge from individual models to the multilingual model with knowledge distillation,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "which has been studied for model compression and knowledge transfer and well matches our setting", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "of multilingual translation. It usually starts by training a big/deep teacher model (or ensemble of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "multiple models), and then train a small/shallow student model to mimic the behaviors of the teacher", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "model, such as its hidden representation (Yim et al., 2017; Romero et al., 2014), its output probabil-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 161 + ], + "score": 1.0, + "content": "ities (Hinton et al., 2015; Freitag et al., 2017) or directly training on the sentences generated by the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "teacher model in neural machine translation (Kim & Rush, 2016a). The student model can (nearly)", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 505, + 182 + ], + "score": 1.0, + "content": "match the accuracy of the cumbersome teacher model (or the ensemble of multiple models) with", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 200, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 200, + 193 + ], + "score": 1.0, + "content": "knowledge distillation.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 198, + 505, + 318 + ], + "lines": [ + { + "bbox": [ + 105, + 198, + 504, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 504, + 210 + ], + "score": 1.0, + "content": "In this paper, we propose a new method based on knowledge distillation for multilingual transla-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "tion to eliminate the accuracy gap between the multilingual model and individual models. In our", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "method, multiple individual models serve as teachers, each handling a separate language pair, while", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "the student handles all the language pairs in a single model, which is different from the conventional", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 243, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 505, + 254 + ], + "score": 1.0, + "content": "knowledge distillation where the teacher and student models usually handle the same task. We first", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "train the individual models for each translation pair and then we train the multilingual model by", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "score": 1.0, + "content": "matching with the outputs of all the individual models and the ground-truth translation simultane-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 273, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 288 + ], + "score": 1.0, + "content": "ously. After some iterations of training, the multilingual model may get higher translation accuracy", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "than the individual models on some language pairs. Then we remove the distillation loss and keep", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 297, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 310 + ], + "score": 1.0, + "content": "training the multilingual model on these languages pairs with the original log-likelihood loss of the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 308, + 206, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 206, + 319 + ], + "score": 1.0, + "content": "ground-truth translation.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 505, + 380 + ], + "lines": [ + { + "bbox": [ + 106, + 324, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 506, + 338 + ], + "score": 1.0, + "content": "We conduct experiments on three translation datasets: IWSLT with 12 language pairs, WMT with 6", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "language pairs and Ted talk with 44 language pairs. 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Anil et al.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 306, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 505, + 318 + ], + "score": 1.0, + "content": "(2018) propose online distillation to improve the scalability of distributed model training and the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 317, + 180, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 180, + 331 + ], + "score": 1.0, + "content": "training accuracy.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 334, + 505, + 378 + ], + "lines": [ + { + "bbox": [ + 105, + 334, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 346 + ], + "score": 1.0, + "content": "In this paper, we develop the multilingual distillation framework for multilingual NMT. Our work", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "differs from Zhang et al. (2017) and Anil et al. 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This can reduce the memory cost again from", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 271, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 155, + 284 + ], + "score": 1.0, + "content": "the scale of", + "type": "text" + }, + { + "bbox": [ + 155, + 271, + 170, + 284 + ], + "score": 0.88, + "content": "| V |", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "(the vocabulary size) to K. 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We also list the language code according", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "to ISO-639-1 standard4 for the languages used in our experiments in Appendix (Section 2). All the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 471, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 104, + 471, + 505, + 487 + ], + "score": 1.0, + "content": "sentences are first tokenized with moses tokenizer5 and then segmented into subword symbols using", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 484, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 496 + ], + "score": 1.0, + "content": "Byte Pair Encoding (BPE) (Sennrich et al., 2016). 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We use the same model configuration for individual models and the multilingual model.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 317, + 577 + ], + "score": 1.0, + "content": "For IWSLT and Ted talk tasks, the model hidden size", + "type": "text" + }, + { + "bbox": [ + 318, + 564, + 342, + 576 + ], + "score": 0.91, + "content": "d _ { \\mathrm { m o d e l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 564, + 446, + 577 + ], + "score": 1.0, + "content": ", feed-forward hidden size", + "type": "text" + }, + { + "bbox": [ + 447, + 565, + 458, + 575 + ], + "score": 0.87, + "content": "d _ { \\mathrm { f f } }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 564, + 506, + 577 + ], + "score": 1.0, + "content": ", number of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 574, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 588 + ], + "score": 1.0, + "content": "layer are 256, 1024 and 2, while for WMT task, the three parameters are 512, 2048 and 6 respectively", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 587, + 277, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 277, + 598 + ], + "score": 1.0, + "content": "considering its large scale of training data.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 531, + 506, + 598 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 611, + 505, + 677 + ], + "lines": [ + { + "bbox": [ + 106, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "Training and Inference For the multilingual model training, we up sample the data of each lan-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "guage to make all languages have the same size of data. The mini batch size is set to roughly 8192", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "tokens. We train the individual models with 4 NVIDIA Tesla V100 GPU cards and multilingual", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "models with 8 of them. We follow the default parameters of Adam optimizer (Kingma & Ba, 2014)", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "and learning rate schedule in Vaswani et al. (2017). 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For", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 227, + 96 + ], + "score": 1.0, + "content": "knowledge distillation, we set", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 227, + 83, + 284, + 94 + ], + "score": 0.91, + "content": "\\mathcal { T } _ { \\mathrm { c h e c k } } = 3 0 0 0", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 284, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "steps (nearly two training epochs), the accuracy thresh-", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 122, + 106 + ], + "score": 1.0, + "content": "old", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 123, + 94, + 151, + 104 + ], + "score": 0.89, + "content": "\\tau = 1", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 152, + 92, + 316, + 106 + ], + "score": 1.0, + "content": "BLEU score, the distillation coefficient", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 317, + 94, + 353, + 104 + ], + "score": 0.92, + "content": "\\lambda = 0 . 5", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 354, + 92, + 505, + 106 + ], + "score": 1.0, + "content": "and the number of teacher’s outputs", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 138, + 115 + ], + "score": 0.9, + "content": "K = 8", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 139, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "according to the validation performance. 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LanguageIndividualMulti-BaselineMulti-Distillation△
Ar→En31.1929.24 (-1.95)31.25 (+0.06)+2.01
Cs-→En28.0426.09 (-1.95)27.09 (-0.95)+1.00
De→En33.0732.74 (-0.33)34.02 (+0.95)+1.28
He-→En37.4235.18 (-2.24)37.33 (-0.09)+2.15
N1-→En35.9436.54 (+0.60)37.69 (+1.75)+1.15
Pt-→En44.3043.49 (-0.81)44.69 (+0.39)+1.20
Ro-→En36.9236.41 1 (-0.51)38.01 (+1.09)+1.60
Ru→En23.0423.12 (+0.08)23.76 (+0.72)+0.64
Th→En18.2419.33 3 (+1.09)19.90 (+1.66)+0.57
Tr→En22.7422.42 (-0.32)23.75 (+1.01)+1.33
Vi→En26.0626.37 (+0.31)27.04 4 (+0.98)+0.67
Zh→En19.4418.82 2 (-0.62)19.52 (+0.08)+0.70
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LanguageIndividualMulti-BaselineMulti-Distillation△
En→Ar13.6712.73 (-0.94)13.80 (+0.13)+1.07
En→Cs17.8117.33 (-0.48)18.69 (+0.88)+1.37
En→De26.1325.16 (-0.97)26.76 (+0.63)+1.60
En→He24.1522.73 (-1.42)24.42 (+0.27)+1.69
En-→Nl30.8829.51 (-1.37)30.52 (-0.36)+1.01
En→Pt37.6335.93 (-1.70)37.23 (-0.40)+1.30
En→Ro27.2325.68 (-1.55)27.11 (-0.12)+1.42
En→Ru17.4016.26 (-1.14)17.42 (+0.02)+1.16
En→Th26.4527.18 (+0.73)27.62 (+1.17)+0.45
En→Tr12.4711.63 (-0.84)12.84 (+0.37)+1.21
En→Vi27.8828.04 (+0.16)28.69 (+0.81)+0.65
En→Zh10.9510.12 (-0.83)10.41 (-0.54)+0.29
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The BLEU scores in ()", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 435, + 532 + ], + "score": 1.0, + "content": "represent the difference between the multilingual model and individual models.", + "type": "text" + }, + { + "bbox": [ + 435, + 520, + 445, + 530 + ], + "score": 0.7, + "content": "\\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "represents the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 531, + 390, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 390, + 542 + ], + "score": 1.0, + "content": "improvements of our multi-distillation method over the multi-baseline.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "title", + "bbox": [ + 107, + 561, + 170, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 171, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 171, + 574 + ], + "score": 1.0, + "content": "4.2 RESULTS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 581, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 594 + ], + "score": 1.0, + "content": "Results on IWSLT Multilingual NMT usually consists of three settings: many-to-one, one-to-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "many and many-to-many. As many-many translation can be bridged though many-to-one and one-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 604, + 506, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 616 + ], + "score": 1.0, + "content": "to-many setting, we just conduct the experiments on many-to-one and one-to-many settings. We first", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 615, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 236, + 627 + ], + "score": 1.0, + "content": "show the results of 12 languages", + "type": "text" + }, + { + "bbox": [ + 237, + 616, + 247, + 625 + ], + "score": 0.79, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 615, + 505, + 627 + ], + "score": 1.0, + "content": "English translations on the IWLST dataset are shown in Table 1.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 625, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 505, + 638 + ], + "score": 1.0, + "content": "There are 3 methods for comparison: 1) Individual, each language pair with a separate model; 2)", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 636, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 505, + 650 + ], + "score": 1.0, + "content": "Multi-Baseline, the baseline multilingual model, simply training all the language pairs in one model;", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 648, + 504, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 504, + 660 + ], + "score": 1.0, + "content": "3) Multi-Distillation, our multilingual model with knowledge distillation. We have several obser-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 657, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 505, + 673 + ], + "score": 1.0, + "content": "vations. First, the multilingual baseline performs worse than individual models on most languages.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 668, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 683 + ], + "score": 1.0, + "content": "The only exception is the languages with small training data, which benefit from data augmentation", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 681, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 693 + ], + "score": 1.0, + "content": "in multilingual training. 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LanguageIndividualMulti-BaselineMulti-Distillation△
Ar→En31.1929.24 (-1.95)31.25 (+0.06)+2.01
Cs-→En28.0426.09 (-1.95)27.09 (-0.95)+1.00
De→En33.0732.74 (-0.33)34.02 (+0.95)+1.28
He-→En37.4235.18 (-2.24)37.33 (-0.09)+2.15
N1-→En35.9436.54 (+0.60)37.69 (+1.75)+1.15
Pt-→En44.3043.49 (-0.81)44.69 (+0.39)+1.20
Ro-→En36.9236.41 1 (-0.51)38.01 (+1.09)+1.60
Ru→En23.0423.12 (+0.08)23.76 (+0.72)+0.64
Th→En18.2419.33 3 (+1.09)19.90 (+1.66)+0.57
Tr→En22.7422.42 (-0.32)23.75 (+1.01)+1.33
Vi→En26.0626.37 (+0.31)27.04 4 (+0.98)+0.67
Zh→En19.4418.82 2 (-0.62)19.52 (+0.08)+0.70
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LanguageIndividualMulti-BaselineMulti-Distillation△
En→Ar13.6712.73 (-0.94)13.80 (+0.13)+1.07
En→Cs17.8117.33 (-0.48)18.69 (+0.88)+1.37
En→De26.1325.16 (-0.97)26.76 (+0.63)+1.60
En→He24.1522.73 (-1.42)24.42 (+0.27)+1.69
En-→Nl30.8829.51 (-1.37)30.52 (-0.36)+1.01
En→Pt37.6335.93 (-1.70)37.23 (-0.40)+1.30
En→Ro27.2325.68 (-1.55)27.11 (-0.12)+1.42
En→Ru17.4016.26 (-1.14)17.42 (+0.02)+1.16
En→Th26.4527.18 (+0.73)27.62 (+1.17)+0.45
En→Tr12.4711.63 (-0.84)12.84 (+0.37)+1.21
En→Vi27.8828.04 (+0.16)28.69 (+0.81)+0.65
En→Zh10.9510.12 (-0.83)10.41 (-0.54)+0.29
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We still improve over the multilingual", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 455, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 455, + 166 + ], + "score": 1.0, + "content": "baseline by nearly 1 BLEU score, which demonstrates the effectiveness of our method.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4 + }, + { + "type": "table", + "bbox": [ + 145, + 177, + 466, + 260 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 145, + 177, + 466, + 260 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 145, + 177, + 466, + 260 + ], + "spans": [ + { + "bbox": [ + 145, + 177, + 466, + 260 + ], + "score": 0.978, + "html": "
LanguageIndividualMulti-BaselineMulti-Distillation△
Cs-En25.2923.82 (-1.47)25.37 (+0.08)+1.55
De-En34.4434.21 (-0.23)36.22 (+1.78)+2.01
Fi-En21.2322.99 (+1.76)24.32 (+3.09)+1.33
Lv-En16.2616.25 (-0.01)18.43 (+2.17)+2.18
Ro-En35.8135.04 (-0.77)36.51 (+0.70)+1.47
Ru-En29.3928.92 (-0.47)30.82 (+1.43)+1.90
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LanguageIndividualMulti-BaselineMulti-Distillation△
En-Cs22.5821.39 (-1.19)23.10 (+0.62)+1.81
En-De31.4030.08 3 (-1.32)31.42 (+0.02)+1.34
En-Fi22.0819.52 (-2.56)21.56 (-0.52)+2.04
En-Lv14.9214.51 (-0.41)15.32 (+0.40)+0.81
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It can be seen that selective distillation performs better on 13 out of 16 lan-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 315, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 328 + ], + "score": 1.0, + "content": "guages, with large BLEU score improvements, which demonstrates the effectiveness of the selective", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 326, + 155, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 155, + 339 + ], + "score": 1.0, + "content": "distillation.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 271, + 506, + 339 + ] + }, + { + "type": "table", + "bbox": [ + 120, + 345, + 491, + 456 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 120, + 345, + 491, + 456 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 120, + 345, + 491, + 456 + ], + "spans": [ + { + "bbox": [ + 120, + 345, + 491, + 456 + ], + "score": 0.984, + "html": "
BgEtFiFrGlHiHyKa
distillation all the time28.0712.6415.1333.6930.2818.8619.8814.04
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distillation all the time8.5032.1014.0222.1017.2225.0530.4537.88
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We", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 350, + 542 + ], + "score": 1.0, + "content": "analyze whether there is accuracy difference between the top-", + "type": "text" + }, + { + "bbox": [ + 350, + 530, + 359, + 540 + ], + "score": 0.27, + "content": "\\mathbf { \\nabla } \\cdot \\mathbf { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "distribution and the full distribution.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 540, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 335, + 553 + ], + "score": 1.0, + "content": "We conduct experiments on IWSLT dataset with varying", + "type": "text" + }, + { + "bbox": [ + 335, + 541, + 346, + 551 + ], + "score": 0.78, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 540, + 390, + 553 + ], + "score": 1.0, + "content": "(from 1 to", + "type": "text" + }, + { + "bbox": [ + 390, + 541, + 405, + 552 + ], + "score": 0.88, + "content": "| V |", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 540, + 435, + 553 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 436, + 541, + 450, + 552 + ], + "score": 0.9, + "content": "| V |", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 540, + 505, + 553 + ], + "score": 1.0, + "content": "is the vocab-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 550, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 565 + ], + "score": 1.0, + "content": "ulary size), and just show the BLEU scores on the validation set of De-En translation due to space", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 563, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 374, + 575 + ], + "score": 1.0, + "content": "limitation, as illustrated in Table 7. It can be seen that increasing", + "type": "text" + }, + { + "bbox": [ + 374, + 563, + 385, + 573 + ], + "score": 0.75, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 563, + 505, + 575 + ], + "score": 1.0, + "content": "from 1 to 8 will improve the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 573, + 458, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 198, + 586 + ], + "score": 1.0, + "content": "accuracy, while bigger", + "type": "text" + }, + { + "bbox": [ + 199, + 574, + 209, + 583 + ], + "score": 0.75, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 573, + 413, + 586 + ], + "score": 1.0, + "content": "will bring no gains, even with the full distribution", + "type": "text" + }, + { + "bbox": [ + 413, + 573, + 453, + 585 + ], + "score": 0.89, + "content": "( K = | V | )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 573, + 458, + 586 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 507, + 505, + 586 + ] + }, + { + "type": "table", + "bbox": [ + 136, + 595, + 477, + 628 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 136, + 595, + 477, + 628 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 136, + 595, + 477, + 628 + ], + "spans": [ + { + "bbox": [ + 136, + 595, + 477, + 628 + ], + "score": 0.96, + "html": "
Top-K1248163264128IVI
BLEU33.4533.8634.4734.7634.6634.6834.5434.4734.49
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LanguageArCsDeN1RoRuThTrVi
Individual31.1928.0433.0735.9436.9223.0418.2422.7426.06
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", + "type": "table", + "image_path": "e496db88ba87864dc1365cf9a64865225114731f3fe10468f2e835f31548fb65.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 117, + 80, + 494, + 99.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 117, + 99.33333333333333, + 494, + 118.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 117, + 118.66666666666666, + 494, + 138.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 107, + 146, + 503, + 168 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 145, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 106, + 145, + 505, + 158 + ], + "score": 1.0, + "content": "Table 8: BLEU score improvements of the individual models with back distillation on the IWSLT", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 155, + 141, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 141, + 170 + ], + "score": 1.0, + "content": "dataset.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 190, + 504, + 214 + ], + "lines": [ + { + "bbox": [ + 105, + 190, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 505, + 203 + ], + "score": 1.0, + "content": "as shown in Table 10. The other 3 languages (He, Pt, Zh) cannot get improvements because the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 202, + 478, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 478, + 214 + ], + "score": 1.0, + "content": "improved multilingual model performs very close to individual models, as shown in Table 1.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 108, + 226, + 504, + 282 + ], + "lines": [ + { + "bbox": [ + 106, + 226, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 505, + 240 + ], + "score": 1.0, + "content": "Comparison with Sequence-Level Knowledge Distillation We conduct experiments to compare", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 237, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 250 + ], + "score": 1.0, + "content": "the word-level knowledge distillation (the exact method used in our paper) with sequence-level", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 248, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 262 + ], + "score": 1.0, + "content": "knowledge distillation(Kim & Rush, 2016b) on IWSLT dataset. As shown in Table 9, sequence-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "score": 1.0, + "content": "level knowledge distillation results in consistently inferior accuracy on all languages compared with", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 270, + 314, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 314, + 284 + ], + "score": 1.0, + "content": "word-level knowledge distillation used in our work.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9 + }, + { + "type": "table", + "bbox": [ + 181, + 293, + 429, + 435 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 181, + 293, + 429, + 435 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 181, + 293, + 429, + 435 + ], + "spans": [ + { + "bbox": [ + 181, + 293, + 429, + 435 + ], + "score": 0.978, + "html": "
LanguageSequence-levelWord-level (OurMethod)△
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LanguageArCsDeN1RoRuThTrVi
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We also show three", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "test BLEU curves on three translation pairs (Figure 1b: Ar-En, Figure 1c: Cs-En, Figure 1d: De-En,", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "which are randomly picked from the 12 languages pairs on the IWSLT dataset). 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Experiments", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "on three translation datasets with up to 44 languages demonstrate the multilingual model based on", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 395, + 504, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 484, + 407 + ], + "score": 1.0, + "content": "our proposed method can nearly match or even outperform the individual models, with just", + "type": "text" + }, + { + "bbox": [ + 484, + 395, + 504, + 407 + ], + "score": 0.84, + "content": "1 / N", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 407, + 319, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 319, + 419 + ], + "score": 1.0, + "content": "model parameters (N is up to 44 in our experiments).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 423, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 422, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 505, + 436 + ], + "score": 1.0, + "content": "In the future, we will conduct more deep analyses about how distillation helps the multilingual", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "model training. 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The data sizes of the training set for each", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 182, + 291, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 143, + 193 + ], + "score": 1.0, + "content": "language", + "type": "text" + }, + { + "bbox": [ + 143, + 183, + 153, + 191 + ], + "score": 0.82, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 182, + 291, + 193 + ], + "score": 1.0, + "content": "English pair are listed in Table 11.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 148, + 505, + 193 + ] + }, + { + "type": "table", + "bbox": [ + 177, + 201, + 434, + 263 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 177, + 201, + 434, + 263 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 177, + 201, + 434, + 263 + ], + "spans": [ + { + "bbox": [ + 177, + 201, + 434, + 263 + ], + "score": 0.977, + "html": "
LanguageArCsDeHeN1Pt
Training Data174K114K167K180K174K167K
LanguageRoRuThTrViZh
Training Data177K173K83K150K131K209K
", + "type": "table", + "image_path": "c03e6691604534996e6f5ac772ee9d52a46386363e82446498a61391212b920f.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 177, + 201, + 434, + 221.66666666666666 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 177, + 221.66666666666666, + 434, + 242.33333333333331 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 177, + 242.33333333333331, + 434, + 263.0 + ], + "spans": [], + "index": 9 + } + ] + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 133, + 270, + 477, + 282 + ], + "lines": [ + { + "bbox": [ + 132, + 269, + 479, + 284 + ], + "spans": [ + { + "bbox": [ + 132, + 269, + 342, + 284 + ], + "score": 1.0, + "content": "Table 11: The training data size on the 12 languages", + "type": "text" + }, + { + "bbox": [ + 342, + 272, + 353, + 281 + ], + "score": 0.83, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 269, + 479, + 284 + ], + "score": 1.0, + "content": "English on the IWSLT dataset.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 132, + 269, + 479, + 284 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 293, + 506, + 348 + ], + "lines": [ + { + "bbox": [ + 106, + 292, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 230, + 306 + ], + "score": 1.0, + "content": "WMT: We collect 6 languages", + "type": "text" + }, + { + "bbox": [ + 230, + 294, + 240, + 303 + ], + "score": 0.81, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 292, + 506, + 306 + ], + "score": 1.0, + "content": "English translation pairs from WMT translation task9. We use 5", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 304, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 143, + 316 + ], + "score": 1.0, + "content": "language", + "type": "text" + }, + { + "bbox": [ + 143, + 305, + 153, + 315 + ], + "score": 0.81, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 304, + 505, + 316 + ], + "score": 1.0, + "content": "English translation pairs from WMT 2016 dataset: Cs-En, De-En, Fi-En, Ro-En, Ru-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "En and one other translation pair from WMT 2017 dataset: Lv-En. We use the official released", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 326, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 462, + 339 + ], + "score": 1.0, + "content": "validation and test sets for each language pair. The training data sizes of each language", + "type": "text" + }, + { + "bbox": [ + 463, + 327, + 473, + 336 + ], + "score": 0.82, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 326, + 506, + 339 + ], + "score": 1.0, + "content": "English", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 337, + 231, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 231, + 348 + ], + "score": 1.0, + "content": "pair are shown in the Table 12.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 292, + 506, + 348 + ] + }, + { + "type": "table", + "bbox": [ + 179, + 357, + 431, + 391 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 179, + 357, + 431, + 391 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 179, + 357, + 431, + 391 + ], + "spans": [ + { + "bbox": [ + 179, + 357, + 431, + 391 + ], + "score": 0.962, + "html": "
LanguageCsDeFiLvRoRu
Training Data1.0M4.5M2.5M4.5M2.2M2.1M
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LanguageArBgCsDaDeE1EsEtFaFiFrca
Training Data214K174k103k45k168k134k196k11k151k24k20k
LanguageFrGlHeHiHrHuHyIdItJaKa
Training Data192K10K212K19K122K147K21K87K205K204K13K
LanguageKoKuLtMkMyNbNIPlPtbrPtRo
Training Data206K10K42K25K21K16K184K176K185K52K180K
LanguageRuSkS1SqSrSvThTrUkViZh
Training Data208K61K20K45K137K57K98K182K108K172K200K
", + "type": "table", + "image_path": "46e89d9b78b5e8969ab45f89aa283a9d330eaff05c618bad1c7ebfebf4396610.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 106, + 474, + 515, + 512.6666666666666 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 106, + 512.6666666666666, + 515, + 551.3333333333333 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 106, + 551.3333333333333, + 515, + 589.9999999999999 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 131, + 598, + 479, + 610 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 131, + 597, + 480, + 611 + ], + "spans": [ + { + "bbox": [ + 131, + 597, + 340, + 611 + ], + "score": 1.0, + "content": "Table 13: The training data size on the 44 languages", + "type": "text" + }, + { + "bbox": [ + 340, + 599, + 352, + 608 + ], + "score": 0.83, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 597, + 480, + 611 + ], + "score": 1.0, + "content": "English on the Ted talk dataset.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + } + ], + "index": 26.0 + }, + { + "type": "title", + "bbox": [ + 107, + 637, + 278, + 650 + ], + "lines": [ + { + "bbox": [ + 104, + 635, + 280, + 652 + ], + "spans": [ + { + "bbox": [ + 104, + 635, + 280, + 652 + ], + "score": 1.0, + "content": "2 LANGUAGE NAME AND CODE", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 661, + 505, + 684 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 675 + ], + "score": 1.0, + "content": "The language names and their corresponding language codes according to ISO 639-1 standard11 are", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 672, + 180, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 180, + 684 + ], + "score": 1.0, + "content": "listed in Table 14.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 659, + 506, + 684 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 127, + 80, + 486, + 225 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 127, + 80, + 486, + 225 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 127, + 80, + 486, + 225 + ], + "spans": [ + { + "bbox": [ + 127, + 80, + 486, + 225 + ], + "score": 0.983, + "html": "
LanguageCodeLanguageCodeLanguageCodeLanguageCode
ArabicArBulgarianBgCzechCsDanishDa
GermanDeGreekElEnglishEnSpanishEs
PersianFaFinnishFiFrenchFrGalicianGl
HebrewHeHindiHiCroatianHrHungarianHu
ArmenianHyIndonesianIdItalianItJapaneseJa
GeorgianKaKoreanKoKurdishKuLithuanianLt
LatvianLvMacedonianMkBurmeseMyNorwegianNb
DutchN1PolishPlPortuguesePtRomanianRo
RussianRuSlovakSkSlovenianS1AlbanianSq
SerbianSrSwedishSvThaiThTurkishTr
UkrainianUkVietnameseViChineseZh
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LanguageArBgCsDaDeE1EsEtFa
Individual31.0738.6426.4238.2134.6336.6941.207.4326.67
Multilingual (Baseline)27.8427.7627.1740.4132.8536.0439.8014.8624.93
Multilingual (Our method)29.5729.1828.3042.2334.5337.4941.4315.6326.76
LanguageFiFrcaFrGlHeHiHrHuHy
Individual10.7818.5239.6212.6436.8110.8434.1424.6712.30
Multilingual (Baseline)16.1233.0838.2730.3232.9619.9334.3922.7620.25
Multilingual (Our method)17.2234.3239.7531.935.2221.0035.624.5621.17
LanguageIdItJaKaKoKuLtMkMy
Individual29.2038.0613.317.0618.545.6318.1921.937.53
Multilingual (Baseline)29.0836.0212.3316.7116.7111.8320.9631.8513.85
Multilingual (Our method)30.5637.5013.2818.2618.1413.3822.6532.6515.16
LanguageNbNIPIPtbrPtRoRuSkSI
Individual27.2835.8522.9844.2833.8134.0724.3625.6711.80
Multilingual (Baseline)39.8833.9723.5042.9640.5933.0324.0228.9722.52
Multilingual (Our method)41.3535.6524.3044.4142.5734.7325.0129.9023.67
LanguageSqSrSvThTrUkViZh
Individual29.7032.1334.5320.9524.4625.7626.3812.56
Multilingual (Baseline)33.0532.2735.9221.5021.7926.8225.7618.81
Multilingual (Our method)34.7333.7136.9222.1223.6727.8026.5319.39
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LanguageCodeLanguageCodeLanguageCodeLanguageCode
ArabicArBulgarianBgCzechCsDanishDa
GermanDeGreekElEnglishEnSpanishEs
PersianFaFinnishFiFrenchFrGalicianGl
HebrewHeHindiHiCroatianHrHungarianHu
ArmenianHyIndonesianIdItalianItJapaneseJa
GeorgianKaKoreanKoKurdishKuLithuanianLt
LatvianLvMacedonianMkBurmeseMyNorwegianNb
DutchN1PolishPlPortuguesePtRomanianRo
RussianRuSlovakSkSlovenianS1AlbanianSq
SerbianSrSwedishSvThaiThTurkishTr
UkrainianUkVietnameseViChineseZh
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Multilingual (Our method)29.5729.1828.3042.2334.5337.4941.4315.6326.76
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Individual10.7818.5239.6212.6436.8110.8434.1424.6712.30
Multilingual (Baseline)16.1233.0838.2730.3232.9619.9334.3922.7620.25
Multilingual (Our method)17.2234.3239.7531.935.2221.0035.624.5621.17
