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+# DEEP LEARNING FOR PHYSICAL PROCESSES: INCORPORATING PRIOR SCIENTIFIC KNOWLEDGE
+
+Emmanuel de Bezenac ´ ∗, Arthur Pajot ∗, Patrick Gallinari emmanuel.de-bezenac, arthur.pajot, patrick.gallinari @lip6.fr Sorbonne Universites, UMR 7606, LIP6, F-75005 Paris, France ´
+
+# ABSTRACT
+
+We consider the use of Deep Learning methods for modeling complex phenomena like those occurring in natural physical processes. With the large amount of data gathered on these phenomena the data intensive paradigm could begin to challenge more traditional approaches elaborated over the years in fields like maths or physics. However, despite considerable successes in a variety of application domains, the machine learning field is not yet ready to handle the level of complexity required by such problems. Using an example application, namely Sea Surface Temperature Prediction, we show how general background knowledge gained from the physics could be used as a guideline for designing efficient Deep Learning models. In order to motivate the approach and to assess its generality we demonstrate a formal link between the solution of a class of differential equations underlying a large family of physical phenomena and the proposed model. Experiments and comparison with series of baselines including a state of the art numerical approach is then provided.
+
+# 1 INTRODUCTION
+
+A physical process is a sustained phenomenon marked by gradual changes through a series of states occurring in the physical world. Physicists and environmental scientists attempt to model these processes in a principled way through analytic descriptions of the scientist’s prior knowledge of the underlying processes. Conservation laws, physical principles or phenomenological behaviors are generally formalized using differential equations. This physical paradigm has been, and still is the main framework for modeling complex natural phenomena like e.g. those involved in climate. With the availability of very large datasets captured via different types of sensors, this physical modeling paradigm is being challenged by the statistical Machine Learning (ML) paradigm, which offers a prior-agnostic approach. However, despite impressive successes in a variety of domains as demonstrated by the deployment of Deep Learning methods in fields such as vision, language, speech, etc, the statistical approach is not yet ready to challenge the physical paradigm for modeling complex natural phenomena, or at least it has not demonstrated how to. This is a new challenge for this field and an emerging research direction in the ML community. We believe that knowledge and techniques accumulated for modeling physical processes in well developed fields such as maths or physics could be useful as a guideline to design efficient learning systems and conversely, that the ML paradigm could open new directions for modeling such complex phenomena. In this paper we then raise two issues: 1) are modern ML techniques ready to be used to model complex physical phenomena, and 2) how general knowledge gained from the physical modeling paradigm could help designing efficient ML models.
+
+In this work, we tackle these questions by considering a specific physical modeling problem: forecasting sea surface temperature (SST). SST plays a significant role in analyzing and assessing the dynamics of weather and other biological systems. Accurately modeling and predicting such dynamics is critical in various applications such as weather forecasting, or planning of coastal activities. Since 1982, weather satellites have made huge quantities of very high resolution SST data available Bernstein (1982). Standard physical methods for forecasting SST use coupled ocean-atmosphere prediction systems, based on the Navier Stokes equations. These models rely on multiple physical hypotheses and do not optimally exploit the information available in the data. On the other hand, despite the availability of large amounts of data, direct applications of ML methods do not lead to competitive state of the art results, as will be seen in section 4.
+
+We use SST as a typical and representative problem of intermediate complexity. Our goal is not to offer one more solution to this problem, but to use it as an illustration for advancing on the challenges mentioned above. The way we handle this problem is general enough to be transfered to a more general class of transport problems.
+
+We propose a Deep Neural Network (NN) model, inspired from general physical motivations which offers a new approach for solving this family of problems. We first motivate our approach by introducing in section 2 the solution of a general class of partial differential equations (PDE) which is a core component of a large family of transport and propagation phenomena in physics. This general solution is used as a guideline for introducing a Deep Learning architecture for SST prediction which is described in section 3. Experiments and comparison with a series of baselines is introduced in section 4. A review of related work is finally presented in section 5.
+
+The main contributions of this work are: 1) an example showing how to incorporate general physical background for designing a NN aimed at modeling a relatively complex prediction task. We believe the approach to be general enough to be used for a family of transport problems obeying general advection-diffusion principles. 2) formal links between our model’s prediction and the solution of a general advection diffusion PDE 3) an unsupervised model for estimating motion fields, given a sequence of images. 4) a proof, on a relatively complex physical modeling problem, that full data intensive approaches based on deep architectures can be competitive with state of the art dedicated numerical methods.