LanguageIdItJaKaKoKuLtMkMy
Individual29.2038.0613.317.0618.545.6318.1921.937.53
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Multilingual (Our method)41.3535.6524.3044.4142.5734.7325.0129.9023.67
LanguageSqSrSvThTrUkViZh
Individual29.7032.1334.5320.9524.4625.7626.3812.56
Multilingual (Baseline)33.0532.2735.9221.5021.7926.8225.7618.81
Multilingual (Our method)34.7333.7136.9222.1223.6727.8026.5319.39
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LanguageIndividualMulti-BaselineMulti-Distillation△
Ar→En31.1929.24 (-1.95)31.25 (+0.06)+2.01
Cs-→En28.0426.09 (-1.95)27.09 (-0.95)+1.00
De→En33.0732.74 (-0.33)34.02 (+0.95)+1.28
He-→En37.4235.18 (-2.24)37.33 (-0.09)+2.15
N1-→En35.9436.54 (+0.60)37.69 (+1.75)+1.15
Pt-→En44.3043.49 (-0.81)44.69 (+0.39)+1.20
Ro-→En36.9236.41 1 (-0.51)38.01 (+1.09)+1.60
Ru→En23.0423.12 (+0.08)23.76 (+0.72)+0.64
Th→En18.2419.33 3 (+1.09)19.90 (+1.66)+0.57
Tr→En22.7422.42 (-0.32)23.75 (+1.01)+1.33
Vi→En26.0626.37 (+0.31)27.04 4 (+0.98)+0.67
Zh→En19.4418.82 2 (-0.62)19.52 (+0.08)+0.70
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LanguageIndividualMulti-BaselineMulti-Distillation△
En→Ar13.6712.73 (-0.94)13.80 (+0.13)+1.07
En→Cs17.8117.33 (-0.48)18.69 (+0.88)+1.37
En→De26.1325.16 (-0.97)26.76 (+0.63)+1.60
En→He24.1522.73 (-1.42)24.42 (+0.27)+1.69
En-→Nl30.8829.51 (-1.37)30.52 (-0.36)+1.01
En→Pt37.6335.93 (-1.70)37.23 (-0.40)+1.30
En→Ro27.2325.68 (-1.55)27.11 (-0.12)+1.42
En→Ru17.4016.26 (-1.14)17.42 (+0.02)+1.16
En→Th26.4527.18 (+0.73)27.62 (+1.17)+0.45
En→Tr12.4711.63 (-0.84)12.84 (+0.37)+1.21
En→Vi27.8828.04 (+0.16)28.69 (+0.81)+0.65
En→Zh10.9510.12 (-0.83)10.41 (-0.54)+0.29
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BgEtFiFrGlHiHyKa
distillation all the time28.0712.6415.1333.6930.2818.8619.8814.04
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LanguageArCsDeN1RoRuThTrVi
Individual31.1928.0433.0735.9436.9223.0418.2422.7426.06
+Back Distillation31.3929.4433.7136.8637.2823.3619.4223.5827.17
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LanguageSequence-levelWord-level (OurMethod)△
En-Ar12.7913.801.01
En-Cs17.0118.691.68
En-De25.8926.760.87
En-He22.9224.421.50
En-N129.9930.520.53
En-Pt36.1237.231.10
En-Ro25.7527.111.36
En-Ru16.3817.421.04
En-Th27.5227.620.10
En-Tr11.1112.841.73
En-Vi28.0828.690.61
En-Zh10.2510.410.16
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LanguageArCsDeN1RoRuThTrVi
Individual31.1928.0433.0735.9436.9223.0418.2422.7426.06
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LanguageFrGlHeHiHrHuHyIdItJaKa
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LanguageKoKuLtMkMyNbNIPlPtbrPtRo
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LanguageRuSkS1SqSrSvThTrUkViZh
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LanguageArCsDeHeN1Pt
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LanguageRoRuThTrViZh
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LanguageArBgCsDaDeE1EsEtFa
Individual31.0738.6426.4238.2134.6336.6941.207.4326.67
Multilingual (Baseline)27.8427.7627.1740.4132.8536.0439.8014.8624.93
Multilingual (Our method)29.5729.1828.3042.2334.5337.4941.4315.6326.76
LanguageFiFrcaFrGlHeHiHrHuHy
Individual10.7818.5239.6212.6436.8110.8434.1424.6712.30
Multilingual (Baseline)16.1233.0838.2730.3232.9619.9334.3922.7620.25
Multilingual (Our method)17.2234.3239.7531.935.2221.0035.624.5621.17
LanguageIdItJaKaKoKuLtMkMy
Individual29.2038.0613.317.0618.545.6318.1921.937.53
Multilingual (Baseline)29.0836.0212.3316.7116.7111.8320.9631.8513.85
Multilingual (Our method)30.5637.5013.2818.2618.1413.3822.6532.6515.16
LanguageNbNIPIPtbrPtRoRuSkSI
Individual27.2835.8522.9844.2833.8134.0724.3625.6711.80
Multilingual (Baseline)39.8833.9723.5042.9640.5933.0324.0228.9722.52
Multilingual (Our method)41.3535.6524.3044.4142.5734.7325.0129.9023.67
LanguageSqSrSvThTrUkViZh
Individual29.7032.1334.5320.9524.4625.7626.3812.56
Multilingual (Baseline)33.0532.2735.9221.5021.7926.8225.7618.81
Multilingual (Our method)34.7333.7136.9222.1223.6727.8026.5319.39
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ArabicArBulgarianBgCzechCsDanishDa
GermanDeGreekElEnglishEnSpanishEs
PersianFaFinnishFiFrenchFrGalicianGl
HebrewHeHindiHiCroatianHrHungarianHu
ArmenianHyIndonesianIdItalianItJapaneseJa
GeorgianKaKoreanKoKurdishKuLithuanianLt
LatvianLvMacedonianMkBurmeseMyNorwegianNb
DutchN1PolishPlPortuguesePtRomanianRo
RussianRuSlovakSkSlovenianS1AlbanianSq
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This requires a skillful choice of communication topology, which", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 355, + 364, + 367 + ], + "spans": [ + { + "bbox": [ + 141, + 355, + 364, + 367 + ], + "score": 1.0, + "content": "is an under-studied topic in decentralized optimization.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11, + "bbox_fs": [ + 140, + 289, + 470, + 367 + ] + }, + { + "type": "text", + "bbox": [ + 143, + 368, + 469, + 521 + ], + "lines": [ + { + "bbox": [ + 142, + 369, + 470, + 380 + ], + "spans": [ + { + "bbox": [ + 142, + 369, + 470, + 380 + ], + "score": 1.0, + "content": "In this paper, we study so-called exponential graphs where every node is con-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 379, + 470, + 392 + ], + "spans": [ + { + "bbox": [ + 141, + 379, + 183, + 392 + ], + "score": 1.0, + "content": "nected to", + "type": "text" + }, + { + "bbox": [ + 183, + 379, + 226, + 391 + ], + "score": 0.92, + "content": "O ( \\log ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 379, + 290, + 392 + ], + "score": 1.0, + "content": "neighbors and", + "type": "text" + }, + { + "bbox": [ + 290, + 381, + 298, + 389 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 379, + 470, + 392 + ], + "score": 1.0, + "content": "is the total number of nodes. This work", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 389, + 470, + 403 + ], + "spans": [ + { + "bbox": [ + 141, + 389, + 470, + 403 + ], + "score": 1.0, + "content": "proves such graphs can lead to both fast communication and effective averaging", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 400, + 469, + 414 + ], + "spans": [ + { + "bbox": [ + 141, + 400, + 353, + 414 + ], + "score": 1.0, + "content": "simultaneously. We also discover that a sequence of", + "type": "text" + }, + { + "bbox": [ + 353, + 401, + 381, + 413 + ], + "score": 0.92, + "content": "\\log ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 400, + 469, + 414 + ], + "score": 1.0, + "content": "one-peer exponential", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 412, + 470, + 425 + ], + "spans": [ + { + "bbox": [ + 141, + 412, + 470, + 425 + ], + "score": 1.0, + "content": "graphs, in which each node communicates to one single neighbor per iteration,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 423, + 470, + 435 + ], + "spans": [ + { + "bbox": [ + 141, + 423, + 470, + 435 + ], + "score": 1.0, + "content": "can together achieve exact averaging. This favorable property enables one-peer", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 434, + 471, + 446 + ], + "spans": [ + { + "bbox": [ + 141, + 434, + 471, + 446 + ], + "score": 1.0, + "content": "exponential graph to average as effective as its static counterpart but commu-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 444, + 470, + 457 + ], + "spans": [ + { + "bbox": [ + 141, + 444, + 470, + 457 + ], + "score": 1.0, + "content": "nicates more efficiently. We apply these exponential graphs in decentralized", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 142, + 456, + 469, + 467 + ], + "spans": [ + { + "bbox": [ + 142, + 456, + 469, + 467 + ], + "score": 1.0, + "content": "(momentum) SGD to obtain the state-of-the-art balance between per-iteration", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 467, + 470, + 478 + ], + "spans": [ + { + "bbox": [ + 141, + 467, + 470, + 478 + ], + "score": 1.0, + "content": "communication and iteration complexity among all commonly-used topologies.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 142, + 478, + 470, + 488 + ], + "spans": [ + { + "bbox": [ + 142, + 478, + 470, + 488 + ], + "score": 1.0, + "content": "Experimental results on a variety of tasks and models demonstrate that decen-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 488, + 470, + 501 + ], + "spans": [ + { + "bbox": [ + 141, + 488, + 470, + 501 + ], + "score": 1.0, + "content": "tralized (momentum) SGD over exponential graphs promises both fast and high-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 499, + 470, + 511 + ], + "spans": [ + { + "bbox": [ + 141, + 499, + 470, + 511 + ], + "score": 1.0, + "content": "quality training. Our code is implemented through BlueFog and available at", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 509, + 462, + 523 + ], + "spans": [ + { + "bbox": [ + 141, + 509, + 462, + 523 + ], + "score": 1.0, + "content": "https://github.com/Bluefog-Lib/NeurIPS2021-Exponential-Graph.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 21.5, + "bbox_fs": [ + 141, + 369, + 471, + 523 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 547, + 190, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 546, + 192, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 192, + 563 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 567, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 566, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 580 + ], + "score": 1.0, + "content": "Efficient distributed training methods across multiple computing nodes are critical for large-scale", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "score": 1.0, + "content": "modern deep learning tasks. Parallel stochastic gradient descent (SGD) is a widely-used approach,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "score": 1.0, + "content": "which, at each iteration, computes a globally averaged gradient either using Parameter-Server [28]", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "score": 1.0, + "content": "or All-Reduce [47]. Such global coordination across all nodes in parallel SGD results in either", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 610, + 478, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 478, + 624 + ], + "score": 1.0, + "content": "significant bandwidth cost or high latency, which can notably hamper the training scalability.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 566, + 506, + 624 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 625, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 505, + 641 + ], + "score": 1.0, + "content": "Decentralized SGD [45, 11, 30, 3] based on partial averaging has been one of the promising", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "alternatives to parallel SGD in distributed deep training. Partial averaging, as opposed to the global", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "averaging exploited in parallel SGD, only requires each node to compute the locally averaged model", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "within its neighborhood. Decentralized SGD does not involve any global operations, so it has much", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 670, + 507, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 507, + 684 + ], + "score": 1.0, + "content": "lower communication overhead per iteration. The fewer neighbors each node needs to communicate,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 682, + 419, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 419, + 693 + ], + "score": 1.0, + "content": "the more efficient the per-iteration communication is in decentralized SGD.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 625, + 507, + 693 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 114, + 504, + 167 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 70, + 506, + 111 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 68, + 506, + 83 + ], + "spans": [ + { + "bbox": [ + 105, + 68, + 506, + 83 + ], + "score": 1.0, + "content": "Table 1: Comparison between decentralized (momentum) SGD over (some) various commonly-used topologes.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 79, + 506, + 92 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 506, + 92 + ], + "score": 1.0, + "content": "The table assumes homogeneous data distributions across all nodes (which is practical for deep training within", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 91, + 505, + 101 + ], + "spans": [ + { + "bbox": [ + 106, + 91, + 505, + 101 + ], + "score": 1.0, + "content": "a data-center). The comparison for data-heterogeneous scenarios, and with more other topologies, is listed in", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 99, + 506, + 112 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 506, + 112 + ], + "score": 1.0, + "content": "Appendix C. The smaller the transient iteration complexity is, the faster decentralized algorithms will converge.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "table_body", + "bbox": [ + 106, + 114, + 504, + 167 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 114, + 504, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 114, + 504, + 167 + ], + "score": 0.974, + "html": "
TopologyRingGridRand-GraphRand-MatchStatic Exp One-peer Exp
Per-iter Comm.(2)2(4)（）(1)Ω(log2(n))(1)
Trans. Iters.Ω(n7)Ω(n5)(n3)1Ω(n³ log2(n))Ω(n³log²2(n)）
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While", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 104, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "it can asymptotically achieve the same convergence linear speedup as parallel SGD [30, 3, 25, 64],", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "i.e., the training speed increases proportionally to the number of computing nodes (see the definition", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 208, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 506, + 221 + ], + "score": 1.0, + "content": "in Sec. 2), decentralized SGD requires more iterations to reach that stage due to the ineffectiveness", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 220, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 506, + 232 + ], + "score": 1.0, + "content": "to aggregate information using partial averaging. We refer those iterations before decentralized", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "SGD reaches its linear speedup stage as transient iterations (see the definition in Sec. 2), which", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 242, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 506, + 254 + ], + "score": 1.0, + "content": "is an important metric to measure the influence of partial-averaging [48, 65] on convergence rate", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "of decentralized SGD. The less effective the partial averaging is, the more transient iterations", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "decentralized SGD needs to take. Fig. 1 illustrates the transient iterations of decentralized SGD for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "the logistic regression problem. It is observed that decentralized SGD can asymptotically converge as", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 285, + 489, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 489, + 298 + ], + "score": 1.0, + "content": "fast as parallel SGD, but it requires more iterations (i.e., transient iterations) to reach that stage.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 345, + 432 + ], + "lines": [ + { + "bbox": [ + 106, + 302, + 345, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 345, + 313 + ], + "score": 1.0, + "content": "Per-iteration communication and transient iterations in de-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 309, + 345, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 345, + 327 + ], + "score": 1.0, + "content": "centralized SGD are determined by the network topology", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 324, + 344, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 344, + 336 + ], + "score": 1.0, + "content": "(we also use graph interchangeably with topology). The", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 335, + 344, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 344, + 346 + ], + "score": 1.0, + "content": "maximum degree of the graph decides the communication", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 345, + 345, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 345, + 357 + ], + "score": 1.0, + "content": "cost while the connectivity influences the transient itera-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 357, + 345, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 345, + 368 + ], + "score": 1.0, + "content": "tion complexity. Generally speaking, a sparsely-connected", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 367, + 345, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 345, + 379 + ], + "score": 1.0, + "content": "topology communicates cheaply but endows decentralized", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 378, + 345, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 345, + 390 + ], + "score": 1.0, + "content": "SGD with more transient iterations due to the less effec-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 389, + 345, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 345, + 401 + ], + "score": 1.0, + "content": "tive information aggregation. A skillful choice of network", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 399, + 346, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 346, + 412 + ], + "score": 1.0, + "content": "topology, which is critical to achieve balance between per-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 410, + 346, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 346, + 424 + ], + "score": 1.0, + "content": "iteration communication and transient iteration complexity,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 421, + 224, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 224, + 434 + ], + "score": 1.0, + "content": "is under-studied in literature.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 23.5 + }, + { + "type": "image", + "bbox": [ + 353, + 304, + 502, + 421 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 353, + 304, + 502, + 421 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 353, + 304, + 502, + 421 + ], + "spans": [ + { + "bbox": [ + 353, + 304, + 502, + 421 + ], + "score": 0.967, + "type": "image", + "image_path": "fe4fdd71eb359ff24cb52a419b362226d510cbefbbb83aaa18627819772c3f6a.jpg" + } + ] + } + ], + "index": 28.0, + "virtual_lines": [ + { + "bbox": [ + 353, + 304, + 502, + 362.5 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 353, + 362.5, + 502, + 421.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 352, + 423, + 504, + 443 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 352, + 421, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 352, + 421, + 505, + 433 + ], + "score": 1.0, + "content": "Figure 1: Illustration of transient iters.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 352, + 433, + 501, + 444 + ], + "spans": [ + { + "bbox": [ + 352, + 433, + 501, + 444 + ], + "score": 1.0, + "content": "Experimental setting is in Appendix D.5.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.0 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 438, + 345, + 449 + ], + "lines": [ + { + "bbox": [ + 106, + 437, + 345, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 345, + 452 + ], + "score": 1.0, + "content": "This work studies exponential graphs which are empiri-", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 449, + 505, + 537 + ], + "lines": [ + { + "bbox": [ + 106, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "cally successful [3, 61, 27, 14, 67] but less theoretically understood in deep training. Exponential", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 460, + 503, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 463, + 472 + ], + "score": 1.0, + "content": "graphs have two variants. In a static exponential graph, each node communicates to", + "type": "text" + }, + { + "bbox": [ + 464, + 460, + 503, + 472 + ], + "score": 0.92, + "content": "\\lceil \\log _ { 2 } ( n ) \\rceil", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 471, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 483 + ], + "score": 1.0, + "content": "neighbors (see Sec. 3 and Fig. 2). In one-peer exponential graph, however, each node cycles through", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "all its neighbors, communicating, only, to a single neighbor per iteration (see Sec. 4 and Fig. 2). This", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "score": 1.0, + "content": "paper will first clarify the connectivity and averaging effectiveness of these exponential graphs, and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 504, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 505, + 515 + ], + "score": 1.0, + "content": "then apply them to decentralized momentum SGD to obtain the state-of-the-art balance between", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 515, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 506, + 527 + ], + "score": 1.0, + "content": "per-iteration communication and transient iteration complexity among all commonly-used topologies.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 524, + 312, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 312, + 538 + ], + "score": 1.0, + "content": "Our main results (as well as our contributions) are:", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 117, + 541, + 506, + 700 + ], + "lines": [ + { + "bbox": [ + 116, + 540, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 116, + 540, + 505, + 554 + ], + "score": 1.0, + "content": "• We prove that the spectral gap, which is used to measure the connectivity of the graph (see the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 125, + 551, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 125, + 551, + 412, + 565 + ], + "score": 1.0, + "content": "definition in Sec. 2), of the static exponential graph is upper bounded by", + "type": "text" + }, + { + "bbox": [ + 412, + 552, + 471, + 564 + ], + "score": 0.92, + "content": "O ( 1 / \\log _ { 2 } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 551, + 505, + 565 + ], + "score": 1.0, + "content": ". Before", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 126, + 563, + 441, + 575 + ], + "spans": [ + { + "bbox": [ + 126, + 563, + 371, + 575 + ], + "score": 1.0, + "content": "us, many literatures (e.g. [27]) claimed its upper bound to be", + "type": "text" + }, + { + "bbox": [ + 371, + 563, + 393, + 575 + ], + "score": 0.9, + "content": "O ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 563, + 441, + 575 + ], + "score": 1.0, + "content": "incorrectly.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 122, + 573, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 122, + 573, + 506, + 588 + ], + "score": 1.0, + "content": "Since one-peer exponential graphs are time-varying, it is difficult to derive their spectral gaps.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 125, + 585, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 125, + 585, + 254, + 599 + ], + "score": 1.0, + "content": "However, we establish that any", + 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graph, though much sparser", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 126, + 620, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 126, + 620, + 506, + 632 + ], + "score": 1.0, + "content": "than its static counterpart, surprisingly endows decentralized momentum SGD with the same", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 125, + 631, + 447, + 642 + ], + "spans": [ + { + "bbox": [ + 125, + 631, + 447, + 642 + ], + "score": 1.0, + "content": "convergence rate as static exponential graph in terms of the best-known bounds.", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 117, + 642, + 507, + 656 + ], + "spans": [ + { + "bbox": [ + 117, + 643, + 297, + 656 + ], + "score": 1.0, + "content": "• We derive that exponential graphs achieve", + "type": "text" + }, + { + "bbox": [ + 298, + 642, + 322, + 656 + ], + "score": 0.93, + "content": "\\tilde { \\Omega } ( 1 ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 643, 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}, + { + "bbox": [ + 174, + 708, + 273, + 724 + ], + "score": 1.0, + "content": "hides all logarithm factors.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 114, + 504, + 167 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 70, + 506, + 111 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 68, + 506, + 83 + ], + "spans": [ + { + "bbox": [ + 105, + 68, + 506, + 83 + ], + "score": 1.0, + "content": "Table 1: Comparison between decentralized (momentum) SGD over (some) various commonly-used topologes.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 79, + 506, + 92 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 506, 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The comparison for data-heterogeneous scenarios, and with more other topologies, is listed in", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 99, + 506, + 112 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 506, + 112 + ], + "score": 1.0, + "content": "Appendix C. The smaller the transient iteration complexity is, the faster decentralized algorithms will converge.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "table_body", + "bbox": [ + 106, + 114, + 504, + 167 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 114, + 504, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 114, + 504, + 167 + ], + "score": 0.974, + "html": "
TopologyRingGridRand-GraphRand-MatchStatic Exp One-peer Exp
Per-iter Comm.(2)2(4)（）(1)Ω(log2(n))(1)
Trans. Iters.Ω(n7)Ω(n5)(n3)1Ω(n³ log2(n))Ω(n³log²2(n)）
", + "type": "table", + "image_path": "d3af7ab584670180f688246b0582c6e8441ecec5bc9b5fa0528926f6c35470cc.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 106, + 114, + 504, + 131.66666666666666 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 106, + 131.66666666666666, + 504, + 149.33333333333331 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 106, + 149.33333333333331, + 504, + 166.99999999999997 + ], + "spans": [], + "index": 6 + } + ] + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 106, + 176, + 506, + 297 + ], + "lines": [ + { + "bbox": [ + 105, + 176, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 188 + ], + "score": 1.0, + "content": "The reduced communication in decentralized SGD comes with a cost: slower convergence. While", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 104, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "it can asymptotically achieve the same convergence linear speedup as parallel SGD [30, 3, 25, 64],", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "i.e., the training speed increases proportionally to the number of computing nodes (see the definition", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 208, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 506, + 221 + ], + "score": 1.0, + "content": "in Sec. 2), decentralized SGD requires more iterations to reach that stage due to the ineffectiveness", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 220, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 506, + 232 + ], + "score": 1.0, + "content": "to aggregate information using partial averaging. We refer those iterations before decentralized", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "SGD reaches its linear speedup stage as transient iterations (see the definition in Sec. 2), which", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 242, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 506, + 254 + ], + "score": 1.0, + "content": "is an important metric to measure the influence of partial-averaging [48, 65] on convergence rate", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "of decentralized SGD. The less effective the partial averaging is, the more transient iterations", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "decentralized SGD needs to take. Fig. 1 illustrates the transient iterations of decentralized SGD for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "the logistic regression problem. It is observed that decentralized SGD can asymptotically converge as", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 285, + 489, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 489, + 298 + ], + "score": 1.0, + "content": "fast as parallel SGD, but it requires more iterations (i.e., transient iterations) to reach that stage.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 12, + "bbox_fs": [ + 104, + 176, + 506, + 298 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 345, + 432 + ], + "lines": [ + { + "bbox": [ + 106, + 302, + 345, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 345, + 313 + ], + "score": 1.0, + "content": "Per-iteration communication and transient iterations in de-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 309, + 345, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 345, + 327 + ], + "score": 1.0, + "content": "centralized SGD are determined by the network topology", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 324, + 344, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 344, + 336 + ], + "score": 1.0, + "content": "(we also use graph interchangeably with topology). The", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 335, + 344, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 344, + 346 + ], + "score": 1.0, + "content": "maximum degree of the graph decides the communication", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 345, + 345, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 345, + 357 + ], + "score": 1.0, + "content": "cost while the connectivity influences the transient itera-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 357, + 345, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 345, + 368 + ], + "score": 1.0, + "content": "tion complexity. Generally speaking, a sparsely-connected", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 367, + 345, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 345, + 379 + ], + "score": 1.0, + "content": "topology communicates cheaply but endows decentralized", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 378, + 345, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 345, + 390 + ], + "score": 1.0, + "content": "SGD with more transient iterations due to the less effec-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 389, + 345, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 345, + 401 + ], + "score": 1.0, + "content": "tive information aggregation. A skillful choice of network", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 399, + 346, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 346, + 412 + ], + "score": 1.0, + "content": "topology, which is critical to achieve balance between per-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 410, + 346, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 346, + 424 + ], + "score": 1.0, + "content": "iteration communication and transient iteration complexity,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 421, + 224, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 224, + 434 + ], + "score": 1.0, + "content": "is under-studied in literature.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 302, + 346, + 434 + ] + }, + { + "type": "image", + "bbox": [ + 353, + 304, + 502, + 421 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 353, + 304, + 502, + 421 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 353, + 304, + 502, + 421 + ], + "spans": [ + { + "bbox": [ + 353, + 304, + 502, + 421 + ], + "score": 0.967, + "type": "image", + "image_path": "fe4fdd71eb359ff24cb52a419b362226d510cbefbbb83aaa18627819772c3f6a.jpg" + } + ] + } + ], + "index": 28.0, + "virtual_lines": [ + { + "bbox": [ + 353, + 304, + 502, + 362.5 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 353, + 362.5, + 502, + 421.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 352, + 423, + 504, + 443 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 352, + 421, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 352, + 421, + 505, + 433 + ], + "score": 1.0, + "content": "Figure 1: Illustration of transient iters.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 352, + 433, + 501, + 444 + ], + "spans": [ + { + "bbox": [ + 352, + 433, + 501, + 444 + ], + "score": 1.0, + "content": "Experimental setting is in Appendix D.5.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.0 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 438, + 345, + 449 + ], + "lines": [ + { + "bbox": [ + 106, + 437, + 345, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 345, + 452 + ], + "score": 1.0, + "content": "This work studies exponential graphs which are empiri-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "cally successful [3, 61, 27, 14, 67] but less theoretically understood in deep training. Exponential", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 460, + 503, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 463, + 472 + ], + "score": 1.0, + "content": "graphs have two variants. In a static exponential graph, each node communicates to", + "type": "text" + }, + { + "bbox": [ + 464, + 460, + 503, + 472 + ], + "score": 0.92, + "content": "\\lceil \\log _ { 2 } ( n ) \\rceil", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 471, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 483 + ], + "score": 1.0, + "content": "neighbors (see Sec. 3 and Fig. 2). In one-peer exponential graph, however, each node cycles through", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "all its neighbors, communicating, only, to a single neighbor per iteration (see Sec. 4 and Fig. 2). This", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "score": 1.0, + "content": "paper will first clarify the connectivity and averaging effectiveness of these exponential graphs, and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 504, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 505, + 515 + ], + "score": 1.0, + "content": "then apply them to decentralized momentum SGD to obtain the state-of-the-art balance between", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 515, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 506, + 527 + ], + "score": 1.0, + "content": "per-iteration communication and transient iteration complexity among all commonly-used topologies.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 524, + 312, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 312, + 538 + ], + "score": 1.0, + "content": "Our main results (as well as our contributions) are:", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 33, + "bbox_fs": [ + 106, + 437, + 345, + 452 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 449, + 505, + 537 + ], + "lines": [], + "index": 38.5, + "bbox_fs": [ + 105, + 448, + 506, + 538 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 117, + 541, + 506, + 700 + ], + "lines": [ + { + "bbox": [ + 116, + 540, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 116, + 540, + 505, + 554 + ], + "score": 1.0, + "content": "• We prove that the spectral gap, which is used to measure the connectivity of the graph (see the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 125, + 551, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 125, + 551, + 412, + 565 + ], + "score": 1.0, + "content": "definition in Sec. 2), of the static exponential graph is upper bounded by", + "type": "text" + }, + { + "bbox": [ + 412, + 552, + 471, + 564 + ], + "score": 0.92, + "content": "O ( 1 / \\log _ { 2 } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 551, + 505, + 565 + ], + "score": 1.0, + "content": ". Before", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 126, + 563, + 441, + 575 + ], + "spans": [ + { + "bbox": [ + 126, + 563, + 371, + 575 + ], + "score": 1.0, + "content": "us, many literatures (e.g. [27]) claimed its upper bound to be", + "type": "text" + }, + { + "bbox": [ + 371, + 563, + 393, + 575 + ], + "score": 0.9, + "content": "O ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 563, + 441, + 575 + ], + "score": 1.0, + "content": "incorrectly.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 122, + 573, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 122, + 573, + 506, + 588 + ], + "score": 1.0, + "content": "Since one-peer exponential graphs are time-varying, it is difficult to derive their spectral gaps.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 125, + 585, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 125, + 585, + 254, + 599 + ], + "score": 1.0, + "content": "However, we establish that any", + 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Transient iterations are referred to those iterations before an algorithm reaches", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 266, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 244, + 278 + ], + "score": 1.0, + "content": "linear-speedup stage, that is when", + "type": "text" + }, + { + "bbox": [ + 244, + 266, + 253, + 275 + ], + "score": 0.81, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 266, + 356, + 278 + ], + "score": 1.0, + "content": "is relatively small so non-", + "type": "text" + }, + { + "bbox": [ + 357, + 266, + 371, + 276 + ], + "score": 0.83, + "content": "\\mathbf { \\nabla } \\cdot n T", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 266, + 505, + 278 + ], + "score": 1.0, + "content": "terms still dominate the rate (see", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 493, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 318, + 290 + ], + "score": 1.0, + "content": "illustration in Appendix C). To reach linear speedup,", + "type": "text" + }, + { + "bbox": [ + 319, + 277, + 327, + 286 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 276, + 493, + 290 + ], + "score": 1.0, + "content": "has to satisfy (derivation in Appendix C)", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 111, + 293, + 489, + 322 + ], + "lines": [ + { + "bbox": [ + 107, + 284, + 489, + 329 + ], + "spans": [ + { + "bbox": [ + 107, + 284, + 188, + 329 + ], + "score": 1.0, + "content": "Homogeneous dat", + "type": "text" + }, + { + "bbox": [ + 189, + 293, + 489, + 322 + ], + "score": 0.7, + "content": " \\mathrm { 1 : } \\quad T = \\Omega \\left( { \\frac { n ^ { 3 } } { ( 1 - \\rho ) ^ { 2 } } } \\right) \\qquad { \\mathrm { H e t e r o g e n e o u s ~ d a t a : } } \\quad T = \\Omega \\left( { \\frac { n ^ { 3 } } { ( 1 - \\rho ) ^ { 4 } } } \\right)", + "type": "inline_equation", + "image_path": "52c9e3543bdb97c169056c920c4d71601baa7cb14c8a805f4c76f39aa38df2fa.jpg" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 338, + 502, + 350 + ], + "lines": [ + { + "bbox": [ + 106, + 337, + 502, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 502, + 351 + ], + "score": 1.0, + "content": "which corresponds to the transient iteration complexity in the homo/hetero-geneous data scenarios.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 107, + 363, + 194, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 195, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 195, + 376 + ], + "score": 1.0, + "content": "2.1 Related Works", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 383, + 505, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 383, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 397 + ], + "score": 1.0, + "content": "Decentralized deep training. Decentralized optimization originates from the control and signal", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "processing community. The first decentralized algorithms on general optimization problems include", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 403, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 420 + ], + "score": 1.0, + "content": "decentralized gradient descent [45], diffusion [11, 51] and dual averaging [18]. In the deep learning", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 416, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 506, + 429 + ], + "score": 1.0, + "content": "regime, decentralize SGD, which was established in [30] to achieve the same linear speedup as parallel", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "SGD in convergence rate, has attracted a lot of attentions. Many efforts have been made to extend the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "algorithm to directed topologies [3, 42], time-varying topologies [25, 42], asynchronous settings [31],", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "score": 1.0, + "content": "and data-heterogeneous scenarios [57, 62, 32, 67]. Techniques such as quantization/compression", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "[2, 8, 26, 24, 58, 36], periodic updates [55, 25, 64], and lazy communication [37, 38, 13] were also", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 470, + 419, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 419, + 483 + ], + "score": 1.0, + "content": "integrated into decentralized SGD to further reduce communiation overheads.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 487, + 506, + 640 + ], + "lines": [ + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "Topology influence. The influence of network topology on decentralized SGD was extensively", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "studied in [25, 51, 66, 45, 42, 27]. All these works indicate that a well-connected topology will", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 507, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 522 + ], + "score": 1.0, + "content": "significantly accelerate decentralized SGD. Two directions have been explored to relieve the influence", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "of network topology. One line of research proposes new algorithms that are less sensitive to topologies.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 530, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 506, + 544 + ], + "score": 1.0, + "content": "For example, [66, 23, 65, 57, 1] removed data heterogeneity with bias-correction techniques in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 540, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 556 + ], + "score": 1.0, + "content": "[68, 29, 62, 40, 69], and [14, 61, 7, 27] utilized periodic global averaging or multiple partial averaging", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 552, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 566 + ], + "score": 1.0, + "content": "steps. All these methods have improved topology dependence. The other line is to investigate", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 563, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 506, + 576 + ], + "score": 1.0, + "content": "topologies that enable communication-efficient decentralized optimization. [43, 15] examined various", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "topologies (such as ring, grid, torus, expander, etc.) on averaging effectiveness, which, however,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "score": 1.0, + "content": "are either communication-costly or averaging-ineffective compared to exponential graphs studied", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "in this paper. [41, 6, 9, 10] studied random graphs (such as Erdos-Renyi random graph and random", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "score": 1.0, + "content": "geometric graph) in which each edge is activated randomly. The randomness of the edge activation", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "score": 1.0, + "content": "can cause a highly unbalanced degrees of each node in the graph, which may significantly affect the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 630, + 276, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 276, + 641 + ], + "score": 1.0, + "content": "efficiency in per-iteration communication.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 507, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 507, + 658 + ], + "score": 1.0, + "content": "Algorithms with time-varying topologies. Many previous works have studied decentralized al-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "gorithms with time-varying topologies. [42] and [44] examined the convergence of decentralized", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "(deterministic) gradient descent and gradient tracking under convex scenarios. [17, 52] investigated", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 104, + 677, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 104, + 677, + 506, + 692 + ], + "score": 1.0, + "content": "gradient tracking under non-convex scenarios, but it did not clarify the influence of the time-varying", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "graphs on convergence rate. In the stochastic scenario, [25] illustrates how decentralized SGD is", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "influenced by time-varying topologies in the non-convex scenario. However, its analysis cannot be", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 711, + 412, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 412, + 722 + ], + "score": 1.0, + "content": "directly extended to the decentralized momentum SGD studied in this work.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 46 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "4", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 506, + 128 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 394, + 86 + ], + "score": 1.0, + "content": "Communication overhead. According to [5], global averaging across", + "type": "text" + }, + { + "bbox": [ + 394, + 75, + 402, + 83 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 72, + 483, + 86 + ], + "score": 1.0, + "content": "nodes either incurs", + "type": "text" + }, + { + "bbox": [ + 483, + 74, + 505, + 84 + ], + "score": 0.89, + "content": "\\Omega ( n )", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 264, + 96 + ], + "score": 1.0, + "content": "bandwidth cost via Parameter-Server, or", + "type": "text" + }, + { + "bbox": [ + 265, + 84, + 286, + 96 + ], + "score": 0.91, + "content": "\\Omega ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 84, + 483, + 96 + ], + "score": 1.0, + "content": "latency via Ring-Allreduce. In either way, it takes", + "type": "text" + }, + { + "bbox": [ + 483, + 85, + 505, + 96 + ], + "score": 0.9, + "content": "\\Omega ( n )", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 413, + 107 + ], + "score": 1.0, + "content": "per-iteration communication time, which is proportional to the network size", + "type": "text" + }, + { + "bbox": [ + 414, + 96, + 420, + 104 + ], + "score": 0.69, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 94, + 506, + 107 + ], + "score": 1.0, + "content": ". As to decentralized", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 503, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 413, + 118 + ], + "score": 1.0, + "content": "methods, we will similarly assume the per-iteration communication time to be", + "type": "text" + }, + { + "bbox": [ + 413, + 106, + 421, + 117 + ], + "score": 0.6, + "content": "\\Omega", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 106, + 503, + 118 + ], + "score": 1.0, + "content": "(maximum degree).", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "Convergence. Under Assumptions A.1–A.4, DmSGD with static topology will converge at [64, 25]:", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 72, + 506, + 128 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 133, + 431, + 168 + ], + "lines": [ + { + "bbox": [ + 170, + 133, + 431, + 168 + ], + "spans": [ + { + "bbox": [ + 170, + 133, + 431, + 168 + ], + "score": 0.93, + "content": "\\frac { 1 } { T } \\sum _ { k = 1 } ^ { T } \\mathbb { E } \\| \\nabla f ( \\bar { \\mathbf { x } } ^ { ( k ) } ) \\| ^ { 2 } = O \\left( \\frac { \\sigma ^ { 2 } } { \\sqrt { n T } } + \\frac { n \\sigma ^ { 2 } } { T ( 1 - \\rho ) } + \\frac { n b ^ { 2 } } { T ( 1 - \\rho ) ^ { 2 } } \\right)", + "type": "interline_equation", + "image_path": "889b47e2e014abbc668e3d92f729cc53dcce44d66cee9db343102df5ee7ca76c.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 170, + 133, + 431, + 144.66666666666666 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 170, + 144.66666666666666, + 431, + 156.33333333333331 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 170, + 156.33333333333331, + 431, + 167.99999999999997 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 109, + 173, + 504, + 199 + ], + "lines": [ + { + "bbox": [ + 102, + 168, + 507, + 196 + ], + "spans": [ + { + "bbox": [ + 102, + 168, + 145, + 196 + ], + "score": 1.0, + "content": "in which", + "type": "text" + }, + { + "bbox": [ + 145, + 173, + 229, + 190 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\bar { x } ^ { ( k ) } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } x _ { i } ^ { ( k ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 168, + 507, + 196 + ], + "score": 1.0, + "content": ". It is worth noting that no analysis in literature, to our knowledge,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 186, + 395, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 395, + 201 + ], + "score": 1.0, + "content": "exists for DmSGD over time-varying topologies with non-convex costs.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 102, + 168, + 507, + 201 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 204, + 505, + 250 + ], + "lines": [ + { + "bbox": [ + 105, + 204, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 210, + 218 + ], + "score": 1.0, + "content": "Linear speedup. When", + "type": "text" + }, + { + "bbox": [ + 210, + 206, + 219, + 215 + ], + "score": 0.81, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 205, + 363, + 218 + ], + "score": 1.0, + "content": "is sufficiently large, the first term", + "type": "text" + }, + { + "bbox": [ + 364, + 204, + 397, + 217 + ], + "score": 0.91, + "content": "1 / \\sqrt { n T }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 205, + 505, + 218 + ], + "score": 1.0, + "content": "dominates (3). This also", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 216, + 506, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 382, + 229 + ], + "score": 1.0, + "content": "applies to parallel SGD. Decentralized and parall SGDs all require", + "type": "text" + }, + { + "bbox": [ + 382, + 216, + 451, + 228 + ], + "score": 0.93, + "content": "T = \\Omega ( 1 / ( n \\epsilon ^ { 2 } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 216, + 506, + 229 + ], + "score": 1.0, + "content": "iterations to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 209, + 239 + ], + "score": 1.0, + "content": "reach a desired accuracy", + "type": "text" + }, + { + "bbox": [ + 210, + 230, + 215, + 237 + ], + "score": 0.65, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 227, + 360, + 239 + ], + "score": 1.0, + "content": ", which is inversely proportional to", + "type": "text" + }, + { + "bbox": [ + 360, + 230, + 367, + 237 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 227, + 505, + 239 + ], + "score": 1.0, + "content": ". Therefore, an algorithm is in its", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 491, + 251 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 200, + 251 + ], + "score": 1.0, + "content": "linear-speedup stage at", + "type": "text" + }, + { + "bbox": [ + 200, + 239, + 208, + 248 + ], + "score": 0.45, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 237, + 295, + 251 + ], + "score": 1.0, + "content": "th iteration if, for this", + "type": "text" + }, + { + "bbox": [ + 295, + 239, + 304, + 248 + ], + "score": 0.8, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 237, + 383, + 251 + ], + "score": 1.0, + "content": ", the term involving", + "type": "text" + }, + { + "bbox": [ + 383, + 238, + 398, + 248 + ], + "score": 0.85, + "content": "n T", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 237, + 491, + 251 + ], + "score": 1.0, + "content": "is dominating the rate.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 204, + 506, + 251 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 254, + 505, + 289 + ], + "lines": [ + { + "bbox": [ + 106, + 255, + 504, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 504, + 266 + ], + "score": 1.0, + "content": "Transient iterations. Transient iterations are referred to those iterations before an algorithm reaches", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 266, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 244, + 278 + ], + "score": 1.0, + "content": "linear-speedup stage, that is when", + "type": "text" + }, + { + "bbox": [ + 244, + 266, + 253, + 275 + ], + "score": 0.81, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 266, + 356, + 278 + ], + "score": 1.0, + "content": "is relatively small so non-", + "type": "text" + }, + { + "bbox": [ + 357, + 266, + 371, + 276 + ], + "score": 0.83, + "content": "\\mathbf { \\nabla } \\cdot n T", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 266, + 505, + 278 + ], + "score": 1.0, + "content": "terms still dominate the rate (see", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 493, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 318, + 290 + ], + "score": 1.0, + "content": "illustration in Appendix C). To reach linear speedup,", + "type": "text" + }, + { + "bbox": [ + 319, + 277, + 327, + 286 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 276, + 493, + 290 + ], + "score": 1.0, + "content": "has to satisfy (derivation in Appendix C)", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 255, + 505, + 290 + ] + }, + { + "type": "text", + "bbox": [ + 111, + 293, + 489, + 322 + ], + "lines": [ + { + "bbox": [ + 107, + 284, + 489, + 329 + ], + "spans": [ + { + "bbox": [ + 107, + 284, + 188, + 329 + ], + "score": 1.0, + "content": "Homogeneous dat", + "type": "text" + }, + { + "bbox": [ + 189, + 293, + 489, + 322 + ], + "score": 0.7, + "content": " \\mathrm { 1 : } \\quad T = \\Omega \\left( { \\frac { n ^ { 3 } } { ( 1 - \\rho ) ^ { 2 } } } \\right) \\qquad { \\mathrm { H e t e r o g e n e o u s ~ d a t a : } } \\quad T = \\Omega \\left( { \\frac { n ^ { 3 } } { ( 1 - \\rho ) ^ { 4 } } } \\right)", + "type": "inline_equation", + "image_path": "52c9e3543bdb97c169056c920c4d71601baa7cb14c8a805f4c76f39aa38df2fa.jpg" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 107, + 284, + 489, + 329 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 338, + 502, + 350 + ], + "lines": [ + { + "bbox": [ + 106, + 337, + 502, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 502, + 351 + ], + "score": 1.0, + "content": "which corresponds to the transient iteration complexity in the homo/hetero-geneous data scenarios.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 106, + 337, + 502, + 351 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 363, + 194, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 195, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 195, + 376 + ], + "score": 1.0, + "content": "2.1 Related Works", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 383, + 505, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 383, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 397 + ], + "score": 1.0, + "content": "Decentralized deep training. Decentralized optimization originates from the control and signal", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "processing community. The first decentralized algorithms on general optimization problems include", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 403, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 420 + ], + "score": 1.0, + "content": "decentralized gradient descent [45], diffusion [11, 51] and dual averaging [18]. In the deep learning", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 416, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 506, + 429 + ], + "score": 1.0, + "content": "regime, decentralize SGD, which was established in [30] to achieve the same linear speedup as parallel", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "SGD in convergence rate, has attracted a lot of attentions. Many efforts have been made to extend the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "algorithm to directed topologies [3, 42], time-varying topologies [25, 42], asynchronous settings [31],", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "score": 1.0, + "content": "and data-heterogeneous scenarios [57, 62, 32, 67]. Techniques such as quantization/compression", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "[2, 8, 26, 24, 58, 36], periodic updates [55, 25, 64], and lazy communication [37, 38, 13] were also", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 470, + 419, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 419, + 483 + ], + "score": 1.0, + "content": "integrated into decentralized SGD to further reduce communiation overheads.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 383, + 506, + 483 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 487, + 506, + 640 + ], + "lines": [ + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "Topology influence. The influence of network topology on decentralized SGD was extensively", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "studied in [25, 51, 66, 45, 42, 27]. All these works indicate that a well-connected topology will", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 507, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 522 + ], + "score": 1.0, + "content": "significantly accelerate decentralized SGD. Two directions have been explored to relieve the influence", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "of network topology. One line of research proposes new algorithms that are less sensitive to topologies.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 530, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 506, + 544 + ], + "score": 1.0, + "content": "For example, [66, 23, 65, 57, 1] removed data heterogeneity with bias-correction techniques in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 540, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 556 + ], + "score": 1.0, + "content": "[68, 29, 62, 40, 69], and [14, 61, 7, 27] utilized periodic global averaging or multiple partial averaging", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 552, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 566 + ], + "score": 1.0, + "content": "steps. All these methods have improved topology dependence. The other line is to investigate", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 563, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 506, + 576 + ], + "score": 1.0, + "content": "topologies that enable communication-efficient decentralized optimization. [43, 15] examined various", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "topologies (such as ring, grid, torus, expander, etc.) on averaging effectiveness, which, however,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "score": 1.0, + "content": "are either communication-costly or averaging-ineffective compared to exponential graphs studied", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "in this paper. [41, 6, 9, 10] studied random graphs (such as Erdos-Renyi random graph and random", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "score": 1.0, + "content": "geometric graph) in which each edge is activated randomly. The randomness of the edge activation", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "score": 1.0, + "content": "can cause a highly unbalanced degrees of each node in the graph, which may significantly affect the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 630, + 276, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 276, + 641 + ], + "score": 1.0, + "content": "efficiency in per-iteration communication.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 487, + 506, + 641 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 507, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 507, + 658 + ], + "score": 1.0, + "content": "Algorithms with time-varying topologies. Many previous works have studied decentralized al-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "gorithms with time-varying topologies. [42] and [44] examined the convergence of decentralized", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "(deterministic) gradient descent and gradient tracking under convex scenarios. [17, 52] investigated", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 104, + 677, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 104, + 677, + 506, + 692 + ], + "score": 1.0, + "content": "gradient tracking under non-convex scenarios, but it did not clarify the influence of the time-varying", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "graphs on convergence rate. In the stochastic scenario, [25] illustrates how decentralized SGD is", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "influenced by time-varying topologies in the non-convex scenario. However, its analysis cannot be", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 711, + 412, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 412, + 722 + ], + "score": 1.0, + "content": "directly extended to the decentralized momentum SGD studied in this work.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 46, + "bbox_fs": [ + 104, + 645, + 507, + 722 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 150 + ], + "lines": [ + { + "bbox": [ + 105, + 70, + 507, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 507, + 86 + ], + "score": 1.0, + "content": "Another related work is the Matcha method [60] based on disjoint matching decomposition sampling.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 82, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 97 + ], + "score": 1.0, + "content": "While similar to Matcha, decentralized SGD over one-peer exponential graphs has several fundamental", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 507, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 507, + 108 + ], + "score": 1.0, + "content": "differences. First, one-peer exponential graph is directed while Matcha only supports undirected", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 506, + 119 + ], + "score": 1.0, + "content": "and symmetric matching decomposition. Second, the favorable periodic exact-average property of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 506, + 130 + ], + "score": 1.0, + "content": "one-peer exponential graphs only holds when sampled cyclicly. However, Matcha only supports", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "independent and random matching samples in analysis. For these reasons, Matcha cannot cover", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 475, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 475, + 150 + ], + "score": 1.0, + "content": "one-peer exponential graphs (especially when momentum is utilized in decentralized SGD).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 188 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "Note. This paper considers deep training within high-performance data-center clusters, in which", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "all GPUs are connected with high-bandwidth channels and the network topology can be fully", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "controlled. It is not for the wireless network setting in which the topology cannot be changed freely.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "title", + "bbox": [ + 106, + 202, + 341, + 217 + ], + "lines": [ + { + "bbox": [ + 104, + 201, + 342, + 221 + ], + "spans": [ + { + "bbox": [ + 104, + 201, + 342, + 221 + ], + "score": 1.0, + "content": "3 Spectral Gap of Static Exponential Graph", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 505, + 272 + ], + "lines": [ + { + "bbox": [ + 106, + 228, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 505, + 239 + ], + "score": 1.0, + "content": "As discussed above, the graph maximum degree decides the per-iteration communication cost while", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 238, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 506, + 251 + ], + "score": 1.0, + "content": "the spectral gap determines the transient iteration complexity (see (4)). It is critical to seek topologies", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 249, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 290, + 262 + ], + "score": 1.0, + "content": "that are both sparse and with large spectral gap", + "type": "text" + }, + { + "bbox": [ + 291, + 250, + 313, + 261 + ], + "score": 0.89, + "content": "1 - \\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 249, + 505, + 262 + ], + "score": 1.0, + "content": "simultaneously. In this section, we will establish", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 260, + 492, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 492, + 273 + ], + "score": 1.0, + "content": "that the static exponential graph, which was first introduced in [3, 30], is one of such topologies.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 276, + 505, + 334 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 388, + 291 + ], + "score": 1.0, + "content": "In a static exponential graph, each node is assigned an index from 0 to", + "type": "text" + }, + { + "bbox": [ + 388, + 277, + 413, + 288 + ], + "score": 0.86, + "content": "n - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 275, + 506, + 291 + ], + "score": 1.0, + "content": "and will communicate", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 286, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 190, + 302 + ], + "score": 1.0, + "content": "to neighbors that are", + "type": "text" + }, + { + "bbox": [ + 191, + 288, + 286, + 300 + ], + "score": 0.89, + "content": "2 ^ { \\bar { 0 } } , 2 ^ { \\bar { 1 } } , \\cdot \\cdot \\cdot , 2 ^ { \\lfloor \\log _ { 2 } ( n - 1 ) \\rfloor }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 286, + 506, + 302 + ], + "score": 1.0, + "content": "hops away. The left plot in Fig. 2 illustrates a directed", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 298, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 104, + 298, + 351, + 314 + ], + "score": 1.0, + "content": "6-node exponential network topology. With maximum degree", + "type": "text" + }, + { + "bbox": [ + 352, + 299, + 391, + 312 + ], + "score": 0.93, + "content": "\\lceil \\log _ { 2 } ^ { - } ( n ) \\rceil", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 298, + 506, + 314 + ], + "score": 1.0, + "content": "neighbors, partial averaging", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 311, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 272, + 324 + ], + "score": 1.0, + "content": "over the static exponential graph will take", + "type": "text" + }, + { + "bbox": [ + 273, + 311, + 320, + 323 + ], + "score": 0.92, + "content": "\\Omega ( \\log _ { 2 } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 311, + 506, + 324 + ], + "score": 1.0, + "content": "communication time per iteration. However, it", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 320, + 340, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 340, + 336 + ], + "score": 1.0, + "content": "remains unclear what the spectral gap is for this topology.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 106, + 337, + 428, + 349 + ], + "lines": [ + { + "bbox": [ + 106, + 336, + 429, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 429, + 351 + ], + "score": 1.0, + "content": "Weight matrix associated with static exponential graph is defined as follows:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 167, + 353, + 444, + 383 + ], + "lines": [ + { + "bbox": [ + 167, + 353, + 444, + 383 + ], + "spans": [ + { + "bbox": [ + 167, + 353, + 444, + 383 + ], + "score": 0.87, + "content": "w _ { i j } ^ { \\mathrm { e x p } } = \\left\\{ \\begin{array} { l l } { \\frac { 1 } { \\lceil \\log _ { 2 } ( n ) \\rceil + 1 } } & { \\mathrm { i f ~ } \\log _ { 2 } ( \\bmod ( j - i , n ) ) \\mathrm { ~ i s ~ a n ~ i n t e g e r ~ o r ~ } i = j } \\\\ { 0 } & { \\mathrm { o t h e r w i s e . } } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "d7f7945e9a0431213e4924d8005c385e7f329885e5b23bcf8e9432976d27d0ed.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 167, + 353, + 444, + 363.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 167, + 363.0, + 444, + 373.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 167, + 373.0, + 444, + 383.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 385, + 503, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 386, + 503, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 503, + 397 + ], + "score": 1.0, + "content": "An example weight matrix associated with the static exponential graph in Fig. 2 is in Appendix A.1", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 396, + 441, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 317, + 410 + ], + "score": 1.0, + "content": "The following proposition evaluates the spectral gap", + "type": "text" + }, + { + "bbox": [ + 317, + 397, + 341, + 408 + ], + "score": 0.88, + "content": "1 - \\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 396, + 441, + 410 + ], + "score": 1.0, + "content": "for weight matrix in (5).", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 416, + 503, + 439 + ], + "lines": [ + { + "bbox": [ + 106, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "Proposition 1 (SPECTRAL GAP OF STATIC EXPO) The spectral gap of matrix (5), which can also", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 426, + 498, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 498, + 441 + ], + "score": 1.0, + "content": "be interpreted as the second largest magnitude of eigenvalues, satisfies (Proof is in Appendix A.2)", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "interline_equation", + "bbox": [ + 184, + 443, + 426, + 497 + ], + "lines": [ + { + "bbox": [ + 184, + 443, + 426, + 497 + ], + "spans": [ + { + "bbox": [ + 184, + 443, + 426, + 497 + ], + "score": 0.89, + "content": "1 - \\rho ( W ^ { \\mathrm { e x p } } ) \\left\\{ \\begin{array} { l l } { \\displaystyle = \\frac { 2 } { 1 + \\lceil \\log _ { 2 } ( n ) \\rceil } , w h e n n i s e \\nu e n n u m b e r } \\\\ { \\displaystyle < \\frac { 2 } { 1 + \\lceil \\log _ { 2 } ( n ) \\rceil } , w h e n n i s o d d n u m b e r } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "df43abe5d38b5897cb54a44aa43c1a3a05c6647aa510f36ffa4c76def30f5e3e.jpg" + } + ] + } + ], + "index": 29.5, + "virtual_lines": [ + { + "bbox": [ + 184, + 443, + 426, + 456.5 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 184, + 456.5, + 426, + 470.