+
+# 2 PHYSICAL MOTIVATION
+
+Forecasting consists in predicting future temperature maps using past records. Temperatures are acquired via satellite imagery. If we focus on a specific area, we can formulate the problem as prediction of future temperature images of this area using past images. The classical approach to forecasting SST consists in using numerical models representing prior knowledge on the conservation laws and physical principles, which take the form of PDEs. These models are then coupled with SST data using assimilation techniques in order to adjust to initial conditions. It is then integrated forward in time to predict SST evolution. For the sea surface, temperature variation is related to a fluid transport problem. In fluids, transport occurs through the combination of two principles: advection and diffusion. During advection, a fluid transports some conserved quantity $I$ (the temperature for SST) or material via bulk motion, i.e.for small variations $\Delta x , \Delta t$ , conservation is expressed as:
+
+$$
+I ( x , t ) = I ( x + \Delta x , t + \Delta t )
+$$
+
+applying a first order approximation of the right hand side and moving the resulting terms to the left hand side of equation 1, we obtain the advection equation, known also as the Brightness Constancy Constraint Equation (BCCE):
+
+$$
+\frac { \partial I } { \partial t } + ( w . \nabla ) I = 0
+$$
+
+where $\nabla$ denotes the gradient operator, and $w$ the motion vector $\textstyle { \frac { \Delta x } { \Delta t } }$ . This equation describes the temporal evolution of quantity $I$ for displacement $w$ . Note that this equation is also the basis for many variational methods for Optical Flow. To retrieve the motion, numerical schemes are applied, and the resulting system of equations, along with a an additional constraint on $w$ is solved for $w$ . This motion can then be used to forecast the future value of $I$ .
+
+$$
+\frac { \partial I } { \partial t } + ( w . \nabla ) I = D \nabla ^ { 2 } I
+$$
+
+$\nabla ^ { 2 }$ denotes the Laplacian operator and $D$ the diffusion coefficient. Note that when $D \to 0$ , we recover the advection equation 2.
+
+This equation describes a large family of physical processes (e.g. fluid dynamics, heat conduction, wind dynamics, etc). Let us now state a result, characterizing the general solutions of equation 3.
+
+Theorem 1. 1 For any initial condition $I _ { 0 } \in L ^ { 1 } ( \mathbb { R } ^ { 2 } )$ with $I _ { 0 } ( \pm \infty ) = 0$ , there exists a unique global solution $I ( x , t )$ to the advection-diffusion equation 3:
+
+$$
+I ( x , t ) = \int _ { \mathbb { R } ^ { 2 } } k ( x - w , y ) I _ { 0 } ( y ) d y
+$$
+
+where $\begin{array} { r } { k ( u , v ) = \frac { 1 } { 4 \pi D t } e ^ { - \frac { 1 } { 4 D t } \left. u - v \right. ^ { 2 } } } \end{array}$ is a radial basis function kernel, or alternatively, a 2 dimensional Gaussian probability density with mean $u$ and variance $2 D t$ .
+
+Equation 4 provides a principled way to calculate $I ( x , t )$ for any time $t$ using the initial condition $I _ { 0 }$ , provided the motion $w$ and the diffusion coefficient $D$ are known. It states that quantity $I ( x , t )$ can be computed from the initial condition $I _ { 0 }$ via a convolution with a Gaussian probability density function. In other words, if $I$ was used as a model for the evolution of the SST and the surface’s underlying advecting mechanisms were known, future surface temperatures could be predicted from previous ones. Unfortunately neither the initial conditions, the motion vector nor the diffusion coefficient are known. They have to be estimated from the data. Inspired from the general form of solution 4, we propose a ML method, expressed as a Deep Learning architecture for predicting SST. This model will learn to predict a motion field analog to the $w$ in equation 4, which will be used to predict future images.
+
+# 3 MODEL
+
+![](images/723d416121047ed73a5ae5137105467c9d588884d82dff94268dd9f826d12a3b.jpg)  
+Figure 1: Motion is estimated from the input images $( I _ { t - k - 1 : t } )$ with a convolutional neural network (top left CDNN component). A warping scheme then displaces the last input image along this motion estimate to produce the future image. The error signal is calculated using the target future image $I _ { t + 1 }$ , and is backprogated through the warping scheme to correct the CDNN. To produce multiple time-step forecasts, the predicted image is fed back in the CDNN in an autoregressive manner.
+
+The model consists of two main components, as illustrated in Figure 1. One predicts the motion field from a sequence of past input images, this is convolutional-deconvolutional (CDNN) module on the top of figure 1, and the other warps the last input image using the motion field from the first component, in order to produce an image forecast. The entire system is trained in an end-to-end fashion, using only the supervision from the target SST image. By doing so, we are able to produce an interpretable latent state which corresponds in our problem to the velocity field advecting the temperatures.
+
+Let us first introduce some notations. Each SST image $I _ { t }$ is acquired on a bounded rectangle of $\mathbb { R } ^ { 2 }$ , named $\Omega$ . We denote $I _ { t } ( x )$ and $w _ { t } ( x )$ the sea surface temperature and the two-dimensional motion vector at time $t \in \mathbb { R }$ at position $x \in \Omega$ . $I _ { t } : \Omega \to \mathbb { R }$ and $\dot { w } _ { t } : \Omega \to \mathbb { R } ^ { 2 }$ represent the temperatures and the motion vector field at time $t$ defined on $\Omega$ . When time $t$ and position $x$ are available from the context, we will drop the subscript $t$ from $w _ { t } ( x )$ and $I _ { t } ( x )$ , along with $x$ for clarity. Given a sequence of $k$ consecutive SST images $\{ I _ { t - k - 1 } , . . . , I _ { t } \}$ (also denoted as $I _ { t - k - 1 : t } )$ , our goal is to predict the next image $I _ { t + 1 }$ .