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 184, + 470.0, + 426, + 483.5 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 184, + 483.5, + 426, + 497.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 500, + 319, + 515 + ], + "lines": [ + { + "bbox": [ + 104, + 497, + 320, + 516 + ], + "spans": [ + { + "bbox": [ + 104, + 497, + 190, + 516 + ], + "score": 1.0, + "content": "In addition, we have", + "type": "text" + }, + { + "bbox": [ + 191, + 500, + 316, + 515 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\| W ^ { \\mathrm { e x p } } - \\frac { 1 } { n } \\mathbb { 1 } \\mathbb { 1 } ^ { T } \\| _ { 2 } = \\rho ( W ^ { \\mathrm { e x p } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 497, + 320, + 516 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 523, + 502, + 551 + ], + "lines": [ + { + "bbox": [ + 104, + 522, + 502, + 539 + ], + "spans": [ + { + "bbox": [ + 104, + 522, + 299, + 539 + ], + "score": 1.0, + "content": "Remark 1 For a general non-symmetric matrix", + "type": "text" + }, + { + "bbox": [ + 299, + 525, + 311, + 535 + ], + "score": 0.65, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 522, + 402, + 539 + ], + "score": 1.0, + "content": ", it typically holds that", + "type": "text" + }, + { + "bbox": [ + 402, + 524, + 502, + 538 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\| W - \\frac { 1 } { n } \\pm \\Im \\Im ^ { T } \\| _ { 2 } \\neq \\rho ( W ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 534, + 429, + 552 + ], + "spans": [ + { + "bbox": [ + 104, + 534, + 209, + 552 + ], + "score": 1.0, + "content": "Proposition 1 establishes", + "type": "text" + }, + { + "bbox": [ + 210, + 537, + 336, + 551 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\| W ^ { \\mathrm { e x p } } - \\frac { 1 } { n } \\mathbb { 1 } \\mathbb { 1 } ^ { T } \\| _ { 2 } = \\rho ( W ^ { \\mathrm { e x p } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 534, + 429, + 552 + ], + "score": 1.0, + "content": "for exponential graph.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 106, + 562, + 505, + 630 + ], + "lines": [ + { + "bbox": [ + 105, + 562, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 315, + 577 + ], + "score": 1.0, + "content": "Remark 2 The hypercube graph is established in", + "type": "text" + }, + { + "bbox": [ + 315, + 564, + 331, + 575 + ], + "score": 0.33, + "content": "I 5 9", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 562, + 506, + 577 + ], + "score": 1.0, + "content": ", Chapter 16] to have the spectral gap as", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 570, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 274, + 586 + ], + "score": 0.92, + "content": "1 - \\rho ( W ^ { \\mathrm { H y p e r C u b e } } ) = 2 / ( 1 + \\log _ { 2 } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 570, + 506, + 589 + ], + "score": 1.0, + "content": ". While such spectral gap is on the same order as the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "score": 1.0, + "content": "exponential graph, there are two fundamental differences between these two graphs: (a) the hypercube", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 314, + 609 + ], + "score": 1.0, + "content": "graph has to be undirected and the corresponding", + "type": "text" + }, + { + "bbox": [ + 315, + 596, + 327, + 607 + ], + "score": 0.48, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "is symmetric; (b) the number of vertices of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 607, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 266, + 620 + ], + "score": 1.0, + "content": "hypercube must be a power of 2, i.e.,", + "type": "text" + }, + { + "bbox": [ + 266, + 608, + 300, + 618 + ], + "score": 0.9, + "content": "n = 2 ^ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 607, + 408, + 620 + ], + "score": 1.0, + "content": "for some positive integer", + "type": "text" + }, + { + "bbox": [ + 408, + 610, + 414, + 617 + ], + "score": 0.62, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 607, + 505, + 620 + ], + "score": 1.0, + "content": ". In comparision, the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 618, + 378, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 378, + 631 + ], + "score": 1.0, + "content": "exponential graph is more flexible in the size of the graph structure.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 504, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "Remark 3 Proposition 1 clarifies the spectral gap of the static exponential graph. Many literatures", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 654, + 470, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 355, + 667 + ], + "score": 1.0, + "content": "before this work (such as [27]) claimed the spectral gap to be", + "type": "text" + }, + { + "bbox": [ + 355, + 655, + 376, + 667 + ], + "score": 0.85, + "content": "O ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 654, + 470, + 667 + ], + "score": 1.0, + "content": ", which is not accurate.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 107, + 675, + 504, + 720 + ], + "lines": [ + { + "bbox": [ + 105, + 675, + 506, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 506, + 688 + ], + "score": 1.0, + "content": "The theoretical analysis of Proposition 1 is non-trivial. To evaluate the spectral gap, for any network", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 687, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 124, + 699 + ], + "score": 1.0, + "content": "size", + "type": "text" + }, + { + "bbox": [ + 124, + 689, + 132, + 696 + ], + "score": 0.64, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 687, + 505, + 699 + ], + "score": 1.0, + "content": ", we have to derive the analytical expression for each eigenvalue using Fourier transform and", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 697, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 505, + 711 + ], + "score": 1.0, + "content": "calculate the magnitudes. The most tricky part is to assert which eigenvalue expression attains the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 192, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 192, + 721 + ], + "score": 1.0, + "content": "second largest value.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 150 + ], + "lines": [ + { + "bbox": [ + 105, + 70, + 507, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 507, + 86 + ], + "score": 1.0, + "content": "Another related work is the Matcha method [60] based on disjoint matching decomposition sampling.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 82, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 97 + ], + "score": 1.0, + "content": "While similar to Matcha, decentralized SGD over one-peer exponential graphs has several fundamental", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 507, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 507, + 108 + ], + "score": 1.0, + "content": "differences. First, one-peer exponential graph is directed while Matcha only supports undirected", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 506, + 119 + ], + "score": 1.0, + "content": "and symmetric matching decomposition. Second, the favorable periodic exact-average property of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 506, + 130 + ], + "score": 1.0, + "content": "one-peer exponential graphs only holds when sampled cyclicly. However, Matcha only supports", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "independent and random matching samples in analysis. For these reasons, Matcha cannot cover", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 475, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 475, + 150 + ], + "score": 1.0, + "content": "one-peer exponential graphs (especially when momentum is utilized in decentralized SGD).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 70, + 507, + 150 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 188 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "Note. This paper considers deep training within high-performance data-center clusters, in which", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "all GPUs are connected with high-bandwidth channels and the network topology can be fully", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "controlled. It is not for the wireless network setting in which the topology cannot be changed freely.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 154, + 506, + 189 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 202, + 341, + 217 + ], + "lines": [ + { + "bbox": [ + 104, + 201, + 342, + 221 + ], + "spans": [ + { + "bbox": [ + 104, + 201, + 342, + 221 + ], + "score": 1.0, + "content": "3 Spectral Gap of Static Exponential Graph", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 505, + 272 + ], + "lines": [ + { + "bbox": [ + 106, + 228, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 505, + 239 + ], + "score": 1.0, + "content": "As discussed above, the graph maximum degree decides the per-iteration communication cost while", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 238, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 506, + 251 + ], + "score": 1.0, + "content": "the spectral gap determines the transient iteration complexity (see (4)). It is critical to seek topologies", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 249, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 290, + 262 + ], + "score": 1.0, + "content": "that are both sparse and with large spectral gap", + "type": "text" + }, + { + "bbox": [ + 291, + 250, + 313, + 261 + ], + "score": 0.89, + "content": "1 - \\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 249, + 505, + 262 + ], + "score": 1.0, + "content": "simultaneously. In this section, we will establish", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 260, + 492, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 492, + 273 + ], + "score": 1.0, + "content": "that the static exponential graph, which was first introduced in [3, 30], is one of such topologies.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 228, + 506, + 273 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 276, + 505, + 334 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 388, + 291 + ], + "score": 1.0, + "content": "In a static exponential graph, each node is assigned an index from 0 to", + "type": "text" + }, + { + "bbox": [ + 388, + 277, + 413, + 288 + ], + "score": 0.86, + "content": "n - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 275, + 506, + 291 + ], + "score": 1.0, + "content": "and will communicate", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 286, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 190, + 302 + ], + "score": 1.0, + "content": "to neighbors that are", + "type": "text" + }, + { + "bbox": [ + 191, + 288, + 286, + 300 + ], + "score": 0.89, + "content": "2 ^ { \\bar { 0 } } , 2 ^ { \\bar { 1 } } , \\cdot \\cdot \\cdot , 2 ^ { \\lfloor \\log _ { 2 } ( n - 1 ) \\rfloor }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 286, + 506, + 302 + ], + "score": 1.0, + "content": "hops away. The left plot in Fig. 2 illustrates a directed", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 298, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 104, + 298, + 351, + 314 + ], + "score": 1.0, + "content": "6-node exponential network topology. With maximum degree", + "type": "text" + }, + { + "bbox": [ + 352, + 299, + 391, + 312 + ], + "score": 0.93, + "content": "\\lceil \\log _ { 2 } ^ { - } ( n ) \\rceil", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 298, + 506, + 314 + ], + "score": 1.0, + "content": "neighbors, partial averaging", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 311, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 272, + 324 + ], + "score": 1.0, + "content": "over the static exponential graph will take", + "type": "text" + }, + { + "bbox": [ + 273, + 311, + 320, + 323 + ], + "score": 0.92, + "content": "\\Omega ( \\log _ { 2 } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 311, + 506, + 324 + ], + "score": 1.0, + "content": "communication time per iteration. However, it", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 320, + 340, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 340, + 336 + ], + "score": 1.0, + "content": "remains unclear what the spectral gap is for this topology.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17, + "bbox_fs": [ + 104, + 275, + 506, + 336 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 337, + 428, + 349 + ], + "lines": [ + { + "bbox": [ + 106, + 336, + 429, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 429, + 351 + ], + "score": 1.0, + "content": "Weight matrix associated with static exponential graph is defined as follows:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 167, + 353, + 444, + 383 + ], + "lines": [ + { + "bbox": [ + 167, + 353, + 444, + 383 + ], + "spans": [ + { + "bbox": [ + 167, + 353, + 444, + 383 + ], + "score": 0.87, + "content": "w _ { i j } ^ { \\mathrm { e x p } } = \\left\\{ \\begin{array} { l l } { \\frac { 1 } { \\lceil \\log _ { 2 } ( n ) \\rceil + 1 } } & { \\mathrm { i f ~ } \\log _ { 2 } ( \\bmod ( j - i , n ) ) \\mathrm { ~ i s ~ a n ~ i n t e g e r ~ o r ~ } i = j } \\\\ { 0 } & { \\mathrm { o t h e r w i s e . } } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "d7f7945e9a0431213e4924d8005c385e7f329885e5b23bcf8e9432976d27d0ed.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 167, + 353, + 444, + 363.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 167, + 363.0, + 444, + 373.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 167, + 373.0, + 444, + 383.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 385, + 503, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 386, + 503, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 503, + 397 + ], + "score": 1.0, + "content": "An example weight matrix associated with the static exponential graph in Fig. 2 is in Appendix A.1", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 396, + 441, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 317, + 410 + ], + "score": 1.0, + "content": "The following proposition evaluates the spectral gap", + "type": "text" + }, + { + "bbox": [ + 317, + 397, + 341, + 408 + ], + "score": 0.88, + "content": "1 - \\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 396, + 441, + 410 + ], + "score": 1.0, + "content": "for weight matrix in (5).", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 386, + 503, + 410 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 416, + 503, + 439 + ], + "lines": [ + { + "bbox": [ + 106, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "Proposition 1 (SPECTRAL GAP OF STATIC EXPO) The spectral gap of matrix (5), which can also", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 426, + 498, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 498, + 441 + ], + "score": 1.0, + "content": "be interpreted as the second largest magnitude of eigenvalues, satisfies (Proof is in Appendix A.2)", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 416, + 505, + 441 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 184, + 443, + 426, + 497 + ], + "lines": [ + { + "bbox": [ + 184, + 443, + 426, + 497 + ], + "spans": [ + { + "bbox": [ + 184, + 443, + 426, + 497 + ], + "score": 0.89, + "content": "1 - \\rho ( W ^ { \\mathrm { e x p } } ) \\left\\{ \\begin{array} { l l } { \\displaystyle = \\frac { 2 } { 1 + \\lceil \\log _ { 2 } ( n ) \\rceil } , w h e n n i s e \\nu e n n u m b e r } \\\\ { \\displaystyle < \\frac { 2 } { 1 + \\lceil \\log _ { 2 } ( n ) \\rceil } , w h e n n i s o d d n u m b e r } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "df43abe5d38b5897cb54a44aa43c1a3a05c6647aa510f36ffa4c76def30f5e3e.jpg" + } + ] + } + ], + "index": 29.5, + "virtual_lines": [ + { + "bbox": [ + 184, + 443, + 426, + 456.5 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 184, + 456.5, + 426, + 470.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 184, + 470.0, + 426, + 483.5 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 184, + 483.5, + 426, + 497.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 500, + 319, + 515 + ], + "lines": [ + { + "bbox": [ + 104, + 497, + 320, + 516 + ], + "spans": [ + { + "bbox": [ + 104, + 497, + 190, + 516 + ], + "score": 1.0, + "content": "In addition, we have", + "type": "text" + }, + { + "bbox": [ + 191, + 500, + 316, + 515 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\| W ^ { \\mathrm { e x p } } - \\frac { 1 } { n } \\mathbb { 1 } \\mathbb { 1 } ^ { T } \\| _ { 2 } = \\rho ( W ^ { \\mathrm { e x p } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 497, + 320, + 516 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 104, + 497, + 320, + 516 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 523, + 502, + 551 + ], + "lines": [ + { + "bbox": [ + 104, + 522, + 502, + 539 + ], + "spans": [ + { + "bbox": [ + 104, + 522, + 299, + 539 + ], + "score": 1.0, + "content": "Remark 1 For a general non-symmetric matrix", + "type": "text" + }, + { + "bbox": [ + 299, + 525, + 311, + 535 + ], + "score": 0.65, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 522, + 402, + 539 + ], + "score": 1.0, + "content": ", it typically holds that", + "type": "text" + }, + { + "bbox": [ + 402, + 524, + 502, + 538 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\| W - \\frac { 1 } { n } \\pm \\Im \\Im ^ { T } \\| _ { 2 } \\neq \\rho ( W ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 534, + 429, + 552 + ], + "spans": [ + { + "bbox": [ + 104, + 534, + 209, + 552 + ], + "score": 1.0, + "content": "Proposition 1 establishes", + "type": "text" + }, + { + "bbox": [ + 210, + 537, + 336, + 551 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\| W ^ { \\mathrm { e x p } } - \\frac { 1 } { n } \\mathbb { 1 } \\mathbb { 1 } ^ { T } \\| _ { 2 } = \\rho ( W ^ { \\mathrm { e x p } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 534, + 429, + 552 + ], + "score": 1.0, + "content": "for exponential graph.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 104, + 522, + 502, + 552 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 562, + 505, + 630 + ], + "lines": [ + { + "bbox": [ + 105, + 562, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 315, + 577 + ], + "score": 1.0, + "content": "Remark 2 The hypercube graph is established in", + "type": "text" + }, + { + "bbox": [ + 315, + 564, + 331, + 575 + ], + "score": 0.33, + "content": "I 5 9", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 562, + 506, + 577 + ], + "score": 1.0, + "content": ", Chapter 16] to have the spectral gap as", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 570, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 274, + 586 + ], + "score": 0.92, + "content": "1 - \\rho ( W ^ { \\mathrm { H y p e r C u b e } } ) = 2 / ( 1 + \\log _ { 2 } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 570, + 506, + 589 + ], + "score": 1.0, + "content": ". While such spectral gap is on the same order as the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "score": 1.0, + "content": "exponential graph, there are two fundamental differences between these two graphs: (a) the hypercube", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 314, + 609 + ], + "score": 1.0, + "content": "graph has to be undirected and the corresponding", + "type": "text" + }, + { + "bbox": [ + 315, + 596, + 327, + 607 + ], + "score": 0.48, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "is symmetric; (b) the number of vertices of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 607, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 266, + 620 + ], + "score": 1.0, + "content": "hypercube must be a power of 2, i.e.,", + "type": "text" + }, + { + "bbox": [ + 266, + 608, + 300, + 618 + ], + "score": 0.9, + "content": "n = 2 ^ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 607, + 408, + 620 + ], + "score": 1.0, + "content": "for some positive integer", + "type": "text" + }, + { + "bbox": [ + 408, + 610, + 414, + 617 + ], + "score": 0.62, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 607, + 505, + 620 + ], + "score": 1.0, + "content": ". In comparision, the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 618, + 378, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 378, + 631 + ], + "score": 1.0, + "content": "exponential graph is more flexible in the size of the graph structure.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 562, + 506, + 631 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 504, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "Remark 3 Proposition 1 clarifies the spectral gap of the static exponential graph. Many literatures", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 654, + 470, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 355, + 667 + ], + "score": 1.0, + "content": "before this work (such as [27]) claimed the spectral gap to be", + "type": "text" + }, + { + "bbox": [ + 355, + 655, + 376, + 667 + ], + "score": 0.85, + "content": "O ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 654, + 470, + 667 + ], + "score": 1.0, + "content": ", which is not accurate.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 643, + 505, + 667 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 675, + 504, + 720 + ], + "lines": [ + { + "bbox": [ + 105, + 675, + 506, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 506, + 688 + ], + "score": 1.0, + "content": "The theoretical analysis of Proposition 1 is non-trivial. To evaluate the spectral gap, for any network", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 687, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 124, + 699 + ], + "score": 1.0, + "content": "size", + "type": "text" + }, + { + "bbox": [ + 124, + 689, + 132, + 696 + ], + "score": 0.64, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 687, + 505, + 699 + ], + "score": 1.0, + "content": ", we have to derive the analytical expression for each eigenvalue using Fourier transform and", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 697, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 505, + 711 + ], + "score": 1.0, + "content": "calculate the magnitudes. The most tricky part is to assert which eigenvalue expression attains the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 192, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 192, + 721 + ], + "score": 1.0, + "content": "second largest value.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 675, + 506, + 721 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 337, + 161 + ], + "lines": [ + { + "bbox": [ + 106, + 71, + 339, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 71, + 339, + 86 + ], + "score": 1.0, + "content": "We now numerically validate the established spectral gap.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 338, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 338, + 96 + ], + "score": 1.0, + "content": "In Fig. 3, we plotted the spectral gap of the static exponen-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 337, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 165, + 106 + ], + "score": 1.0, + "content": "tial graph with", + "type": "text" + }, + { + "bbox": [ + 165, + 96, + 173, + 104 + ], + "score": 0.73, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 93, + 337, + 106 + ], + "score": 1.0, + "content": "ranging from 4 to 290. It is observed that", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 337, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 172, + 118 + ], + "score": 1.0, + "content": "the derived gap", + "type": "text" + }, + { + "bbox": [ + 172, + 105, + 284, + 118 + ], + "score": 0.92, + "content": "\\rho = 1 - 2 / ( 1 + \\lceil \\log _ { 2 } ( n ) \\rceil )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 105, + 337, + 118 + ], + "score": 1.0, + "content": "is very tight", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 337, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 337, + 128 + ], + "score": 1.0, + "content": "(see the black dashed line). In fact, it exactly matches the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 337, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 228, + 139 + ], + "score": 1.0, + "content": "numerical spectral gap when", + "type": "text" + }, + { + "bbox": [ + 228, + 129, + 235, + 137 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 127, + 337, + 139 + ], + "score": 1.0, + "content": "is even. Moreover, it is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 137, + 336, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 336, + 151 + ], + "score": 1.0, + "content": "also observed the spectral gap of static exponential graph", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 148, + 270, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 270, + 162 + ], + "score": 1.0, + "content": "is much smaller than that of ring or grid.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 338, + 221 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 338, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 338, + 177 + ], + "score": 1.0, + "content": "Finally, we compare the spectral gap and maximum de-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 177, + 337, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 337, + 189 + ], + "score": 1.0, + "content": "gree of the static exponential graph with all other common", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 187, + 338, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 338, + 199 + ], + "score": 1.0, + "content": "graphs in Appendix A.3. It is observed that static exponen-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 198, + 338, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 338, + 211 + ], + "score": 1.0, + "content": "tial graph, while with a sightly larger maximum degree,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 209, + 338, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 338, + 222 + ], + "score": 1.0, + "content": "has a significantly smaller spectral gap than ring and grid.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "image", + "bbox": [ + 345, + 80, + 501, + 205 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 345, + 80, + 501, + 205 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 345, + 80, + 501, + 205 + ], + "spans": [ + { + "bbox": [ + 345, + 80, + 501, + 205 + ], + "score": 0.97, + "type": "image", + "image_path": "c6e6fcda0a87523c8d253a37abda487eba34df530571865620d798dd903940d9.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 345, + 80, + 501, + 93.88888888888889 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 345, + 93.88888888888889, + 501, + 107.77777777777777 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 345, + 107.77777777777777, + 501, + 121.66666666666666 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 345, + 121.66666666666666, + 501, + 135.55555555555554 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 345, + 135.55555555555554, + 501, + 149.44444444444443 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 345, + 149.44444444444443, + 501, + 163.33333333333331 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 345, + 163.33333333333331, + 501, + 177.2222222222222 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 345, + 177.2222222222222, + 501, + 191.1111111111111 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 345, + 191.1111111111111, + 501, + 204.99999999999997 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 345, + 208, + 502, + 219 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 344, + 206, + 504, + 220 + ], + "spans": [ + { + "bbox": [ + 344, + 206, + 504, + 220 + ], + "score": 1.0, + "content": "Figure 3: Spectral gap of some topologies.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + } + ], + "index": 17.0 + }, + { + "type": "title", + "bbox": [ + 105, + 235, + 458, + 249 + ], + "lines": [ + { + "bbox": [ + 103, + 232, + 460, + 253 + ], + "spans": [ + { + "bbox": [ + 103, + 232, + 460, + 253 + ], + "score": 1.0, + "content": "4 One-Peer Exponential Graph Achieves Periodic Exact-Averaging", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 253, + 506, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 231, + 268 + ], + "score": 1.0, + "content": "Static exponential graph incurs", + "type": "text" + }, + { + "bbox": [ + 231, + 254, + 278, + 266 + ], + "score": 0.92, + "content": "\\Omega ( \\log _ { 2 } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 253, + 506, + 268 + ], + "score": 1.0, + "content": "communication overhead per iteration. To overcome this", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 265, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 506, + 278 + ], + "score": 1.0, + "content": "issue, [3] proposes to decompose the static exponential graph into a sequence of one-peer graphs,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "score": 1.0, + "content": "in which each node cycles through all its neighbors, communicating, only, to a single neighbor per", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 420, + 299 + ], + "score": 1.0, + "content": "iteration, see the right plot in Fig. 2. Apparently, each one-peer realization incurs", + "type": "text" + }, + { + "bbox": [ + 420, + 286, + 441, + 299 + ], + "score": 0.89, + "content": "\\Omega ( 1 ) { \\bar { } }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 287, + 505, + 299 + ], + "score": 1.0, + "content": "communication", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 297, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 311 + ], + "score": 1.0, + "content": "cost, which matches with ring or grid. Since each realization is sparser than the static graph, one may", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 308, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 322 + ], + "score": 1.0, + "content": "expect DmSGD with one-peer exponential graphs are less effective in aggregating information. In", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 318, + 475, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 475, + 333 + ], + "score": 1.0, + "content": "the following, we will establish an interesting result: one-peer is very effective in averaging.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 105, + 335, + 462, + 348 + ], + "lines": [ + { + "bbox": [ + 105, + 335, + 463, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 259, + 349 + ], + "score": 1.0, + "content": "Time-varying weight matrix. We let", + "type": "text" + }, + { + "bbox": [ + 259, + 335, + 319, + 348 + ], + "score": 0.93, + "content": "\\tau = \\lceil \\log _ { 2 } ( n ) \\rceil", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 335, + 445, + 349 + ], + "score": 1.0, + "content": ". The weight matrix at iteration", + "type": "text" + }, + { + "bbox": [ + 446, + 336, + 452, + 345 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 335, + 463, + 349 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "interline_equation", + "bbox": [ + 201, + 351, + 408, + 393 + ], + "lines": [ + { + "bbox": [ + 201, + 351, + 408, + 393 + ], + "spans": [ + { + "bbox": [ + 201, + 351, + 408, + 393 + ], + "score": 0.93, + "content": "w _ { i j } ^ { ( k ) } = \\left\\{ \\begin{array} { l l } { \\frac { 1 } { 2 } } & { \\mathrm { i f } \\log _ { 2 } ( \\bmod ( j - i , n ) ) = \\bmod ( k , \\tau ) } \\\\ { \\frac { 1 } { 2 } } & { \\mathrm { i f } i = j } \\\\ { 0 } & { \\mathrm { o t h e r w i s e } . } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "56a16c32ec5031afb71990c27e3978a2fc6659592e1690f7340e1c19e38dc926.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 201, + 351, + 408, + 365.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 201, + 365.0, + 408, + 379.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 201, + 379.0, + 408, + 393.0 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 393, + 505, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "The weight matrix for each realization of the one-peer exponential graphs in Fig. 2 is in Appendix", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "score": 1.0, + "content": "B.1. Since each node communicates to one single neighbor per iteration, the resulting weight matrix", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 415, + 497, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 497, + 427 + ], + "score": 1.0, + "content": "is very sparse, with only one non-zero element in the non-diagonal positions per row and column.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 428, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "Periodic exact-averaging. The periodic exact-averaging property, which was observed by [3]", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 439, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 506, + 451 + ], + "score": 1.0, + "content": "without theoretical justifications, is fundamental to clarify the averaging effectiveness of one-peer", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 450, + 494, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 421, + 462 + ], + "score": 1.0, + "content": "exponential graphs. The following lemma proves that the property holds when", + "type": "text" + }, + { + "bbox": [ + 421, + 452, + 429, + 460 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 450, + 494, + 462 + ], + "score": 1.0, + "content": "is a power of 2.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 106, + 467, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 104, + 466, + 507, + 481 + ], + "spans": [ + { + "bbox": [ + 104, + 467, + 326, + 481 + ], + "score": 1.0, + "content": "Lemma 1 (PERIODIC EXACT AVERAGING) Suppose", + "type": "text" + }, + { + "bbox": [ + 326, + 468, + 377, + 480 + ], + "score": 0.94, + "content": "\\tau = \\log _ { 2 } ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 467, + 471, + 481 + ], + "score": 1.0, + "content": "is a positive integer. If", + "type": "text" + }, + { + "bbox": [ + 472, + 466, + 494, + 479 + ], + "score": 0.89, + "content": "W ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 467, + 507, + 481 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "the weight matrix generated by (7) over the one-peer exponential graphs, it then holds that each", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 107, + 488, + 470, + 502 + ], + "spans": [ + { + "bbox": [ + 107, + 489, + 129, + 500 + ], + "score": 0.88, + "content": "W ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 488, + 230, + 502 + ], + "score": 1.0, + "content": "is doubly-stochastic, i.e.", + "type": "text" + }, + { + "bbox": [ + 230, + 489, + 279, + 500 + ], + "score": 0.91, + "content": "\\mathbf { \\dot { W } } ^ { ( \\dot { k } ) } \\mathbb { 1 } = \\mathbb { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 488, + 298, + 502 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 298, + 489, + 358, + 500 + ], + "score": 0.94, + "content": "\\mathbb { 1 } ^ { \\dot { T } } W ^ { ( k ) } \\dot { = } \\mathbb { 1 } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 488, + 470, + 502 + ], + "score": 1.0, + "content": ". Furthermore, it holds that", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42 + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 502, + 397, + 525 + ], + "lines": [ + { + "bbox": [ + 213, + 502, + 397, + 525 + ], + "spans": [ + { + "bbox": [ + 213, + 502, + 397, + 525 + ], + "score": 0.9, + "content": "W ^ { ( k + \\ell ) } W ^ { ( k + \\ell - 1 ) } \\cdot \\cdot \\cdot W ^ { ( k + 1 ) } W ^ { ( k ) } = \\frac { 1 } { n } \\mathbb { 1 } \\mathbb { 1 } ^ { T }", + "type": "interline_equation", + "image_path": "93d9e92849d6527fabfb3d35361e2ebf966007939d90d1cf725e2f83ea0da3e2.jpg" + } + ] + } + ], + "index": 44, + "virtual_lines": [ + { + "bbox": [ + 213, + 502, + 397, + 525 + ], + "spans": [], + "index": 44 + } + ] + }, + { + "type": "text", + "bbox": [ + 100, + 527, + 462, + 538 + ], + "lines": [ + { + "bbox": [ + 104, + 526, + 465, + 541 + ], + "spans": [ + { + "bbox": [ + 104, + 526, + 168, + 541 + ], + "score": 1.0, + "content": "for any integer", + "type": "text" + }, + { + "bbox": [ + 168, + 527, + 193, + 538 + ], + "score": 0.9, + "content": "k \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 526, + 211, + 541 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 212, + 527, + 235, + 538 + ], + "score": 0.88, + "content": "\\ell \\geq \\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 526, + 465, + 541 + ], + "score": 1.0, + "content": ". And equivalently, the consensus residue form holds that", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "interline_equation", + "bbox": [ + 169, + 538, + 443, + 562 + ], + "lines": [ + { + "bbox": [ + 169, + 538, + 443, + 562 + ], + "spans": [ + { + "bbox": [ + 169, + 538, + 443, + 562 + ], + "score": 0.92, + "content": "\\left( W ^ { ( k + \\ell ) } - { \\frac { 1 } { n } } \\mathbb { 1 } \\mathbb { 1 } ^ { T } \\right) \\left( W ^ { ( k + \\ell - 1 ) } - { \\frac { 1 } { n } } \\mathbb { 1 } \\mathbb { 1 } ^ { T } \\right) \\cdots \\left( W ^ { ( k ) } - { \\frac { 1 } { n } } \\mathbb { 1 } \\mathbb { 1 } ^ { T } \\right) = 0", + "type": "interline_equation", + "image_path": "92e776d87465cc56b7e95f6dc42438afbfb1db43db516432fa0809fad098c814.jpg" + } + ] + } + ], + "index": 46, + "virtual_lines": [ + { + "bbox": [ + 169, + 538, + 443, + 562 + ], + "spans": [], + "index": 46 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 564, + 214, + 576 + ], + "lines": [ + { + "bbox": [ + 104, + 563, + 215, + 577 + ], + "spans": [ + { + "bbox": [ + 104, + 563, + 215, + 577 + ], + "score": 1.0, + "content": "(Proof is in Appendix B.2).", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 106, + 584, + 505, + 618 + ], + "lines": [ + { + "bbox": [ + 105, + 583, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 235, + 597 + ], + "score": 1.0, + "content": "Remark 4 The assumption that", + "type": "text" + }, + { + "bbox": [ + 235, + 585, + 267, + 596 + ], + "score": 0.92, + "content": "\\log _ { 2 } ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 583, + 506, + 597 + ], + "score": 1.0, + "content": "is a positive integer seems necessary. We numerically tested", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 328, + 608 + ], + "score": 1.0, + "content": "various one-peer exponential graphs with non-integer", + "type": "text" + }, + { + "bbox": [ + 328, + 596, + 360, + 608 + ], + "score": 0.91, + "content": "\\log _ { 2 } ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 595, + 505, + 608 + ], + "score": 1.0, + "content": ". None of them is endowed with the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 606, + 239, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 239, + 620 + ], + "score": 1.0, + "content": "periodic exact-average property.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49 + }, + { + "type": "text", + "bbox": [ + 106, + 626, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 180, + 639 + ], + "score": 1.0, + "content": "Remark 5 When", + "type": "text" + }, + { + "bbox": [ + 180, + 626, + 212, + 639 + ], + "score": 0.92, + "content": "\\log _ { 2 } ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "is a positive integer and each realization of the one-peer exponential", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "score": 1.0, + "content": "graph is sampled without replacement, it is easy to verify that the periodic exact-averaging property", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 648, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 663 + ], + "score": 1.0, + "content": "still holds. However, if each realization is sampled with replacement, the periodic exact-averaging", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 104, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 104, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "property generally does not hold unless all realizations are occasionally sampled without repeating.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 52.5 + }, + { + "type": "text", + "bbox": [ + 107, + 682, + 505, + 717 + ], + "lines": [ + { + "bbox": [ + 105, + 682, + 506, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 506, + 696 + ], + "score": 1.0, + "content": "Remark 6 It is worth noting that an one-peer variant of the hypercube graph is established to", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 106, + 694, + 505, + 707 + ], + "spans": [ + { + "bbox": [ + 106, + 694, + 227, + 707 + ], + "score": 1.0, + "content": "achieve exact averaging with", + "type": "text" + }, + { + "bbox": [ + 227, + 694, + 279, + 706 + ], + "score": 0.91, + "content": "\\tau = \\log _ { n } ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 694, + 505, + 707 + ], + "score": 1.0, + "content": "steps [54]. Such one-peer hypercube is undirected and", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 705, + 506, + 717 + ], + "spans": [ + { + "bbox": [ + 105, + 705, + 506, + 717 + ], + "score": 1.0, + "content": "symmetric, which is different from the one-peer exponential graph which is directed and asymmetric.", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 56 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 337, + 161 + ], + "lines": [ + { + "bbox": [ + 106, + 71, + 339, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 71, + 339, + 86 + ], + "score": 1.0, + "content": "We now numerically validate the established spectral gap.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 338, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 338, + 96 + ], + "score": 1.0, + "content": "In Fig. 3, we plotted the spectral gap of the static exponen-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 337, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 165, + 106 + ], + "score": 1.0, + "content": "tial graph with", + "type": "text" + }, + { + "bbox": [ + 165, + 96, + 173, + 104 + ], + "score": 0.73, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 93, + 337, + 106 + ], + "score": 1.0, + "content": "ranging from 4 to 290. It is observed that", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 337, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 172, + 118 + ], + "score": 1.0, + "content": "the derived gap", + "type": "text" + }, + { + "bbox": [ + 172, + 105, + 284, + 118 + ], + "score": 0.92, + "content": "\\rho = 1 - 2 / ( 1 + \\lceil \\log _ { 2 } ( n ) \\rceil )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 105, + 337, + 118 + ], + "score": 1.0, + "content": "is very tight", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 337, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 337, + 128 + ], + "score": 1.0, + "content": "(see the black dashed line). In fact, it exactly matches the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 337, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 228, + 139 + ], + "score": 1.0, + "content": "numerical spectral gap when", + "type": "text" + }, + { + "bbox": [ + 228, + 129, + 235, + 137 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 127, + 337, + 139 + ], + "score": 1.0, + "content": "is even. Moreover, it is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 137, + 336, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 336, + 151 + ], + "score": 1.0, + "content": "also observed the spectral gap of static exponential graph", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 148, + 270, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 270, + 162 + ], + "score": 1.0, + "content": "is much smaller than that of ring or grid.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 71, + 339, + 162 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 338, + 221 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 338, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 338, + 177 + ], + "score": 1.0, + "content": "Finally, we compare the spectral gap and maximum de-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 177, + 337, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 337, + 189 + ], + "score": 1.0, + "content": "gree of the static exponential graph with all other common", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 187, + 338, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 338, + 199 + ], + "score": 1.0, + "content": "graphs in Appendix A.3. It is observed that static exponen-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 198, + 338, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 338, + 211 + ], + "score": 1.0, + "content": "tial graph, while with a sightly larger maximum degree,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 209, + 338, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 338, + 222 + ], + "score": 1.0, + "content": "has a significantly smaller spectral gap than ring and grid.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 165, + 338, + 222 + ] + }, + { + "type": "image", + "bbox": [ + 345, + 80, + 501, + 205 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 345, + 80, + 501, + 205 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 345, + 80, + 501, + 205 + ], + "spans": [ + { + "bbox": [ + 345, + 80, + 501, + 205 + ], + "score": 0.97, + "type": "image", + "image_path": "c6e6fcda0a87523c8d253a37abda487eba34df530571865620d798dd903940d9.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 345, + 80, + 501, + 93.88888888888889 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 345, + 93.88888888888889, + 501, + 107.77777777777777 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 345, + 107.77777777777777, + 501, + 121.66666666666666 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 345, + 121.66666666666666, + 501, + 135.55555555555554 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 345, + 135.55555555555554, + 501, + 149.44444444444443 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 345, + 149.44444444444443, + 501, + 163.33333333333331 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 345, + 163.33333333333331, + 501, + 177.2222222222222 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 345, + 177.2222222222222, + 501, + 191.1111111111111 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 345, + 191.1111111111111, + 501, + 204.99999999999997 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 345, + 208, + 502, + 219 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 344, + 206, + 504, + 220 + ], + "spans": [ + { + "bbox": [ + 344, + 206, + 504, + 220 + ], + "score": 1.0, + "content": "Figure 3: Spectral gap of some topologies.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + } + ], + "index": 17.0 + }, + { + "type": "title", + "bbox": [ + 105, + 235, + 458, + 249 + ], + "lines": [ + { + "bbox": [ + 103, + 232, + 460, + 253 + ], + "spans": [ + { + "bbox": [ + 103, + 232, + 460, + 253 + ], + "score": 1.0, + "content": "4 One-Peer Exponential Graph Achieves Periodic Exact-Averaging", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 253, + 506, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 231, + 268 + ], + "score": 1.0, + "content": "Static exponential graph incurs", + "type": "text" + }, + { + "bbox": [ + 231, + 254, + 278, + 266 + ], + "score": 0.92, + "content": "\\Omega ( \\log _ { 2 } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 253, + 506, + 268 + ], + "score": 1.0, + "content": "communication overhead per iteration. To overcome this", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 265, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 506, + 278 + ], + "score": 1.0, + "content": "issue, [3] proposes to decompose the static exponential graph into a sequence of one-peer graphs,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "score": 1.0, + "content": "in which each node cycles through all its neighbors, communicating, only, to a single neighbor per", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 420, + 299 + ], + "score": 1.0, + "content": "iteration, see the right plot in Fig. 2. Apparently, each one-peer realization incurs", + "type": "text" + }, + { + "bbox": [ + 420, + 286, + 441, + 299 + ], + "score": 0.89, + "content": "\\Omega ( 1 ) { \\bar { } }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 287, + 505, + 299 + ], + "score": 1.0, + "content": "communication", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 297, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 311 + ], + "score": 1.0, + "content": "cost, which matches with ring or grid. Since each realization is sparser than the static graph, one may", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 308, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 322 + ], + "score": 1.0, + "content": "expect DmSGD with one-peer exponential graphs are less effective in aggregating information. In", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 318, + 475, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 475, + 333 + ], + "score": 1.0, + "content": "the following, we will establish an interesting result: one-peer is very effective in averaging.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 253, + 506, + 333 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 335, + 462, + 348 + ], + "lines": [ + { + "bbox": [ + 105, + 335, + 463, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 259, + 349 + ], + "score": 1.0, + "content": "Time-varying weight matrix. We let", + "type": "text" + }, + { + "bbox": [ + 259, + 335, + 319, + 348 + ], + "score": 0.93, + "content": "\\tau = \\lceil \\log _ { 2 } ( n ) \\rceil", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 335, + 445, + 349 + ], + "score": 1.0, + "content": ". The weight matrix at iteration", + "type": "text" + }, + { + "bbox": [ + 446, + 336, + 452, + 345 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 335, + 463, + 349 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 335, + 463, + 349 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 201, + 351, + 408, + 393 + ], + "lines": [ + { + "bbox": [ + 201, + 351, + 408, + 393 + ], + "spans": [ + { + "bbox": [ + 201, + 351, + 408, + 393 + ], + "score": 0.93, + "content": "w _ { i j } ^ { ( k ) } = \\left\\{ \\begin{array} { l l } { \\frac { 1 } { 2 } } & { \\mathrm { i f } \\log _ { 2 } ( \\bmod ( j - i , n ) ) = \\bmod ( k , \\tau ) } \\\\ { \\frac { 1 } { 2 } } & { \\mathrm { i f } i = j } \\\\ { 0 } & { \\mathrm { o t h e r w i s e } . } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "56a16c32ec5031afb71990c27e3978a2fc6659592e1690f7340e1c19e38dc926.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 201, + 351, + 408, + 365.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 201, + 365.0, + 408, + 379.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 201, + 379.0, + 408, + 393.0 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 393, + 505, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "The weight matrix for each realization of the one-peer exponential graphs in Fig. 2 is in Appendix", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "score": 1.0, + "content": "B.1. Since each node communicates to one single neighbor per iteration, the resulting weight matrix", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 415, + 497, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 497, + 427 + ], + "score": 1.0, + "content": "is very sparse, with only one non-zero element in the non-diagonal positions per row and column.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 392, + 505, + 427 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 428, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "Periodic exact-averaging. The periodic exact-averaging property, which was observed by [3]", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 439, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 506, + 451 + ], + "score": 1.0, + "content": "without theoretical justifications, is fundamental to clarify the averaging effectiveness of one-peer", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 450, + 494, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 421, + 462 + ], + "score": 1.0, + "content": "exponential graphs. The following lemma proves that the property holds when", + "type": "text" + }, + { + "bbox": [ + 421, + 452, + 429, + 460 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 450, + 494, + 462 + ], + "score": 1.0, + "content": "is a power of 2.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 428, + 506, + 462 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 467, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 104, + 466, + 507, + 481 + ], + "spans": [ + { + "bbox": [ + 104, + 467, + 326, + 481 + ], + "score": 1.0, + "content": "Lemma 1 (PERIODIC EXACT AVERAGING) Suppose", + "type": "text" + }, + { + "bbox": [ + 326, + 468, + 377, + 480 + ], + "score": 0.94, + "content": "\\tau = \\log _ { 2 } ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 467, + 471, + 481 + ], + "score": 1.0, + "content": "is a positive integer. If", + "type": "text" + }, + { + "bbox": [ + 472, + 466, + 494, + 479 + ], + "score": 0.89, + "content": "W ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 467, + 507, + 481 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "the weight matrix generated by (7) over the one-peer exponential graphs, it then holds that each", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 107, + 488, + 470, + 502 + ], + "spans": [ + { + "bbox": [ + 107, + 489, + 129, + 500 + ], + "score": 0.88, + "content": "W ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 488, + 230, + 502 + ], + "score": 1.0, + "content": "is doubly-stochastic, i.e.", + "type": "text" + }, + { + "bbox": [ + 230, + 489, + 279, + 500 + ], + "score": 0.91, + "content": "\\mathbf { \\dot { W } } ^ { ( \\dot { k } ) } \\mathbb { 1 } = \\mathbb { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 488, + 298, + 502 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 298, + 489, + 358, + 500 + ], + "score": 0.94, + "content": "\\mathbb { 1 } ^ { \\dot { T } } W ^ { ( k ) } \\dot { = } \\mathbb { 1 } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 488, + 470, + 502 + ], + "score": 1.0, + "content": ". Furthermore, it holds that", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42, + "bbox_fs": [ + 104, + 466, + 507, + 502 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 502, + 397, + 525 + ], + "lines": [ + { + "bbox": [ + 213, + 502, + 397, + 525 + ], + "spans": [ + { + "bbox": [ + 213, + 502, + 397, + 525 + ], + "score": 0.9, + "content": "W ^ { ( k + \\ell ) } W ^ { ( k + \\ell - 1 ) } \\cdot \\cdot \\cdot W ^ { ( k + 1 ) } W ^ { ( k ) } = \\frac { 1 } { n } \\mathbb { 1 } \\mathbb { 1 } ^ { T }", + "type": "interline_equation", + "image_path": "93d9e92849d6527fabfb3d35361e2ebf966007939d90d1cf725e2f83ea0da3e2.jpg" + } + ] + } + ], + "index": 44, + "virtual_lines": [ + { + "bbox": [ + 213, + 502, + 397, + 525 + ], + "spans": [], + "index": 44 + } + ] + }, + { + "type": "text", + "bbox": [ + 100, + 527, + 462, + 538 + ], + "lines": [ + { + "bbox": [ + 104, + 526, + 465, + 541 + ], + "spans": [ + { + "bbox": [ + 104, + 526, + 168, + 541 + ], + "score": 1.0, + "content": "for any integer", + "type": "text" + }, + { + "bbox": [ + 168, + 527, + 193, + 538 + ], + "score": 0.9, + "content": "k \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 526, + 211, + 541 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 212, + 527, + 235, + 538 + ], + "score": 0.88, + "content": "\\ell \\geq \\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 526, + 465, + 541 + ], + "score": 1.0, + "content": ". And equivalently, the consensus residue form holds that", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45, + "bbox_fs": [ + 104, + 526, + 465, + 541 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 169, + 538, + 443, + 562 + ], + "lines": [ + { + "bbox": [ + 169, + 538, + 443, + 562 + ], + "spans": [ + { + "bbox": [ + 169, + 538, + 443, + 562 + ], + "score": 0.92, + "content": "\\left( W ^ { ( k + \\ell ) } - { \\frac { 1 } { n } } \\mathbb { 1 } \\mathbb { 1 } ^ { T } \\right) \\left( W ^ { ( k + \\ell - 1 ) } - { \\frac { 1 } { n } } \\mathbb { 1 } \\mathbb { 1 } ^ { T } \\right) \\cdots \\left( W ^ { ( k ) } - { \\frac { 1 } { n } } \\mathbb { 1 } \\mathbb { 1 } ^ { T } \\right) = 0", + "type": "interline_equation", + "image_path": "92e776d87465cc56b7e95f6dc42438afbfb1db43db516432fa0809fad098c814.jpg" + } + ] + } + ], + "index": 46, + "virtual_lines": [ + { + "bbox": [ + 169, + 538, + 443, + 562 + ], + "spans": [], + "index": 46 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 564, + 214, + 576 + ], + "lines": [ + { + "bbox": [ + 104, + 563, + 215, + 577 + ], + "spans": [ + { + "bbox": [ + 104, + 563, + 215, + 577 + ], + "score": 1.0, + "content": "(Proof is in Appendix B.2).", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47, + "bbox_fs": [ + 104, + 563, + 215, + 577 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 584, + 505, + 618 + ], + "lines": [ + { + "bbox": [ + 105, + 583, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 235, + 597 + ], + "score": 1.0, + "content": "Remark 4 The assumption that", + "type": "text" + }, + { + "bbox": [ + 235, + 585, + 267, + 596 + ], + "score": 0.92, + "content": "\\log _ { 2 } ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 583, + 506, + 597 + ], + "score": 1.0, + "content": "is a positive integer seems necessary. We numerically tested", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 328, + 608 + ], + "score": 1.0, + "content": "various one-peer exponential graphs with non-integer", + "type": "text" + }, + { + "bbox": [ + 328, + 596, + 360, + 608 + ], + "score": 0.91, + "content": "\\log _ { 2 } ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 595, + 505, + 608 + ], + "score": 1.0, + "content": ". None of them is endowed with the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 606, + 239, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 239, + 620 + ], + "score": 1.0, + "content": "periodic exact-average property.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49, + "bbox_fs": [ + 105, + 583, + 506, + 620 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 626, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 180, + 639 + ], + "score": 1.0, + "content": "Remark 5 When", + "type": "text" + }, + { + "bbox": [ + 180, + 626, + 212, + 639 + ], + "score": 0.92, + "content": "\\log _ { 2 } ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "is a positive integer and each realization of the one-peer exponential", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "score": 1.0, + "content": "graph is sampled without replacement, it is easy to verify that the periodic exact-averaging property", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 648, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 663 + ], + "score": 1.0, + "content": "still holds. However, if each realization is sampled with replacement, the periodic exact-averaging", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 104, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 104, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "property generally does not hold unless all realizations are occasionally sampled without repeating.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 52.5, + "bbox_fs": [ + 104, + 626, + 506, + 673 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 682, + 505, + 717 + ], + "lines": [ + { + "bbox": [ + 105, + 682, + 506, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 506, + 696 + ], + "score": 1.0, + "content": "Remark 6 It is worth noting that an one-peer variant of the hypercube graph is established to", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 106, + 694, + 505, + 707 + ], + "spans": [ + { + "bbox": [ + 106, + 694, + 227, + 707 + ], + "score": 1.0, + "content": "achieve exact averaging with", + "type": "text" + }, + { + "bbox": [ + 227, + 694, + 279, + 706 + ], + "score": 0.91, + "content": "\\tau = \\log _ { n } ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 694, + 505, + 707 + ], + "score": 1.0, + "content": "steps [54]. Such one-peer hypercube is undirected and", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 705, + 506, + 717 + ], + "spans": [ + { + "bbox": [ + 105, + 705, + 506, + 717 + ], + "score": 1.0, + "content": "symmetric, which is different from the one-peer exponential graph which is directed and asymmetric.", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 56, + "bbox_fs": [ + 105, + 682, + 506, + 717 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 337, + 197 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 337, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 337, + 85 + ], + "score": 1.0, + "content": "We now numerically validate Lemma 1. To this end, we", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 337, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 181, + 96 + ], + "score": 1.0, + "content": "initialize a vector", + "type": "text" + }, + { + "bbox": [ + 181, + 83, + 214, + 94 + ], + "score": 0.91, + "content": "\\bar { x _ { \\mathrm { ~ \\in ~ } } } \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 83, + 337, + 96 + ], + "score": 1.0, + "content": "arbitrarily, and examine how", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 92, + 339, + 109 + ], + "spans": [ + { + "bbox": [ + 107, + 94, + 211, + 109 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\| ( \\Pi _ { \\ell = 0 } ^ { k } W ^ { ( \\ell ) } - \\frac { 1 } { n } \\pmb { 1 } \\pmb { 1 } \\| ^ { T } ) x \\| } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 92, + 308, + 109 + ], + "score": 1.0, + "content": "decreases with iteration", + "type": "text" + }, + { + "bbox": [ + 308, + 96, + 315, + 106 + ], + "score": 0.72, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 92, + 339, + 109 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 107, + 339, + 121 + ], + "spans": [ + { + "bbox": [ + 105, + 107, + 165, + 121 + ], + "score": 1.0, + "content": "weight matrix", + "type": "text" + }, + { + "bbox": [ + 165, + 108, + 188, + 119 + ], + "score": 0.88, + "content": "W ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 107, + 339, + 121 + ], + "score": 1.0, + "content": "is either static or samples from one-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 120, + 338, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 338, + 131 + ], + "score": 1.0, + "content": "peer exponential graph or bipartite random match graph.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 131, + 336, + 143 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 336, + 143 + ], + "score": 1.0, + "content": "In Fig. 4, it is observed that one-peer exponential graphs", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 142, + 338, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 236, + 153 + ], + "score": 1.0, + "content": "can achieve exact average after", + "type": "text" + }, + { + "bbox": [ + 236, + 142, + 268, + 153 + ], + "score": 0.93, + "content": "\\log _ { 2 } ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 142, + 338, + 153 + ], + "score": 1.0, + "content": "steps, which co-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 153, + 337, + 164 + ], + "spans": [ + { + "bbox": [ + 106, + 153, + 337, + 164 + ], + "score": 1.0, + "content": "incides with the results in Lemma 1. In contrast, the static", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 164, + 337, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 337, + 176 + ], + "score": 1.0, + "content": "exponential and bipartite random match graphs can only", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 174, + 338, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 338, + 186 + ], + "score": 1.0, + "content": "achieve the global average asymptotically. The justifica-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 185, + 288, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 288, + 196 + ], + "score": 1.0, + "content": "tion for Remarks 4 and 5 is in Appendix B.3.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5 + }, + { + "type": "title", + "bbox": [ + 106, + 212, + 302, + 226 + ], + "lines": [ + { + "bbox": [ + 104, + 210, + 302, + 229 + ], + "spans": [ + { + "bbox": [ + 104, + 210, + 302, + 229 + ], + "score": 1.0, + "content": "5 DmSGD with Exponential Graphs", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "image", + "bbox": [ + 347, + 79, + 502, + 201 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 347, + 79, + 502, + 201 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 347, + 79, + 502, + 201 + ], + "spans": [ + { + "bbox": [ + 347, + 79, + 502, + 201 + ], + "score": 0.97, + "type": "image", + "image_path": "89129067f685321fb71b3eef9cea4444a11c7818b525fdcfb5d21abc483690c2.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 347, + 79, + 502, + 92.55555555555556 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 347, + 92.55555555555556, + 502, + 106.11111111111111 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 347, + 106.11111111111111, + 502, + 119.66666666666667 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 347, + 119.66666666666667, + 502, + 133.22222222222223 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 347, + 133.22222222222223, + 502, + 146.77777777777777 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 347, + 146.77777777777777, + 502, + 160.33333333333331 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 347, + 160.33333333333331, + 502, + 173.88888888888886 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 347, + 173.88888888888886, + 502, + 187.4444444444444 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 347, + 187.4444444444444, + 502, + 200.99999999999994 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 344, + 204, + 504, + 255 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 344, + 203, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 344, + 203, + 505, + 215 + ], + "score": 1.0, + "content": "Figure 4: Illustration of how consensus", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 343, + 214, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 343, + 214, + 505, + 225 + ], + "score": 1.0, + "content": "residues decay with iterations for various", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 343, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 343, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "graphs. O.E. and S.E. denote one-peer and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 344, + 234, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 344, + 234, + 505, + 245 + ], + "score": 1.0, + "content": "static exponential graphs, and R.M. denotes", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 344, + 244, + 454, + 254 + ], + "spans": [ + { + "bbox": [ + 344, + 244, + 454, + 254 + ], + "score": 1.0, + "content": "bipartite random match graph.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 236, + 336, + 271 + ], + "lines": [ + { + "bbox": [ + 105, + 236, + 337, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 337, + 249 + ], + "score": 1.0, + "content": "With the derived property in Sec. 3 and 4, this section", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 247, + 337, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 337, + 260 + ], + "score": 1.0, + "content": "will examine the convergence of DmSGD with static and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 260, + 223, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 223, + 271 + ], + "score": 1.0, + "content": "one-peer exponential graphs.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 275, + 505, + 298 + ], + "lines": [ + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "score": 1.0, + "content": "DmSGD with static exponential graph. Based on Proposition 1, we can achieve the convergence", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "rate and transient iterations, by following analysis in [64], of DmSGD with static exponential graph.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 106, + 307, + 505, + 328 + ], + "lines": [ + { + "bbox": [ + 103, + 306, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 103, + 306, + 350, + 329 + ], + "score": 1.0, + "content": "Corollary 1 Under Assumptions A.1–A.4, if γ = n(1−β)3√ ,", + "type": "text" + }, + { + "bbox": [ + 346, + 311, + 506, + 327 + ], + "score": 1.0, + "content": "DmSGD (Algorithm 1) will converge at", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "interline_equation", + "bbox": [ + 113, + 334, + 482, + 367 + ], + "lines": [ + { + "bbox": [ + 113, + 334, + 482, + 367 + ], + "spans": [ + { + "bbox": [ + 113, + 334, + 482, + 367 + ], + "score": 0.93, + "content": "\\frac { 1 } { T } \\sum _ { k = 1 } ^ { T } \\mathbb { E } \\| \\nabla f ( \\bar { \\mathbf { x } } ^ { ( k ) } ) \\| ^ { 2 } = O \\left( \\frac { \\sigma ^ { 2 } } { \\sqrt { ( 1 - \\beta ) n T } } + \\frac { n \\log _ { 2 } ( n ) ( 1 - \\beta ) \\sigma ^ { 2 } } { T } + \\frac { n ( 1 - \\beta ) b ^ { 2 } \\log _ { 2 } ^ { 2 } ( n ) } { T } \\right)", + "type": "interline_equation", + "image_path": "cec45730fb2c9210060c288bb0c5b5d3b35dbff42ddaa4c2fbd749f3e6f98735.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 113, + 334, + 482, + 345.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 113, + 345.0, + 482, + 356.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 113, + 356.0, + 482, + 367.0 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 372, + 505, + 397 + ], + "lines": [ + { + "bbox": [ + 104, + 371, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 104, + 371, + 506, + 387 + ], + "score": 1.0, + "content": "Furthermore, the transient iteration complexity of DmSGD over static exponential graph is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 384, + 507, + 398 + ], + "spans": [ + { + "bbox": [ + 107, + 384, + 166, + 397 + ], + "score": 0.92, + "content": "O ( n ^ { 3 } \\log _ { 2 } ^ { 2 } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 384, + 313, + 398 + ], + "score": 1.0, + "content": "for data-homogeneous scenario and", + "type": "text" + }, + { + "bbox": [ + 313, + 384, + 372, + 397 + ], + "score": 0.93, + "content": "O ( n ^ { 3 } \\log _ { 2 } ^ { 4 } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 384, + 507, + 398 + ], + "score": 1.0, + "content": "for data-heterogeneous scenario.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 106, + 406, + 506, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 419 + ], + "score": 1.0, + "content": "DmSGD with one-peer exponential graph. With each realization being sparser than its static", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 417, + 507, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 507, + 430 + ], + "score": 1.0, + "content": "counterpart, one-peer exponential graph is believed to converge slower. However, the periodic exact-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 428, + 507, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 507, + 441 + ], + "score": 1.0, + "content": "averaging property can help DmSGD achieve the same convergence rate as its static counterpart.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "score": 1.0, + "content": "Note that DmSGD with one-peer exponential graph is an one-loop algorithm, see Algorithm 1. The", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 449, + 386, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 386, + 463 + ], + "score": 1.0, + "content": "DmSGD updates start immediately after sampling one weight matrix.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 106, + 471, + 506, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 470, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 205, + 485 + ], + "score": 1.0, + "content": "Theorem 1 We assume", + "type": "text" + }, + { + "bbox": [ + 206, + 471, + 259, + 484 + ], + "score": 0.93, + "content": "\\tau = \\log _ { 2 } ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 470, + 506, + 485 + ], + "score": 1.0, + "content": "is a positive integer, and the time-varying weight matrix is√", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 483, + 502, + 506 + ], + "spans": [ + { + "bbox": [ + 104, + 483, + 442, + 506 + ], + "score": 1.0, + "content": "generated by (7) over one-peer exponential graphs. Under Assumptions A.1–A.4 and", + "type": "text" + }, + { + "bbox": [ + 442, + 484, + 502, + 504 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\gamma = \\frac { \\sqrt { n ( 1 - \\beta ) ^ { 3 } } } { \\sqrt { T } } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 500, + 393, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 189, + 514 + ], + "score": 1.0, + "content": "DmSGD (Algorithm", + "type": "text" + }, + { + "bbox": [ + 189, + 502, + 195, + 511 + ], + "score": 0.31, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 500, + 393, + 514 + ], + "score": 1.0, + "content": ") will converge at (Proof is in Appendix D.1-D.3).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43 + }, + { + "type": "interline_equation", + "bbox": [ + 140, + 518, + 470, + 553 + ], + "lines": [ + { + "bbox": [ + 140, + 518, + 470, + 553 + ], + "spans": [ + { + "bbox": [ + 140, + 518, + 470, + 553 + ], + "score": 0.94, + "content": "\\frac { 1 } { T } \\sum _ { k = 1 } ^ { T } \\mathbb { E } \\| \\nabla f ( \\bar { \\mathbf { x } } ^ { ( k ) } ) \\| ^ { 2 } = O \\left( \\frac { \\sigma ^ { 2 } } { \\sqrt { ( 1 - \\beta ) n T } } + \\frac { n ( 1 - \\beta ) \\sigma ^ { 2 } \\tau } { T } + \\frac { n ( 1 - \\beta ) b ^ { 2 } \\tau ^ { 2 } } { T } \\right) .", + "type": "interline_equation", + "image_path": "9ecb1531fba5d4f7a8a84f38f08ecaaf5800ba181a8c6ae8d9d2c9d471ffb9aa.jpg" + } + ] + } + ], + "index": 46, + "virtual_lines": [ + { + "bbox": [ + 140, + 518, + 470, + 529.6666666666666 + ], + "spans": [], + "index": 45 + }, + { + "bbox": [ + 140, + 529.6666666666666, + 470, + 541.3333333333333 + ], + "spans": [], + "index": 46 + }, + { + "bbox": [ + 140, + 541.3333333333333, + 470, + 552.9999999999999 + ], + "spans": [], + "index": 47 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 558, + 506, + 582 + ], + "lines": [ + { + "bbox": [ + 105, + 557, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 505, + 571 + ], + "score": 1.0, + "content": "Furthermore, the transient iteration complexity of DmSGD over one-peer exponential graph is", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 107, + 568, + 507, + 584 + ], + "spans": [ + { + "bbox": [ + 107, + 569, + 166, + 582 + ], + "score": 0.92, + "content": "O ( n ^ { 3 } \\log _ { 2 } ^ { 2 } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 568, + 313, + 584 + ], + "score": 1.0, + "content": "for data-homogeneous scenario and", + "type": "text" + }, + { + "bbox": [ + 313, + 569, + 372, + 582 + ], + "score": 0.92, + "content": "O ( n ^ { 3 } \\log _ { 2 } ^ { 4 } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 568, + 507, + 584 + ], + "score": 1.0, + "content": "for data-heterogeneous scenario.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48.5 + }, + { + "type": "text", + "bbox": [ + 106, + 590, + 505, + 625 + ], + "lines": [ + { + "bbox": [ + 105, + 590, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 324, + 604 + ], + "score": 1.0, + "content": "Remark 7 Comparing (11) with (10), and noting that", + "type": "text" + }, + { + "bbox": [ + 324, + 591, + 375, + 603 + ], + "score": 0.92, + "content": "\\tau = \\log _ { 2 } ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 590, + 505, + 604 + ], + "score": 1.0, + "content": ", we conclude that DmSGD with", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 603, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 614 + ], + "score": 1.0, + "content": "one-peer graphs converge exactly as fast as with the static counterpart in terms of the established", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 613, + 503, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 503, + 626 + ], + "score": 1.0, + "content": "rate bounds. In addition, both graphs endow DmSGD with the same transient iteration complexity.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51 + }, + { + "type": "text", + "bbox": [ + 106, + 634, + 506, + 668 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 648 + ], + "score": 1.0, + "content": "Remark 8 We can also achieve the convergence rate for decentralized SGD (i.e., DSGD without", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 646, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 383, + 658 + ], + "score": 1.0, + "content": "momentum acceleration) with one-peer exponential graph by setting", + "type": "text" + }, + { + "bbox": [ + 383, + 646, + 409, + 657 + ], + "score": 0.91, + "content": "\\beta = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 646, + 506, + 658 + ], + "score": 1.0, + "content": ". It is easy to verify that", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 104, + 655, + 480, + 669 + ], + "spans": [ + { + "bbox": [ + 104, + 655, + 480, + 669 + ], + "score": 1.0, + "content": "DSGD with one-peer graphs can also converge as fast as with the static exponential graph.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 54 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "score": 1.0, + "content": "Remark 9 The convergence rate and transient iteration complexity of DSGD with general mixing", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "matrices sampling strategy are also studied in [25]. However, the results in reference [25] does", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "not cover the scenario with momentum acceleration. As we show in the proof details, it is highly", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 711, + 240, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 240, + 723 + ], + "score": 1.0, + "content": "non-trivial to handle momentum.", + "type": "text" + } + ], + "index": 59 + } + ], + "index": 57.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 337, + 197 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 337, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 337, + 85 + ], + "score": 1.0, + "content": "We now numerically validate Lemma 1. To this end, we", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 337, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 181, + 96 + ], + "score": 1.0, + "content": "initialize a vector", + "type": "text" + }, + { + "bbox": [ + 181, + 83, + 214, + 94 + ], + "score": 0.91, + "content": "\\bar { x _ { \\mathrm { ~ \\in ~ } } } \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 83, + 337, + 96 + ], + "score": 1.0, + "content": "arbitrarily, and examine how", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 92, + 339, + 109 + ], + "spans": [ + { + "bbox": [ + 107, + 94, + 211, + 109 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\| ( \\Pi _ { \\ell = 0 } ^ { k } W ^ { ( \\ell ) } - \\frac { 1 } { n } \\pmb { 1 } \\pmb { 1 } \\| ^ { T } ) x \\| } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 92, + 308, + 109 + ], + "score": 1.0, + "content": "decreases with iteration", + "type": "text" + }, + { + "bbox": [ + 308, + 96, + 315, + 106 + ], + "score": 0.72, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 92, + 339, + 109 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 107, + 339, + 121 + ], + "spans": [ + { + "bbox": [ + 105, + 107, + 165, + 121 + ], + "score": 1.0, + "content": "weight matrix", + "type": "text" + }, + { + "bbox": [ + 165, + 108, + 188, + 119 + ], + "score": 0.88, + "content": "W ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 107, + 339, + 121 + ], + "score": 1.0, + "content": "is either static or samples from one-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 120, + 338, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 338, + 131 + ], + "score": 1.0, + "content": "peer exponential graph or bipartite random match graph.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 131, + 336, + 143 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 336, + 143 + ], + "score": 1.0, + "content": "In Fig. 4, it is observed that one-peer exponential graphs", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 142, + 338, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 236, + 153 + ], + "score": 1.0, + "content": "can achieve exact average after", + "type": "text" + }, + { + "bbox": [ + 236, + 142, + 268, + 153 + ], + "score": 0.93, + "content": "\\log _ { 2 } ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 142, + 338, + 153 + ], + "score": 1.0, + "content": "steps, which co-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 153, + 337, + 164 + ], + "spans": [ + { + "bbox": [ + 106, + 153, + 337, + 164 + ], + "score": 1.0, + "content": "incides with the results in Lemma 1. In contrast, the static", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 164, + 337, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 337, + 176 + ], + "score": 1.0, + "content": "exponential and bipartite random match graphs can only", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 174, + 338, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 338, + 186 + ], + "score": 1.0, + "content": "achieve the global average asymptotically. The justifica-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 185, + 288, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 288, + 196 + ], + "score": 1.0, + "content": "tion for Remarks 4 and 5 is in Appendix B.3.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 73, + 339, + 196 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 212, + 302, + 226 + ], + "lines": [ + { + "bbox": [ + 104, + 210, + 302, + 229 + ], + "spans": [ + { + "bbox": [ + 104, + 210, + 302, + 229 + ], + "score": 1.0, + "content": "5 DmSGD with Exponential Graphs", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "image", + "bbox": [ + 347, + 79, + 502, + 201 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 347, + 79, + 502, + 201 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 347, + 79, + 502, + 201 + ], + "spans": [ + { + "bbox": [ + 347, + 79, + 502, + 201 + ], + "score": 0.97, + "type": "image", + "image_path": "89129067f685321fb71b3eef9cea4444a11c7818b525fdcfb5d21abc483690c2.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 347, + 79, + 502, + 92.55555555555556 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 347, + 92.55555555555556, + 502, + 106.11111111111111 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 347, + 106.11111111111111, + 502, + 119.66666666666667 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 347, + 119.66666666666667, + 502, + 133.22222222222223 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 347, + 133.22222222222223, + 502, + 146.77777777777777 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 347, + 146.77777777777777, + 502, + 160.33333333333331 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 347, + 160.33333333333331, + 502, + 173.88888888888886 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 347, + 173.88888888888886, + 502, + 187.4444444444444 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 347, + 187.4444444444444, + 502, + 200.99999999999994 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 344, + 204, + 504, + 255 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 344, + 203, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 344, + 203, + 505, + 215 + ], + "score": 1.0, + "content": "Figure 4: Illustration of how consensus", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 343, + 214, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 343, + 214, + 505, + 225 + ], + "score": 1.0, + "content": "residues decay with iterations for various", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 343, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 343, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "graphs. O.E. and S.E. denote one-peer and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 344, + 234, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 344, + 234, + 505, + 245 + ], + "score": 1.0, + "content": "static exponential graphs, and R.M. denotes", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 344, + 244, + 454, + 254 + ], + "spans": [ + { + "bbox": [ + 344, + 244, + 454, + 254 + ], + "score": 1.0, + "content": "bipartite random match graph.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 236, + 336, + 271 + ], + "lines": [ + { + "bbox": [ + 105, + 236, + 337, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 337, + 249 + ], + "score": 1.0, + "content": "With the derived property in Sec. 3 and 4, this section", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 247, + 337, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 337, + 260 + ], + "score": 1.0, + "content": "will examine the convergence of DmSGD with static and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 260, + 223, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 223, + 271 + ], + "score": 1.0, + "content": "one-peer exponential graphs.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 236, + 337, + 271 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 275, + 505, + 298 + ], + "lines": [ + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "score": 1.0, + "content": "DmSGD with static exponential graph. Based on Proposition 1, we can achieve the convergence", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "rate and transient iterations, by following analysis in [64], of DmSGD with static exponential graph.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 274, + 505, + 299 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 307, + 505, + 328 + ], + "lines": [ + { + "bbox": [ + 103, + 306, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 103, + 306, + 350, + 329 + ], + "score": 1.0, + "content": "Corollary 1 Under Assumptions A.1–A.4, if γ = n(1−β)3√ ,", + "type": "text" + }, + { + "bbox": [ + 346, + 311, + 506, + 327 + ], + "score": 1.0, + "content": "DmSGD (Algorithm 1) will converge at", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 103, + 306, + 506, + 329 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 113, + 334, + 482, + 367 + ], + "lines": [ + { + "bbox": [ + 113, + 334, + 482, + 367 + ], + "spans": [ + { + "bbox": [ + 113, + 334, + 482, + 367 + ], + "score": 0.93, + "content": "\\frac { 1 } { T } \\sum _ { k = 1 } ^ { T } \\mathbb { E } \\| \\nabla f ( \\bar { \\mathbf { x } } ^ { ( k ) } ) \\| ^ { 2 } = O \\left( \\frac { \\sigma ^ { 2 } } { \\sqrt { ( 1 - \\beta ) n T } } + \\frac { n \\log _ { 2 } ( n ) ( 1 - \\beta ) \\sigma ^ { 2 } } { T } + \\frac { n ( 1 - \\beta ) b ^ { 2 } \\log _ { 2 } ^ { 2 } ( n ) } { T } \\right)", + "type": "interline_equation", + "image_path": "cec45730fb2c9210060c288bb0c5b5d3b35dbff42ddaa4c2fbd749f3e6f98735.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 113, + 334, + 482, + 345.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 113, + 345.0, + 482, + 356.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 113, + 356.0, + 482, + 367.0 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 372, + 505, + 397 + ], + "lines": [ + { + "bbox": [ + 104, + 371, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 104, + 371, + 506, + 387 + ], + "score": 1.0, + "content": "Furthermore, the transient iteration complexity of DmSGD over static exponential graph is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 384, + 507, + 398 + ], + "spans": [ + { + "bbox": [ + 107, + 384, + 166, + 397 + ], + "score": 0.92, + "content": "O ( n ^ { 3 } \\log _ { 2 } ^ { 2 } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 384, + 313, + 398 + ], + "score": 1.0, + "content": "for data-homogeneous scenario and", + "type": "text" + }, + { + "bbox": [ + 313, + 384, + 372, + 397 + ], + "score": 0.93, + "content": "O ( n ^ { 3 } \\log _ { 2 } ^ { 4 } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 384, + 507, + 398 + ], + "score": 1.0, + "content": "for data-heterogeneous scenario.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 104, + 371, + 507, + 398 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 406, + 506, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 419 + ], + "score": 1.0, + "content": "DmSGD with one-peer exponential graph. With each realization being sparser than its static", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 417, + 507, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 507, + 430 + ], + "score": 1.0, + "content": "counterpart, one-peer exponential graph is believed to converge slower. However, the periodic exact-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 428, + 507, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 507, + 441 + ], + "score": 1.0, + "content": "averaging property can help DmSGD achieve the same convergence rate as its static counterpart.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "score": 1.0, + "content": "Note that DmSGD with one-peer exponential graph is an one-loop algorithm, see Algorithm 1. The", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 449, + 386, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 386, + 463 + ], + "score": 1.0, + "content": "DmSGD updates start immediately after sampling one weight matrix.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 406, + 507, + 463 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 471, + 506, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 470, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 205, + 485 + ], + "score": 1.0, + "content": "Theorem 1 We assume", + "type": "text" + }, + { + "bbox": [ + 206, + 471, + 259, + 484 + ], + "score": 0.93, + "content": "\\tau = \\log _ { 2 } ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 470, + 506, + 485 + ], + "score": 1.0, + "content": "is a positive integer, and the time-varying weight matrix is√", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 483, + 502, + 506 + ], + "spans": [ + { + "bbox": [ + 104, + 483, + 442, + 506 + ], + "score": 1.0, + "content": "generated by (7) over one-peer exponential graphs. Under Assumptions A.1–A.4 and", + "type": "text" + }, + { + "bbox": [ + 442, + 484, + 502, + 504 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\gamma = \\frac { \\sqrt { n ( 1 - \\beta ) ^ { 3 } } } { \\sqrt { T } } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 500, + 393, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 189, + 514 + ], + "score": 1.0, + "content": "DmSGD (Algorithm", + "type": "text" + }, + { + "bbox": [ + 189, + 502, + 195, + 511 + ], + "score": 0.31, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 500, + 393, + 514 + ], + "score": 1.0, + "content": ") will converge at (Proof is in Appendix D.1-D.3).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43, + "bbox_fs": [ + 104, + 470, + 506, + 514 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 140, + 518, + 470, + 553 + ], + "lines": [ + { + "bbox": [ + 140, + 518, + 470, + 553 + ], + "spans": [ + { + "bbox": [ + 140, + 518, + 470, + 553 + ], + "score": 0.94, + "content": "\\frac { 1 } { T } \\sum _ { k = 1 } ^ { T } \\mathbb { E } \\| \\nabla f ( \\bar { \\mathbf { x } } ^ { ( k ) } ) \\| ^ { 2 } = O \\left( \\frac { \\sigma ^ { 2 } } { \\sqrt { ( 1 - \\beta ) n T } } + \\frac { n ( 1 - \\beta ) \\sigma ^ { 2 } \\tau } { T } + \\frac { n ( 1 - \\beta ) b ^ { 2 } \\tau ^ { 2 } } { T } \\right) .", + "type": "interline_equation", + "image_path": "9ecb1531fba5d4f7a8a84f38f08ecaaf5800ba181a8c6ae8d9d2c9d471ffb9aa.jpg" + } + ] + } + ], + "index": 46, + "virtual_lines": [ + { + "bbox": [ + 140, + 518, + 470, + 529.6666666666666 + ], + "spans": [], + "index": 45 + }, + { + "bbox": [ + 140, + 529.6666666666666, + 470, + 541.3333333333333 + ], + "spans": [], + "index": 46 + }, + { + "bbox": [ + 140, + 541.3333333333333, + 470, + 552.9999999999999 + ], + "spans": [], + "index": 47 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 558, + 506, + 582 + ], + "lines": [ + { + "bbox": [ + 105, + 557, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 505, + 571 + ], + "score": 1.0, + "content": "Furthermore, the transient iteration complexity of DmSGD over one-peer exponential graph is", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 107, + 568, + 507, + 584 + ], + "spans": [ + { + "bbox": [ + 107, + 569, + 166, + 582 + ], + "score": 0.92, + "content": "O ( n ^ { 3 } \\log _ { 2 } ^ { 2 } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 568, + 313, + 584 + ], + "score": 1.0, + "content": "for data-homogeneous scenario and", + "type": "text" + }, + { + "bbox": [ + 313, + 569, + 372, + 582 + ], + "score": 0.92, + "content": "O ( n ^ { 3 } \\log _ { 2 } ^ { 4 } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 568, + 507, + 584 + ], + "score": 1.0, + "content": "for data-heterogeneous scenario.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 557, + 507, + 584 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 590, + 505, + 625 + ], + "lines": [ + { + "bbox": [ + 105, + 590, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 324, + 604 + ], + "score": 1.0, + "content": "Remark 7 Comparing (11) with (10), and noting that", + "type": "text" + }, + { + "bbox": [ + 324, + 591, + 375, + 603 + ], + "score": 0.92, + "content": "\\tau = \\log _ { 2 } ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 590, + 505, + 604 + ], + "score": 1.0, + "content": ", we conclude that DmSGD with", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 603, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 614 + ], + "score": 1.0, + "content": "one-peer graphs converge exactly as fast as with the static counterpart in terms of the established", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 613, + 503, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 503, + 626 + ], + "score": 1.0, + "content": "rate bounds. In addition, both graphs endow DmSGD with the same transient iteration complexity.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51, + "bbox_fs": [ + 105, + 590, + 505, + 626 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 634, + 506, + 668 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 648 + ], + "score": 1.0, + "content": "Remark 8 We can also achieve the convergence rate for decentralized SGD (i.e., DSGD without", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 646, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 383, + 658 + ], + "score": 1.0, + "content": "momentum acceleration) with one-peer exponential graph by setting", + "type": "text" + }, + { + "bbox": [ + 383, + 646, + 409, + 657 + ], + "score": 0.91, + "content": "\\beta = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 646, + 506, + 658 + ], + "score": 1.0, + "content": ". It is easy to verify that", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 104, + 655, + 480, + 669 + ], + "spans": [ + { + "bbox": [ + 104, + 655, + 480, + 669 + ], + "score": 1.0, + "content": "DSGD with one-peer graphs can also converge as fast as with the static exponential graph.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 54, + "bbox_fs": [ + 104, + 633, + 506, + 669 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "score": 1.0, + "content": "Remark 9 The convergence rate and transient iteration complexity of DSGD with general mixing", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "matrices sampling strategy are also studied in [25]. However, the results in reference [25] does", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "not cover the scenario with momentum acceleration. As we show in the proof details, it is highly", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 711, + 240, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 240, + 723 + ], + "score": 1.0, + "content": "non-trivial to handle momentum.", + "type": "text" + } + ], + "index": 59 + } + ], + "index": 57.5, + "bbox_fs": [ + 105, + 677, + 505, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "It is worth noting that the analysis for the above theorem is non-trivial. While it targets on the one-peer", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 506, + 96 + ], + "score": 1.0, + "content": "exponential graph, the analysis techniques can be extended to the general time-varying topologies.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "To our best knowledge, it establishes the first result for DSGD with momentum acceleration, over the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "time-varying topologies, and in the non-convex settings. Existing analysis either focuses on DSGD", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "without momentum [25], or DmSGD with static topologies [64]. In addition, the last two terms in", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "(11), actually, can be further tightened by the spectral gap of one-peer exponential graphs. Since the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 138, + 465, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 465, + 151 + ], + "score": 1.0, + "content": "tightened terms are rather complicated, we leave them to the discussion in Appendix D.4.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 254 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 506, + 168 + ], + "score": 1.0, + "content": "State-of-the-art balance between communication and convergence. Table 1 (and tables in Ap-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "pendix D.5) summarize the per-iteration communication time and transient iteration complexity for", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 257, + 189 + ], + "score": 1.0, + "content": "all commonly-used topologies. When", + "type": "text" + }, + { + "bbox": [ + 257, + 178, + 264, + 186 + ], + "score": 0.68, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 177, + 380, + 189 + ], + "score": 1.0, + "content": "is sufficiently large, the term", + "type": "text" + }, + { + "bbox": [ + 380, + 176, + 413, + 188 + ], + "score": 0.92, + "content": "\\log _ { 2 } ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "can be ignored. In this", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 188, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 505, + 200 + ], + "score": 1.0, + "content": "scenario, the exponential graphs (including both static and one-peer variants) achieve state-of-the-art", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 198, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 128, + 211 + ], + "score": 0.89, + "content": "\\tilde { \\Omega } ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 198, + 263, + 213 + ], + "score": 1.0, + "content": "per-iteration communication and", + "type": "text" + }, + { + "bbox": [ + 263, + 198, + 289, + 212 + ], + "score": 0.93, + "content": "\\tilde { \\Omega } ( n ^ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 198, + 407, + 213 + ], + "score": 1.0, + "content": "transient iterations, in which", + "type": "text" + }, + { + "bbox": [ + 407, + 198, + 426, + 212 + ], + "score": 0.91, + "content": "\\tilde { \\Omega } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 198, + 506, + 213 + ], + "score": 1.0, + "content": "hides all logarithm", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 210, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 506, + 223 + ], + "score": 1.0, + "content": "factors. In Appendix D.5, we numerically validate that exponential graphs have smaller transient", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 221, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 506, + 235 + ], + "score": 1.0, + "content": "iteration complexity than ring or grid graph as predicted in Table 1. The comparison between ex-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "score": 1.0, + "content": "ponential graph with random graphs [41, 6, 9, 10] (such as the Erdos-Renyi graph and geometric", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 244, + 297, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 297, + 255 + ], + "score": 1.0, + "content": "random graph) is discussed in Appendix A.3.3.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 108, + 259, + 503, + 293 + ], + "lines": [ + { + "bbox": [ + 106, + 259, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 505, + 272 + ], + "score": 1.0, + "content": "One-peer exponential graph is recommended for decentralized deep training. It is because one-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 271, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 505, + 282 + ], + "score": 1.0, + "content": "peer exponential graph endows DmSGD with the same convergence rate as its static counterpart, but", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 282, + 336, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 336, + 294 + ], + "score": 1.0, + "content": "incurs strictly less communication overhead per iteration.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 107, + 304, + 191, + 318 + ], + "lines": [ + { + "bbox": [ + 104, + 302, + 193, + 321 + ], + "spans": [ + { + "bbox": [ + 104, + 302, + 193, + 321 + ], + "score": 1.0, + "content": "6 Experiments", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "score": 1.0, + "content": "This section will validate our theoretical results by extensive deep learning experiments. First, we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "evaluate how DmSGD with exponential graphs perform against other commonly-used graphs with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "varying network size. Second, we examine whether one-peer exponential graphs achieve the same", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 358, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 506, + 370 + ], + "score": 1.0, + "content": "convergence rate and accuracy as its static counterpart across different tasks, models, and algorithms.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 374, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "score": 1.0, + "content": "Metrics. Training time and validation accuracy are two critical metrics to examine the effectiveness", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "of a distributed training algorithm in deep learning. These two metrics are typically evaluated after", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "the algorithm completes a fixed number of epochs (say, 90 epochs). Training time can reflect the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 407, + 504, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 504, + 419 + ], + "score": 1.0, + "content": "communication efficiency while accuracy, though might not be precise, can roughly measure the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 418, + 502, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 502, + 430 + ], + "score": 1.0, + "content": "convergence rate (or iteration complexity). These two metrics are used in most of our experiments.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26 + }, + { + "type": "title", + "bbox": [ + 107, + 436, + 155, + 447 + ], + "lines": [ + { + "bbox": [ + 105, + 433, + 158, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 158, + 451 + ], + "score": 1.0, + "content": "6.1 Setup", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 451, + 505, + 539 + ], + "lines": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "We implement all decentralized algorithms with PyTorch [46] 1.8.0 using NCCL 2.8.3 (CUDA 10.1)", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "as the communication backend. For parallel SGD, we used PyTorch’s native Distributed Data Parallel", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "score": 1.0, + "content": "(DDP) module. For the implementation of decentralized methods, we utilize BlueFog [63], which is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "a high-performance decentralized deep training framework, to facilitate the topology organization,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "score": 1.0, + "content": "weight matrix generation, and efficient partial averaging. We also follow DDP’s design to enable", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 506, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 517 + ], + "score": 1.0, + "content": "computation and communication overlap. Each server contains 8 V100 GPUs in our cluster and is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 517, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 505, + 528 + ], + "score": 1.0, + "content": "treated as one node. The inter-node network fabrics are 25 Gbps TCP as default, which is a common", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 526, + 253, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 253, + 540 + ], + "score": 1.0, + "content": "distributed training platform setting.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33.5 + }, + { + "type": "title", + "bbox": [ + 107, + 553, + 390, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 392, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 392, + 569 + ], + "score": 1.0, + "content": "6.2 Exponential graphs enable efficient and high-quality training", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 108, + 569, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 567, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 506, + 582 + ], + "score": 1.0, + "content": "In this subsection we evaluate how DmSGD with exponential graphs perform against other commonly-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 580, + 308, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 308, + 591 + ], + "score": 1.0, + "content": "used topologies in the task of image classification.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 107, + 596, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "Implementation. We conduct a series of image classification experiments with the ImageNet-1K", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 607, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 620 + ], + "score": 1.0, + "content": "[16], which consists of 1,281,167 training images and 50,000 validation images in 1000 classes. We", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "train classification models with different topologies and numbers of nodes to verify our theoretical", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 627, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 643 + ], + "score": 1.0, + "content": "findings. The training protocol in [21] is used. In details, we train total 90 epochs. The learning", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 640, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 506, + 653 + ], + "score": 1.0, + "content": "rate is warmed up in the first 5 epochs and is decayed by a factor of 10 at 30, 60 and 80-th epoch.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "score": 1.0, + "content": "The momentum SGD optimizer is used with linear learning rate scaling by default. Experiments are", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 662, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 505, + 675 + ], + "score": 1.0, + "content": "trained in the mixed precision using Pytorch native amp module. We implement DmSGD with all", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 672, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 506, + 685 + ], + "score": 1.0, + "content": "graphs listed in Table 1. The details of each graph is described in Appendix E. For each graph, we test", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 683, + 500, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 500, + 696 + ], + "score": 1.0, + "content": "the training time and validation accuracy for DmSGD with GPU numbers ranging from 32 to 256.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 106, + 700, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "Experiment results. The comparison between different graphs (with varying size) in top-1 validation", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 710, + 453, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 453, + 724 + ], + "score": 1.0, + "content": "accuracy and training time after 90 epochs is listed in Table 2. Major observations are:", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "It is worth noting that the analysis for the above theorem is non-trivial. While it targets on the one-peer", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 506, + 96 + ], + "score": 1.0, + "content": "exponential graph, the analysis techniques can be extended to the general time-varying topologies.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "To our best knowledge, it establishes the first result for DSGD with momentum acceleration, over the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "time-varying topologies, and in the non-convex settings. Existing analysis either focuses on DSGD", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "without momentum [25], or DmSGD with static topologies [64]. In addition, the last two terms in", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "(11), actually, can be further tightened by the spectral gap of one-peer exponential graphs. Since the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 138, + 465, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 465, + 151 + ], + "score": 1.0, + "content": "tightened terms are rather complicated, we leave them to the discussion in Appendix D.4.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 72, + 506, + 151 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 254 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 506, + 168 + ], + "score": 1.0, + "content": "State-of-the-art balance between communication and convergence. Table 1 (and tables in Ap-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "pendix D.5) summarize the per-iteration communication time and transient iteration complexity for", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 257, + 189 + ], + "score": 1.0, + "content": "all commonly-used topologies. When", + "type": "text" + }, + { + "bbox": [ + 257, + 178, + 264, + 186 + ], + "score": 0.68, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 177, + 380, + 189 + ], + "score": 1.0, + "content": "is sufficiently large, the term", + "type": "text" + }, + { + "bbox": [ + 380, + 176, + 413, + 188 + ], + "score": 0.92, + "content": "\\log _ { 2 } ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "can be ignored. In this", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 188, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 505, + 200 + ], + "score": 1.0, + "content": "scenario, the exponential graphs (including both static and one-peer variants) achieve state-of-the-art", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 198, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 128, + 211 + ], + "score": 0.89, + "content": "\\tilde { \\Omega } ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 198, + 263, + 213 + ], + "score": 1.0, + "content": "per-iteration communication and", + "type": "text" + }, + { + "bbox": [ + 263, + 198, + 289, + 212 + ], + "score": 0.93, + "content": "\\tilde { \\Omega } ( n ^ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 198, + 407, + 213 + ], + "score": 1.0, + "content": "transient iterations, in which", + "type": "text" + }, + { + "bbox": [ + 407, + 198, + 426, + 212 + ], + "score": 0.91, + "content": "\\tilde { \\Omega } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 198, + 506, + 213 + ], + "score": 1.0, + "content": "hides all logarithm", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 210, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 506, + 223 + ], + "score": 1.0, + "content": "factors. In Appendix D.5, we numerically validate that exponential graphs have smaller transient", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 221, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 506, + 235 + ], + "score": 1.0, + "content": "iteration complexity than ring or grid graph as predicted in Table 1. The comparison between ex-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "score": 1.0, + "content": "ponential graph with random graphs [41, 6, 9, 10] (such as the Erdos-Renyi graph and geometric", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 244, + 297, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 297, + 255 + ], + "score": 1.0, + "content": "random graph) is discussed in Appendix A.3.3.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 153, + 506, + 255 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 259, + 503, + 293 + ], + "lines": [ + { + "bbox": [ + 106, + 259, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 505, + 272 + ], + "score": 1.0, + "content": "One-peer exponential graph is recommended for decentralized deep training. It is because one-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 271, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 505, + 282 + ], + "score": 1.0, + "content": "peer exponential graph endows DmSGD with the same convergence rate as its static counterpart, but", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 282, + 336, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 336, + 294 + ], + "score": 1.0, + "content": "incurs strictly less communication overhead per iteration.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 106, + 259, + 505, + 294 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 304, + 191, + 318 + ], + "lines": [ + { + "bbox": [ + 104, + 302, + 193, + 321 + ], + "spans": [ + { + "bbox": [ + 104, + 302, + 193, + 321 + ], + "score": 1.0, + "content": "6 Experiments", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "score": 1.0, + "content": "This section will validate our theoretical results by extensive deep learning experiments. First, we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "evaluate how DmSGD with exponential graphs perform against other commonly-used graphs with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "varying network size. Second, we examine whether one-peer exponential graphs achieve the same", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 358, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 506, + 370 + ], + "score": 1.0, + "content": "convergence rate and accuracy as its static counterpart across different tasks, models, and algorithms.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 324, + 506, + 370 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 374, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "score": 1.0, + "content": "Metrics. Training time and validation accuracy are two critical metrics to examine the effectiveness", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "of a distributed training algorithm in deep learning. These two metrics are typically evaluated after", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "the algorithm completes a fixed number of epochs (say, 90 epochs). Training time can reflect the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 407, + 504, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 504, + 419 + ], + "score": 1.0, + "content": "communication efficiency while accuracy, though might not be precise, can roughly measure the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 418, + 502, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 502, + 430 + ], + "score": 1.0, + "content": "convergence rate (or iteration complexity). These two metrics are used in most of our experiments.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 374, + 505, + 430 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 436, + 155, + 447 + ], + "lines": [ + { + "bbox": [ + 105, + 433, + 158, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 158, + 451 + ], + "score": 1.0, + "content": "6.1 Setup", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 451, + 505, + 539 + ], + "lines": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "We implement all decentralized algorithms with PyTorch [46] 1.8.0 using NCCL 2.8.3 (CUDA 10.1)", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "as the communication backend. For parallel SGD, we used PyTorch’s native Distributed Data Parallel", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "score": 1.0, + "content": "(DDP) module. For the implementation of decentralized methods, we utilize BlueFog [63], which is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "a high-performance decentralized deep training framework, to facilitate the topology organization,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "score": 1.0, + "content": "weight matrix generation, and efficient partial averaging. We also follow DDP’s design to enable", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 506, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 517 + ], + "score": 1.0, + "content": "computation and communication overlap. Each server contains 8 V100 GPUs in our cluster and is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 517, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 505, + 528 + ], + "score": 1.0, + "content": "treated as one node. The inter-node network fabrics are 25 Gbps TCP as default, which is a common", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 526, + 253, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 253, + 540 + ], + "score": 1.0, + "content": "distributed training platform setting.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 450, + 506, + 540 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 553, + 390, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 392, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 392, + 569 + ], + "score": 1.0, + "content": "6.2 Exponential graphs enable efficient and high-quality training", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 108, + 569, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 567, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 506, + 582 + ], + "score": 1.0, + "content": "In this subsection we evaluate how DmSGD with exponential graphs perform against other commonly-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 580, + 308, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 308, + 591 + ], + "score": 1.0, + "content": "used topologies in the task of image classification.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 567, + 506, + 591 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 596, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "Implementation. We conduct a series of image classification experiments with the ImageNet-1K", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 607, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 620 + ], + "score": 1.0, + "content": "[16], which consists of 1,281,167 training images and 50,000 validation images in 1000 classes. We", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "train classification models with different topologies and numbers of nodes to verify our theoretical", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 627, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 643 + ], + "score": 1.0, + "content": "findings. The training protocol in [21] is used. In details, we train total 90 epochs. The learning", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 640, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 506, + 653 + ], + "score": 1.0, + "content": "rate is warmed up in the first 5 epochs and is decayed by a factor of 10 at 30, 60 and 80-th epoch.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "score": 1.0, + "content": "The momentum SGD optimizer is used with linear learning rate scaling by default. Experiments are", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 662, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 505, + 675 + ], + "score": 1.0, + "content": "trained in the mixed precision using Pytorch native amp module. We implement DmSGD with all", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 672, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 506, + 685 + ], + "score": 1.0, + "content": "graphs listed in Table 1. The details of each graph is described in Appendix E. For each graph, we test", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 683, + 500, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 500, + 696 + ], + "score": 1.0, + "content": "the training time and validation accuracy for DmSGD with GPU numbers ranging from 32 to 256.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 596, + 506, + 696 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 700, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "Experiment results. The comparison between different graphs (with varying size) in top-1 validation", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 710, + 453, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 453, + 724 + ], + "score": 1.0, + "content": "accuracy and training time after 90 epochs is listed in Table 2. Major observations are:", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50.5, + "bbox_fs": [ + 105, + 699, + 506, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 107, + 69, + 507, + 163 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 111, + 53, + 496, + 65 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 51, + 497, + 66 + ], + "spans": [ + { + "bbox": [ + 114, + 51, + 299, + 66 + ], + "score": 1.0, + "content": "Table 2: Comparison of top-1 validation accuracy", + "type": "text" + }, + { + "bbox": [ + 299, + 54, + 311, + 63 + ], + "score": 0.66, + "content": "\\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 51, + 497, + 66 + ], + "score": 1.0, + "content": "and training time (hours) with different topologies.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 107, + 69, + 507, + 163 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 69, + 507, + 163 + ], + "spans": [ + { + "bbox": [ + 107, + 69, + 507, + 163 + ], + "score": 0.976, + "html": "
NODES TOPOLOGY4(4x8 GPUs)8(8x8 GPUs)16(16x8 GPUs)32(32x8 GPUs)
ACC.TIMEACC.TIMEACC.TIMEACC.TIME
RING76.13 ±0.02311.676.07 ±0.0136.576.08 ±0.0263.375.58 ±0.0211.8
GRID76.08 ±0.00711.676.35 ±0.0376.775.88 ±0.0113.475.76 ±0.0222.0
BI-RAND.MATCH.75.96 ±0.03211.176.26 ±0.0275.776.07 ±0.0122.875.83 ±0.0291.5
RANDOM GRAPH75.97 ±0.02811.576.01 ±0.0337.176.18 ±0.0086.776.24 ±0.0184.7
STATIC EXP.76.21 ±0.02811.676.32 ±0.0376.976.30 ±0.0074.176.28 ±0.0202.5
ONE-PEER EXP.76.28 ±0.06311.176.47 ±0.0355.776.42 ±0.0302.876.30 ±0.0621.5
", + "type": "table", + "image_path": "a92a210f09849421930f1937dfdf8c43b2002eb9695b7c2d4ff974e67878db6f.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 107, + 69, + 507, + 100.33333333333333 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 100.33333333333333, + 507, + 131.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 107, + 131.66666666666666, + 507, + 163.0 + ], + "spans": [], + "index": 3 + } + ] + } + ], + "index": 1.0 + }, + { + "type": "text", + "bbox": [ + 107, + 170, + 505, + 225 + ], + "lines": [ + { + "bbox": [ + 105, + 169, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 505, + 183 + ], + "score": 1.0, + "content": "[1] All graphs (except the random graph) endows DmSGD with training time linear speedup. Among", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "score": 1.0, + "content": "them, bipartite random matching and one-peer exponential graphs achieve the best linear speedup due", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "to their efficient per-iteration communication. However, the accuracy of the matching graph cannot", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 202, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 505, + 216 + ], + "score": 1.0, + "content": "match one-peer exponential graph. The random graph fails to achieve linear speedup because of its", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 214, + 299, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 299, + 226 + ], + "score": 1.0, + "content": "extremely expensive communication overheads.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 258 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 507, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 149, + 238 + ], + "score": 1.0, + "content": "[2] In the", + "type": "text" + }, + { + "bbox": [ + 149, + 226, + 178, + 236 + ], + "score": 0.9, + "content": "3 2 \\times 8", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 225, + 507, + 238 + ], + "score": 1.0, + "content": "GPUs scenario, the training time to finish all 90 epochs can be sorted as follows:", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 236, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 144, + 248 + ], + "score": 1.0, + "content": "one-peer", + "type": "text" + }, + { + "bbox": [ + 144, + 237, + 154, + 246 + ], + "score": 0.63, + "content": "\\approx", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 236, + 217, + 248 + ], + "score": 1.0, + "content": "Bi-RandMatch", + "type": "text" + }, + { + "bbox": [ + 218, + 237, + 291, + 248 + ], + "score": 0.82, + "content": "< \\mathrm { R i n g } < \\mathrm { G r i d } <", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 236, + 365, + 248 + ], + "score": 1.0, + "content": "static exponential", + "type": "text" + }, + { + "bbox": [ + 365, + 237, + 375, + 246 + ], + "score": 0.81, + "content": "<", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 236, + 505, + 248 + ], + "score": 1.0, + "content": "random graph, which coincides", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 247, + 346, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 346, + 259 + ], + "score": 1.0, + "content": "with the per-iteration communication time listed in Table 1.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 259, + 505, + 314 + ], + "lines": [ + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 149, + 271 + ], + "score": 1.0, + "content": "[3] In the", + "type": "text" + }, + { + "bbox": [ + 149, + 259, + 178, + 270 + ], + "score": 0.9, + "content": "3 2 \\times 8", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "GPUs scenario, the training accuracy achieved by each graph after 90 epochs is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 269, + 502, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 237, + 283 + ], + "score": 1.0, + "content": "sorted as follows: random graph", + "type": "text" + }, + { + "bbox": [ + 237, + 271, + 248, + 280 + ], + "score": 0.77, + "content": "\\approx", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 269, + 320, + 283 + ], + "score": 1.0, + "content": "static exponential", + "type": "text" + }, + { + "bbox": [ + 320, + 271, + 331, + 280 + ], + "score": 0.74, + "content": "\\approx", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 269, + 367, + 283 + ], + "score": 1.0, + "content": "one-peer", + "type": "text" + }, + { + "bbox": [ + 368, + 271, + 378, + 280 + ], + "score": 0.8, + "content": ">", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 269, + 439, + 283 + ], + "score": 1.0, + "content": "Bi-RandMatch", + "type": "text" + }, + { + "bbox": [ + 440, + 270, + 502, + 281 + ], + "score": 0.37, + "content": "> \\mathrm { G r i d } > \\mathrm { R i n g }", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "score": 1.0, + "content": "which coincides with the transient iteration complexity listed in Table 1. Note that the random graph", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 291, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 305 + ], + "score": 1.0, + "content": "is rather dense (see the detail in Appendix A.3.1) so it has good accuracy but consumes significant", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 302, + 217, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 217, + 315 + ], + "score": 1.0, + "content": "wall-clock time in training.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 323, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 105, + 321, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 506, + 336 + ], + "score": 1.0, + "content": "With the second and third observations, we can find exponential graphs (especially the one-peer", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 334, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 346 + ], + "score": 1.0, + "content": "exponential graph) can enable both fast and high-quality training performance. We also examined the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 345, + 435, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 272, + 357 + ], + "score": 1.0, + "content": "performance of exponential graphs when", + "type": "text" + }, + { + "bbox": [ + 272, + 347, + 279, + 355 + ], + "score": 0.71, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 345, + 435, + 357 + ], + "score": 1.0, + "content": "is not a power of 2, see Appendix E.2.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 106, + 369, + 371, + 381 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 373, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 373, + 384 + ], + "score": 1.0, + "content": "6.3 One-peer exponential graph v.s. static exponential graph", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 505, + 416 + ], + "lines": [ + { + "bbox": [ + 105, + 383, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 397 + ], + "score": 1.0, + "content": "In this subsection we will focus on the two exponential graphs studied in this paper. In particular, we", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "score": 1.0, + "content": "will validate that one-peer exponential graph endows DmSGD with the same convergence rate as its", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 404, + 501, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 501, + 418 + ], + "score": 1.0, + "content": "static counterpart (i.e., the conclusion in Remark 7) across different tasks, models, and algorithms.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 421, + 505, + 532 + ], + "lines": [ + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "Comparison across models and algorithms. Now we compare one-peer and static exponential", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "score": 1.0, + "content": "graphs with different neural network architectures and algorithms. The task is image classification", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "and the setting is the same as in Sec. 6.2. We test both graphs for ResNet [22], MobileNetv2 [50] and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 454, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 467 + ], + "score": 1.0, + "content": "EfficientNet [56], which are widely-used models in industry. In addition to the DmSGD algorithm", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 465, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 506, + 478 + ], + "score": 1.0, + "content": "(Algorithm 1) studied in this paper, we also examine how exponential graphs perform with other", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 475, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 104, + 475, + 505, + 490 + ], + "score": 1.0, + "content": "commonly-used decentralized momentum method: the vanilla DmSGD [3] which does not exchange", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 487, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 505, + 499 + ], + "score": 1.0, + "content": "momentum between neighbors, and QG-DmSGD [32] which adds a quasi-global momentum to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 496, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 104, + 496, + 438, + 511 + ], + "score": 1.0, + "content": "relieve the influence of data heterogeneity. We do not examine DecentLaM [67] and", + "type": "text" + }, + { + "bbox": [ + 438, + 497, + 451, + 509 + ], + "score": 0.85, + "content": "\\mathrm { D ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 496, + 506, + 511 + ], + "score": 1.0, + "content": "[57] because", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 508, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 522 + ], + "score": 1.0, + "content": "both methods require symmetric weight matrix during the training process which exponential graphs", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "cannot provide. We also list the performance of parallel SGD using global averaging as one baseline.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 106, + 537, + 238, + 667 + ], + "lines": [ + { + "bbox": [ + 106, + 534, + 239, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 239, + 549 + ], + "score": 1.0, + "content": "Table 3 lists the top-1 valida-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 547, + 238, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 238, + 559 + ], + "score": 1.0, + "content": "tion accuracy comparison across", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 558, + 238, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 238, + 570 + ], + "score": 1.0, + "content": "all models and algorithms. In", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 569, + 237, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 237, + 579 + ], + "score": 1.0, + "content": "all scenarios, it is observed that", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 579, + 238, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 238, + 592 + ], + "score": 1.0, + "content": "both graphs can lead to roughly", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 591, + 239, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 239, + 603 + ], + "score": 1.0, + "content": "the same accuracy across mod-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 600, + 239, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 239, + 615 + ], + "score": 1.0, + "content": "els and algorithms. The accuracy", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 612, + 239, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 239, + 624 + ], + "score": 1.0, + "content": "difference (DIFF) is marginal.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 623, + 238, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 238, + 636 + ], + "score": 1.0, + "content": "We also depict the convergence", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 635, + 239, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 239, + 646 + ], + "score": 1.0, + "content": "curves in training loss and ac-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 645, + 238, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 238, + 657 + ], + "score": 1.0, + "content": "curacy for DmSGD with both", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 656, + 238, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 238, + 668 + ], + "score": 1.0, + "content": "graphs in Fig. 5. It shows both", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 40.5 + }, + { + "type": "image", + "bbox": [ + 245, + 549, + 503, + 638 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 245, + 549, + 503, + 638 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 245, + 549, + 503, + 638 + ], + "spans": [ + { + "bbox": [ + 245, + 549, + 503, + 638 + ], + "score": 0.97, + "type": "image", + "image_path": "52f474ec7692e039999038d22622d0f50930b7507149649451ac52bf6583104b.jpg" + } + ] + } + ], + "index": 44, + "virtual_lines": [ + { + "bbox": [ + 245, + 549, + 503, + 578.6666666666666 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 245, + 578.6666666666666, + 503, + 608.3333333333333 + ], + "spans": [], + "index": 44 + }, + { + "bbox": [ + 245, + 608.3333333333333, + 503, + 637.9999999999999 + ], + "spans": [], + "index": 48 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 244, + 639, + 505, + 660 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 244, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 244, + 637, + 506, + 650 + ], + "score": 1.0, + "content": "Figure 5: Convergence curves on the ImageNet (ResNet-50) in terms of", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 245, + 649, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 245, + 649, + 458, + 660 + ], + "score": 1.0, + "content": "training loss and validation top-1 accuracy . Network size is", + "type": "text" + }, + { + "bbox": [ + 459, + 649, + 480, + 658 + ], + "score": 0.89, + "content": "8 \\times 8", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 649, + 506, + 660 + ], + "score": 1.0, + "content": "GPUs.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49.5 + } + ], + "index": 46.75 + }, + { + "type": "text", + "bbox": [ + 107, + 668, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 448, + 680 + ], + "score": 1.0, + "content": "curves evolve closely to each other, indicating that one-peer exponential graph enables", + "type": "text" + }, + { + "bbox": [ + 448, + 667, + 484, + 677 + ], + "score": 0.31, + "content": "\\mathrm { D m } \\mathrm { S G D }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "the same convergence rate as its static counterpart. This is consistent with Theorem 1 and Remark", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "7. Since one-peer is more communication-efficient than static exponential graph (see Table 2), it is", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 104, + 698, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 104, + 698, + 506, + 714 + ], + "score": 1.0, + "content": "recommended to utilize one-peer exponential graph in decentralized deep training. In addition, we", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 710, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 724 + ], + "score": 1.0, + "content": "observe that decentralized methods, while utilizing partial-averaging during training process, has no", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 53 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 107, + 69, + 507, + 163 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 111, + 53, + 496, + 65 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 51, + 497, + 66 + ], + "spans": [ + { + "bbox": [ + 114, + 51, + 299, + 66 + ], + "score": 1.0, + "content": "Table 2: Comparison of top-1 validation accuracy", + "type": "text" + }, + { + "bbox": [ + 299, + 54, + 311, + 63 + ], + "score": 0.66, + "content": "\\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 51, + 497, + 66 + ], + "score": 1.0, + "content": "and training time (hours) with different topologies.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 107, + 69, + 507, + 163 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 69, + 507, + 163 + ], + "spans": [ + { + "bbox": [ + 107, + 69, + 507, + 163 + ], + "score": 0.976, + "html": "
NODES TOPOLOGY4(4x8 GPUs)8(8x8 GPUs)16(16x8 GPUs)32(32x8 GPUs)
ACC.TIMEACC.TIMEACC.TIMEACC.TIME
RING76.13 ±0.02311.676.07 ±0.0136.576.08 ±0.0263.375.58 ±0.0211.8
GRID76.08 ±0.00711.676.35 ±0.0376.775.88 ±0.0113.475.76 ±0.0222.0
BI-RAND.MATCH.75.96 ±0.03211.176.26 ±0.0275.776.07 ±0.0122.875.83 ±0.0291.5
RANDOM GRAPH75.97 ±0.02811.576.01 ±0.0337.176.18 ±0.0086.776.24 ±0.0184.7
STATIC EXP.76.21 ±0.02811.676.32 ±0.0376.976.30 ±0.0074.176.28 ±0.0202.5
ONE-PEER EXP.76.28 ±0.06311.176.47 ±0.0355.776.42 ±0.0302.876.30 ±0.0621.5
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Among", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "score": 1.0, + "content": "them, bipartite random matching and one-peer exponential graphs achieve the best linear speedup due", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "to their efficient per-iteration communication. However, the accuracy of the matching graph cannot", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 202, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 505, + 216 + ], + "score": 1.0, + "content": "match one-peer exponential graph. The random graph fails to achieve linear speedup because of its", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 214, + 299, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 299, + 226 + ], + "score": 1.0, + "content": "extremely expensive communication overheads.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 169, + 506, + 226 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 258 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 507, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 149, + 238 + ], + "score": 1.0, + "content": "[2] In the", + "type": "text" + }, + { + "bbox": [ + 149, + 226, + 178, + 236 + ], + "score": 0.9, + "content": "3 2 \\times 8", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 225, + 507, + 238 + ], + "score": 1.0, + "content": "GPUs scenario, the training time to finish all 90 epochs can be sorted as follows:", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 236, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 144, + 248 + ], + "score": 1.0, + "content": "one-peer", + "type": "text" + }, + { + "bbox": [ + 144, + 237, + 154, + 246 + ], + "score": 0.63, + "content": "\\approx", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 236, + 217, + 248 + ], + "score": 1.0, + "content": "Bi-RandMatch", + "type": "text" + }, + { + "bbox": [ + 218, + 237, + 291, + 248 + ], + "score": 0.82, + "content": "< \\mathrm { R i n g } < \\mathrm { G r i d } <", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 236, + 365, + 248 + ], + "score": 1.0, + "content": "static exponential", + "type": "text" + }, + { + "bbox": [ + 365, + 237, + 375, + 246 + ], + "score": 0.81, + "content": "<", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 236, + 505, + 248 + ], + "score": 1.0, + "content": "random graph, which coincides", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 247, + 346, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 346, + 259 + ], + "score": 1.0, + "content": "with the per-iteration communication time listed in Table 1.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 225, + 507, + 259 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 259, + 505, + 314 + ], + "lines": [ + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 149, + 271 + ], + "score": 1.0, + "content": "[3] In the", + "type": "text" + }, + { + "bbox": [ + 149, + 259, + 178, + 270 + ], + "score": 0.9, + "content": "3 2 \\times 8", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "GPUs scenario, the training accuracy achieved by each graph after 90 epochs is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 269, + 502, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 237, + 283 + ], + "score": 1.0, + "content": "sorted as follows: random graph", + "type": "text" + }, + { + "bbox": [ + 237, + 271, + 248, + 280 + ], + "score": 0.77, + "content": "\\approx", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 269, + 320, + 283 + ], + "score": 1.0, + "content": "static exponential", + "type": "text" + }, + { + "bbox": [ + 320, + 271, + 331, + 280 + ], + "score": 0.74, + "content": "\\approx", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 269, + 367, + 283 + ], + "score": 1.0, + "content": "one-peer", + "type": "text" + }, + { + "bbox": [ + 368, + 271, + 378, + 280 + ], + "score": 0.8, + "content": ">", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 269, + 439, + 283 + ], + "score": 1.0, + "content": "Bi-RandMatch", + "type": "text" + }, + { + "bbox": [ + 440, + 270, + 502, + 281 + ], + "score": 0.37, + "content": "> \\mathrm { G r i d } > \\mathrm { R i n g }", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "score": 1.0, + "content": "which coincides with the transient iteration complexity listed in Table 1. Note that the random graph", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 291, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 305 + ], + "score": 1.0, + "content": "is rather dense (see the detail in Appendix A.3.1) so it has good accuracy but consumes significant", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 302, + 217, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 217, + 315 + ], + "score": 1.0, + "content": "wall-clock time in training.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 258, + 506, + 315 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 323, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 105, + 321, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 506, + 336 + ], + "score": 1.0, + "content": "With the second and third observations, we can find exponential graphs (especially the one-peer", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 334, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 346 + ], + "score": 1.0, + "content": "exponential graph) can enable both fast and high-quality training performance. We also examined the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 345, + 435, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 272, + 357 + ], + "score": 1.0, + "content": "performance of exponential graphs when", + "type": "text" + }, + { + "bbox": [ + 272, + 347, + 279, + 355 + ], + "score": 0.71, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 345, + 435, + 357 + ], + "score": 1.0, + "content": "is not a power of 2, see Appendix E.2.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 321, + 506, + 357 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 369, + 371, + 381 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 373, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 373, + 384 + ], + "score": 1.0, + "content": "6.3 One-peer exponential graph v.s. static exponential graph", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 505, + 416 + ], + "lines": [ + { + "bbox": [ + 105, + 383, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 397 + ], + "score": 1.0, + "content": "In this subsection we will focus on the two exponential graphs studied in this paper. In particular, we", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "score": 1.0, + "content": "will validate that one-peer exponential graph endows DmSGD with the same convergence rate as its", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 404, + 501, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 501, + 418 + ], + "score": 1.0, + "content": "static counterpart (i.e., the conclusion in Remark 7) across different tasks, models, and algorithms.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 383, + 505, + 418 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 421, + 505, + 532 + ], + "lines": [ + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "Comparison across models and algorithms. Now we compare one-peer and static exponential", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "score": 1.0, + "content": "graphs with different neural network architectures and algorithms. The task is image classification", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "and the setting is the same as in Sec. 6.2. We test both graphs for ResNet [22], MobileNetv2 [50] and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 454, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 467 + ], + "score": 1.0, + "content": "EfficientNet [56], which are widely-used models in industry. In addition to the DmSGD algorithm", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 465, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 506, + 478 + ], + "score": 1.0, + "content": "(Algorithm 1) studied in this paper, we also examine how exponential graphs perform with other", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 475, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 104, + 475, + 505, + 490 + ], + "score": 1.0, + "content": "commonly-used decentralized momentum method: the vanilla DmSGD [3] which does not exchange", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 487, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 505, + 499 + ], + "score": 1.0, + "content": "momentum between neighbors, and QG-DmSGD [32] which adds a quasi-global momentum to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 496, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 104, + 496, + 438, + 511 + ], + "score": 1.0, + "content": "relieve the influence of data heterogeneity. We do not examine DecentLaM [67] and", + "type": "text" + }, + { + "bbox": [ + 438, + 497, + 451, + 509 + ], + "score": 0.85, + "content": "\\mathrm { D ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 496, + 506, + 511 + ], + "score": 1.0, + "content": "[57] because", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 508, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 522 + ], + "score": 1.0, + "content": "both methods require symmetric weight matrix during the training process which exponential graphs", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "cannot provide. We also list the performance of parallel SGD using global averaging as one baseline.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28.5, + "bbox_fs": [ + 104, + 421, + 506, + 533 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 537, + 238, + 667 + ], + "lines": [ + { + "bbox": [ + 106, + 534, + 239, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 239, + 549 + ], + "score": 1.0, + "content": "Table 3 lists the top-1 valida-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 547, + 238, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 238, + 559 + ], + "score": 1.0, + "content": "tion accuracy comparison across", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 558, + 238, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 238, + 570 + ], + "score": 1.0, + "content": "all models and algorithms. In", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 569, + 237, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 237, + 579 + ], + "score": 1.0, + "content": "all scenarios, it is observed that", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 579, + 238, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 238, + 592 + ], + "score": 1.0, + "content": "both graphs can lead to roughly", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 591, + 239, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 239, + 603 + ], + "score": 1.0, + "content": "the same accuracy across mod-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 600, + 239, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 239, + 615 + ], + "score": 1.0, + "content": "els and algorithms. The accuracy", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 612, + 239, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 239, + 624 + ], + "score": 1.0, + "content": "difference (DIFF) is marginal.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 623, + 238, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 238, + 636 + ], + "score": 1.0, + "content": "We also depict the convergence", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 635, + 239, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 239, + 646 + ], + "score": 1.0, + "content": "curves in training loss and ac-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 645, + 238, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 238, + 657 + ], + "score": 1.0, + "content": "curacy for DmSGD with both", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 656, + 238, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 238, + 668 + ], + "score": 1.0, + "content": "graphs in Fig. 5. It shows both", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 534, + 239, + 668 + ] + }, + { + "type": "image", + "bbox": [ + 245, + 549, + 503, + 638 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 245, + 549, + 503, + 638 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 245, + 549, + 503, + 638 + ], + "spans": [ + { + "bbox": [ + 245, + 549, + 503, + 638 + ], + "score": 0.97, + "type": "image", + "image_path": "52f474ec7692e039999038d22622d0f50930b7507149649451ac52bf6583104b.jpg" + } + ] + } + ], + "index": 44, + "virtual_lines": [ + { + "bbox": [ + 245, + 549, + 503, + 578.6666666666666 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 245, + 578.6666666666666, + 503, + 608.3333333333333 + ], + "spans": [], + "index": 44 + }, + { + "bbox": [ + 245, + 608.3333333333333, + 503, + 637.9999999999999 + ], + "spans": [], + "index": 48 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 244, + 639, + 505, + 660 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 244, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 244, + 637, + 506, + 650 + ], + "score": 1.0, + "content": "Figure 5: Convergence curves on the ImageNet (ResNet-50) in terms of", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 245, + 649, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 245, + 649, + 458, + 660 + ], + "score": 1.0, + "content": "training loss and validation top-1 accuracy . Network size is", + "type": "text" + }, + { + "bbox": [ + 459, + 649, + 480, + 658 + ], + "score": 0.89, + "content": "8 \\times 8", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 649, + 506, + 660 + ], + "score": 1.0, + "content": "GPUs.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49.5 + } + ], + "index": 46.75 + }, + { + "type": "text", + "bbox": [ + 107, + 668, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 448, + 680 + ], + "score": 1.0, + "content": "curves evolve closely to each other, indicating that one-peer exponential graph enables", + "type": "text" + }, + { + "bbox": [ + 448, + 667, + 484, + 677 + ], + "score": 0.31, + "content": "\\mathrm { D m } \\mathrm { S G D }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "the same convergence rate as its static counterpart. This is consistent with Theorem 1 and Remark", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "7. Since one-peer is more communication-efficient than static exponential graph (see Table 2), it is", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 104, + 698, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 104, + 698, + 506, + 714 + ], + "score": 1.0, + "content": "recommended to utilize one-peer exponential graph in decentralized deep training. In addition, we", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 710, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 724 + ], + "score": 1.0, + "content": "observe that decentralized methods, while utilizing partial-averaging during training process, has no", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 174, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 505, + 187 + ], + "score": 1.0, + "content": "significant accuracy degradation compared parallel SGD. Decentralized SGD can even be superior", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 186, + 154, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 154, + 198 + ], + "score": 1.0, + "content": "sometimes.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 53, + "bbox_fs": [ + 104, + 667, + 506, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 112, + 88, + 497, + 162 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 64, + 503, + 86 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 63, + 505, + 76 + ], + "spans": [ + { + "bbox": [ + 105, + 63, + 505, + 76 + ], + "score": 1.0, + "content": "Table 3: Top-1 validation accuracy and wall-clock time (in hours) comparison with different models and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 74, + 414, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 74, + 414, + 86 + ], + "score": 1.0, + "content": "algorithms on ImageNet dataset over static/one-peer exponential graphs (8x8 GPUs).", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 112, + 88, + 497, + 162 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 88, + 497, + 162 + ], + "spans": [ + { + "bbox": [ + 112, + 88, + 497, + 162 + ], + "score": 0.981, + "html": "
MODEL TOPOLOGYREsNET-50MOBILENET-V2EFFICIENTNET
STATICONE-PEERSTATICONE-PEERSTATICONE-PEER
PARALLEL SGD76.21 (7.0)70.12 (5.8)77.63 (9.0)
VANILLA DMSGD76.14 (6.6)76.06 (5.5)69.98 (5.6)69.81 (4.6)77.62 (8.4)77.48 (6.9)
DMSGD76.50 (6.9)76.52(5.7)69.62 (5.7)69.98 (4.8)77.44 (8.7)77.51 (7.1)
QG-DMSGD76.43 (6.6)76.35(5.6)69.83 (5.6)69.81 (4.6)77.60 (8.4)77.72 (6.9)
", + "type": "table", + "image_path": "d2f76766d684f033377afdcf0e6ce9f12e42200a31d6c4556528efb3bce2b7af.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 112, + 88, + 497, + 112.66666666666667 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 112, + 112.66666666666667, + 497, + 137.33333333333334 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 112, + 137.33333333333334, + 497, + 162.0 + ], + "spans": [], + "index": 4 + } + ] + } + ], + "index": 1.75 + }, + { + "type": "text", + "bbox": [ + 108, + 174, + 504, + 196 + ], + "lines": [ + { + "bbox": [ + 105, + 174, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 505, + 187 + ], + "score": 1.0, + "content": "significant accuracy degradation compared parallel SGD. Decentralized SGD can even be superior", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 186, + 154, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 154, + 198 + ], + "score": 1.0, + "content": "sometimes.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 106, + 202, + 505, + 279 + ], + "lines": [ + { + "bbox": [ + 106, + 201, + 507, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 507, + 215 + ], + "score": 1.0, + "content": "Comparison across different tasks. We next compare the aforementioned algorithms with one-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 213, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 225 + ], + "score": 1.0, + "content": "peer and static exponential graphs in another well-known task: object detection. We will test the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 223, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 505, + 236 + ], + "score": 1.0, + "content": "following widely-used models: Faster-RCNN [49] and RetinaNet [34] on popular PASCAL VOC", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 234, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 505, + 247 + ], + "score": 1.0, + "content": "[19] and COCO [35] datasets. We adopt the MMDetection [12] framework as the building blocks", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "score": 1.0, + "content": "and utilize ResNet-50 with FPN [33] as the backbone network. We choose mean Average Precision", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 257, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 505, + 269 + ], + "score": 1.0, + "content": "(mAP) as the evaluation metric for both datesets. We used 8 GPUs (which are connected by the static", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 267, + 490, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 490, + 280 + ], + "score": 1.0, + "content": "or dynamic exponential topology) and set the total batch size as 64 in all detection experiments.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 284, + 505, + 339 + ], + "lines": [ + { + "bbox": [ + 106, + 283, + 504, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 504, + 295 + ], + "score": 1.0, + "content": "Table 4 compares the performance of decentralized training across different object detection models", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 295, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 505, + 307 + ], + "score": 1.0, + "content": "and datasets. Similar to the above experiment, it is observed that both graphs enable decentralized", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 304, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 317 + ], + "score": 1.0, + "content": "algorithms with almost the same performance in each scenario. This again illustrates the value of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "one-peer exponential graph in deep learning tasks - it endows decentralized deep training with both", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 326, + 286, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 286, + 341 + ], + "score": 1.0, + "content": "fast training speed and satisfactory accuracy.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "table", + "bbox": [ + 107, + 363, + 504, + 446 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 132, + 352, + 474, + 362 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 133, + 351, + 475, + 364 + ], + "spans": [ + { + "bbox": [ + 133, + 351, + 475, + 364 + ], + "score": 1.0, + "content": "Table 4: Comparison of different methods and models on PASCAL VOC and COCO datasets.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "table_body", + "bbox": [ + 107, + 363, + 504, + 446 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 107, + 363, + 504, + 446 + ], + "spans": [ + { + "bbox": [ + 107, + 363, + 504, + 446 + ], + "score": 0.984, + "html": "
DATASET MODEL TOPOLOGYPASCAL VOCCoCo
RETINANETFASTER RCNNRETINANETFASTER RCNN
STATICONE-PEERSTATICONE-PEERSTATICONE-PEERSTATICONE-PEER
PARALLEL SGD79.0-80.3=36.2=37.2=
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With these results, we reveal that one-peer exponential graphs endow DmSGD with the same", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 511, + 504, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 504, + 523 + ], + "score": 1.0, + "content": "convergence rate as their static counterpart. We also establish that exponential graphs achieve nearly", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "minimum per-iteration communication time and transient iteration complexity simultaneously when", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 532, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 107, + 535, + 114, + 542 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 532, + 505, + 545 + ], + "score": 1.0, + "content": "is large. All conclusions are thoroughly examined with industrial-standard benchmarks. As the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "future work, we will investigate symmetric time-varying graphs that can perform as well as one-peer", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 554, + 447, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 317, + 567 + ], + "score": 1.0, + "content": "exponential graph. Symmetric graphs are critical for", + "type": "text" + }, + { + "bbox": [ + 318, + 554, + 330, + 564 + ], + "score": 0.87, + "content": "\\mathrm { D } ^ { \\beth }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 554, + 447, + 567 + ], + "score": 1.0, + "content": "and DecentLaM algorithms.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27.5 + }, + { + "type": "title", + "bbox": [ + 108, + 583, + 207, + 596 + ], + "lines": [ + { + "bbox": [ + 105, + 581, + 208, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 208, + 599 + ], + "score": 1.0, + "content": "Acknowledgements", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 609, + 504, + 631 + ], + "lines": [ + { + "bbox": [ + 105, + 607, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 506, + 622 + ], + "score": 1.0, + "content": "The authors are grateful to Dr. Sai Praneeth Karimireddy from EPFL for the helpful discussions on", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 619, + 192, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 192, + 632 + ], + "score": 1.0, + "content": "the hypercube graph.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "title", + "bbox": [ + 107, + 648, + 163, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 646, + 165, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 165, + 663 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 111, + 669, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 110, + 668, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 110, + 668, + 506, + 682 + ], + "score": 1.0, + "content": "[1] Sulaiman A Alghunaim and Kun Yuan. A unified and refined convergence analysis for non-convex", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 126, + 679, + 356, + 691 + ], + "spans": [ + { + "bbox": [ + 126, + 679, + 356, + 691 + ], + "score": 1.0, + "content": "decentralized learning. arXiv preprint arXiv:2110.09993, 2021.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 111, + 691, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 111, + 691, + 505, + 703 + ], + "score": 1.0, + "content": "[2] Dan Alistarh, Demjan Grubic, Jerry Li, Ryota Tomioka, and Milan Vojnovic. Qsgd: Communication-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 125, + 700, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 125, + 700, + 505, + 714 + ], + "score": 1.0, + "content": "efficient sgd via gradient quantization and encoding. In Advances in Neural Information Processing", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 125, + 712, + 249, + 722 + ], + "spans": [ + { + "bbox": [ + 125, + 712, + 249, + 722 + ], + "score": 1.0, + "content": "Systems, pages 1709–1720, 2017.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "spans": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "score": 1.0, + "content": "10", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 112, + 88, + 497, + 162 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 64, + 503, + 86 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 63, + 505, + 76 + ], + "spans": [ + { + "bbox": [ + 105, + 63, + 505, + 76 + ], + "score": 1.0, + "content": "Table 3: Top-1 validation accuracy and wall-clock time (in hours) comparison with different models and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 74, + 414, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 74, + 414, + 86 + ], + "score": 1.0, + "content": "algorithms on ImageNet dataset over static/one-peer exponential graphs (8x8 GPUs).", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 112, + 88, + 497, + 162 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 88, + 497, + 162 + ], + "spans": [ + { + "bbox": [ + 112, + 88, + 497, + 162 + ], + "score": 0.981, + "html": "
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STATICONE-PEERSTATICONE-PEERSTATICONE-PEER
PARALLEL SGD76.21 (7.0)70.12 (5.8)77.63 (9.0)
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We next compare the aforementioned algorithms with one-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 213, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 225 + ], + "score": 1.0, + "content": "peer and static exponential graphs in another well-known task: object detection. We will test the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 223, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 505, + 236 + ], + "score": 1.0, + "content": "following widely-used models: Faster-RCNN [49] and RetinaNet [34] on popular PASCAL VOC", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 234, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 505, + 247 + ], + "score": 1.0, + "content": "[19] and COCO [35] datasets. We adopt the MMDetection [12] framework as the building blocks", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "score": 1.0, + "content": "and utilize ResNet-50 with FPN [33] as the backbone network. We choose mean Average Precision", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 257, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 505, + 269 + ], + "score": 1.0, + "content": "(mAP) as the evaluation metric for both datesets. We used 8 GPUs (which are connected by the static", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 267, + 490, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 490, + 280 + ], + "score": 1.0, + "content": "or dynamic exponential topology) and set the total batch size as 64 in all detection experiments.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 201, + 507, + 280 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 284, + 505, + 339 + ], + "lines": [ + { + "bbox": [ + 106, + 283, + 504, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 504, + 295 + ], + "score": 1.0, + "content": "Table 4 compares the performance of decentralized training across different object detection models", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 295, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 505, + 307 + ], + "score": 1.0, + "content": "and datasets. Similar to the above experiment, it is observed that both graphs enable decentralized", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 304, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 317 + ], + "score": 1.0, + "content": "algorithms with almost the same performance in each scenario. This again illustrates the value of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "one-peer exponential graph in deep learning tasks - it endows decentralized deep training with both", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 326, + 286, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 286, + 341 + ], + "score": 1.0, + "content": "fast training speed and satisfactory accuracy.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 283, + 506, + 341 + ] + }, + { + "type": "table", + "bbox": [ + 107, + 363, + 504, + 446 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 132, + 352, + 474, + 362 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 133, + 351, + 475, + 364 + ], + "spans": [ + { + "bbox": [ + 133, + 351, + 475, + 364 + ], + "score": 1.0, + "content": "Table 4: Comparison of different methods and models on PASCAL VOC and COCO datasets.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "table_body", + "bbox": [ + 107, + 363, + 504, + 446 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 107, + 363, + 504, + 446 + ], + "spans": [ + { + "bbox": [ + 107, + 363, + 504, + 446 + ], + "score": 0.984, + "html": "
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RETINANETFASTER RCNNRETINANETFASTER RCNN
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PARALLEL SGD79.0-80.3=36.2=37.2=
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With these results, we reveal that one-peer exponential graphs endow DmSGD with the same", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 511, + 504, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 504, + 523 + ], + "score": 1.0, + "content": "convergence rate as their static counterpart. We also establish that exponential graphs achieve nearly", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "minimum per-iteration communication time and transient iteration complexity simultaneously when", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 532, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 107, + 535, + 114, + 542 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 532, + 505, + 545 + ], + "score": 1.0, + "content": "is large. All conclusions are thoroughly examined with industrial-standard benchmarks. As the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "future work, we will investigate symmetric time-varying graphs that can perform as well as one-peer", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 554, + 447, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 317, + 567 + ], + "score": 1.0, + "content": "exponential graph. Symmetric graphs are critical for", + "type": "text" + }, + { + "bbox": [ + 318, + 554, + 330, + 564 + ], + "score": 0.87, + "content": "\\mathrm { D } ^ { \\beth }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 554, + 447, + 567 + ], + "score": 1.0, + "content": "and DecentLaM algorithms.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 477, + 506, + 567 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 583, + 207, + 596 + ], + "lines": [ + { + "bbox": [ + 105, + 581, + 208, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 208, + 599 + ], + "score": 1.0, + "content": "Acknowledgements", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 609, + 504, + 631 + ], + "lines": [ + { + "bbox": [ + 105, + 607, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 506, + 622 + ], + "score": 1.0, + "content": "The authors are grateful to Dr. Sai Praneeth Karimireddy from EPFL for the helpful discussions on", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 619, + 192, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 192, + 632 + ], + "score": 1.0, + "content": "the hypercube graph.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 607, + 506, + 632 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 648, + 163, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 646, + 165, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 165, + 663 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "list", + "bbox": [ + 111, + 669, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 110, + 668, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 110, + 668, + 506, + 682 + ], + "score": 1.0, + "content": "[1] Sulaiman A Alghunaim and Kun Yuan. A unified and refined convergence analysis for non-convex", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 126, + 679, + 356, + 691 + ], + "spans": [ + { + "bbox": [ + 126, + 679, + 356, + 691 + ], + "score": 1.0, + "content": "decentralized learning. arXiv preprint arXiv:2110.09993, 2021.", + "type": "text" + } + ], + "index": 37, + "is_list_end_line": true + }, + { + "bbox": [ + 111, + 691, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 111, + 691, + 505, + 703 + ], + "score": 1.0, + "content": "[2] Dan Alistarh, Demjan Grubic, Jerry Li, Ryota Tomioka, and Milan Vojnovic. Qsgd: Communication-", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 125, + 700, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 125, + 700, + 505, + 714 + ], + "score": 1.0, + "content": "efficient sgd via gradient quantization and encoding. In Advances in Neural Information Processing", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 125, + 712, + 249, + 722 + ], + "spans": [ + { + "bbox": [ + 125, + 712, + 249, + 722 + ], + "score": 1.0, + "content": "Systems, pages 1709–1720, 2017.", + "type": "text" + } + ], + "index": 40, + "is_list_end_line": true + }, + { + "bbox": [ + 212, + 88, + 221, + 96 + ], + "spans": [ + { + "bbox": [ + 212, + 88, + 221, + 96 + ], + "score": 1.0, + "content": "ng.", + "type": "text", + "cross_page": true + } + ], + "index": 0, + "is_list_end_line": true + }, + { + "bbox": [ + 125, + 93, + 150, + 105 + ], + "spans": [ + { + "bbox": [ + 125, + 93, + 150, + 105 + ], + "score": 1.0, + "content": "2019.", + "type": "text", + "cross_page": true + } + ], + "index": 1, + "is_list_end_line": true + }, + { + "bbox": [ + 110, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 110, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "[4] Aditya Balu, Zhanhong Jiang, Sin Yong Tan, Chinmay Hedge, Young M Lee, and Soumik Sarkar. 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TopologyRingGridRand-GraphRand-MatchStatic Exp One-peer Exp
Per-iter Comm.(2)2(4)（）(1)Ω(log2(n))(1)
Trans. Iters.Ω(n7)Ω(n5)(n3)1Ω(n³ log2(n))Ω(n³log²2(n)）
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NODES TOPOLOGY4(4x8 GPUs)8(8x8 GPUs)16(16x8 GPUs)32(32x8 GPUs)
ACC.TIMEACC.TIMEACC.TIMEACC.TIME
RING76.13 ±0.02311.676.07 ±0.0136.576.08 ±0.0263.375.58 ±0.0211.8
GRID76.08 ±0.00711.676.35 ±0.0376.775.88 ±0.0113.475.76 ±0.0222.0
BI-RAND.MATCH.75.96 ±0.03211.176.26 ±0.0275.776.07 ±0.0122.875.83 ±0.0291.5
RANDOM GRAPH75.97 ±0.02811.576.01 ±0.0337.176.18 ±0.0086.776.24 ±0.0184.7
STATIC EXP.76.21 ±0.02811.676.32 ±0.0376.976.30 ±0.0074.176.28 ±0.0202.5
ONE-PEER EXP.76.28 ±0.06311.176.47 ±0.0355.776.42 ±0.0302.876.30 ±0.0621.5
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DATASET MODEL TOPOLOGYPASCAL VOCCoCo
RETINANETFASTER RCNNRETINANETFASTER RCNN
STATICONE-PEERSTATICONE-PEERSTATICONE-PEERSTATICONE-PEER
PARALLEL SGD79.0-80.3=36.2=37.2=
VANILLA DMSGD79.079.180.780.536.336.137.337.2
DMSGD79.179.080.480.536.436.437.137.0
QG-DMSGD79.279.180.880.436.336.237.237.1
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MODEL TOPOLOGYREsNET-50MOBILENET-V2EFFICIENTNET
STATICONE-PEERSTATICONE-PEERSTATICONE-PEER
PARALLEL SGD76.21 (7.0)70.12 (5.8)77.63 (9.0)
VANILLA DMSGD76.14 (6.6)76.06 (5.5)69.98 (5.6)69.81 (4.6)77.62 (8.4)77.48 (6.9)
DMSGD76.50 (6.9)76.52(5.7)69.62 (5.7)69.98 (4.8)77.44 (8.7)77.51 (7.1)
QG-DMSGD76.43 (6.6)76.35(5.6)69.83 (5.6)69.81 (4.6)77.60 (8.4)77.72 (6.9)
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