+
+![](images/fdfbc089060a5d4b341bb93d8bfc0394e30c6e4e3b5ae9494dd7e8d78b641924.jpg)  
+Figure 2: Architecture of the CDNN motion estimation component. For the estimated motion flow $\hat { w } _ { t }$ , colours correspond to the flow orientation and colour intensity to the flow intensity
+
+As indicated in section 2, provided the underlying motion field is known, one can compute SST forecasts. Let us introduce how the motion field is estimated in our architecture. We are looking for a vector field $w$ which when applied to the geometric space $\Omega$ renders $I _ { t }$ close to $I _ { t + 1 }$ , i.e. $I _ { t + 1 } ( \bar { x } ) \simeq$ $I _ { t } ( x + w ( x ) )$ , $\forall x \in \Omega$ . If $I _ { t + 1 }$ were known, we could estimate $w$ , but $I _ { t + 1 }$ is precisely what we are looking for. Instead, we choose to use a convolutional-deconvolutional architecture to predict a motion vector for each pixel. As shown in figure 2, this network makes use of skip connections He et al. (2015), allowing fine grained information from the first layers to flow through in a more direct manner. We use a Batch Normalization layer between each convolution, and Leaky $R e L U$ (with parameter value set to 0.1) non-linearities between convolutions and transposed-convolutions. We used $k = 4$ concatenated images $I _ { t - k - 1 : t }$ as input for training. We have selected this architecture experimentally, testing different state-of-the-art convolution-deconvolution network architectures. Let $\hat { w } \in \mathbb { R } ^ { 2 \times \mathbf { \bar { W } } \times H }$ be the output of the network, where $W$ and $H$ are respectively the width and height of the images, and $\mathit { \Omega } ^ { , }$ corresponds to the two components of the flow at each point of the motion field.
+
+Generally, and this is the case for our problem, we do not have a direct supervision on the motion vector field, since the target motion is usually not available. Using the warping scheme introduced below, we will nonetheless be able to (weakly) supervise $w$ , based on the discrepancy of the warped version of the $I _ { t }$ image and the target image $I _ { t + 1 }$ .
+
+# 3.2 WARPING SCHEME
+
+![](images/f26b7c0fe8bb8e32a7af7bd237402c2fba74592354a184042a168128f6b08f8e.jpg)  
+Figure 3: Warping scheme. To calculate the pixel value for time $t + 1$ at position $x$ , we first compute its previous position at time $t$ , i.e. $x - w$ . We then center a Gaussian in that position in order to obtain a weight value for each pixel in $I _ { t }$ based on its distance with $x - w$ , and compute a weighted average of the pixel values of $I _ { t }$ . This weighted average will correspond to the new pixel value at $x$ in $I _ { t + 1 }$ .
+
+Discretizing the solution of the advection-diffusion equation in section 2 by replacing the integral with a sum, and setting image $I _ { t }$ as the initial condition, we obtain a method to calculate the future image, based on the motion field estimate $\hat { w }$ . The latter is used as a warping scheme:
+
+$$
+\hat { I } _ { t + 1 } ( x ) = \sum _ { y \in \Omega } k ( x - \hat { w } ( x ) , y ) I _ { t } ( y )
+$$
+
+where $\begin{array} { r } { k ( x - \underset { . } { \hat { w } } , y ) = \underset { . } { \frac { 1 } { 4 \pi D \Delta t } } e ^ { - \frac { 1 } { 4 D \Delta t } \| x - \hat { w } - y \| ^ { 2 } } } \end{array}$ is a radial basis function kernel, as in equation 4, parameterized by the diffusion coefficient $D$ and the time step value $\Delta t$ between $t$ and $t + 1$ and $\hat { w }$ is the estimated value of the vector flow $w$ . To calculate the temperature for time $t + 1$ at position $x$ , we compute the scalar product between $k ( x - { \hat { w } } , . )$ , a Gaussian centered in $x - \hat { w }$ , and the previous image $I _ { t }$ . Simply put, it is a weighted average of the temperatures $I _ { t }$ , where the weight values are larger when the pixel’s positions that are closer to $x - \hat { w }$ . Informally, $x - \hat { w }$ corresponds to the pixel’s previous position at time $t$ . See figure 3.
+
+As seen by the relation with the solution of the advection-diffusion equation, the proposed warping mechanism is then clearly adapted to the modeling of phenomena governed by the advectiondiffusion equation. SST forecasting is a particular case, but the proposed scheme can be used for any problems in which advection and diffusion are occurring. Moreover, this warping scheme is entirely differentiable, allowing backpropagation of the error signal to the motion fireld estimating module.
+
+This warping mechanism has been inspired by the Spatial Transformer Network (STN) Jaderberg et al. (2015), originally designed to be incorporated as a layer in a convolutional neural network architecture in order to gain invariance under geometric transformations. Using the notations in Jaderberg et al. (2015), when the inverse geometric transformation $\mathcal { T } _ { \theta }$ of the grid generator step is set to $\mathcal { T } _ { \theta } ( x ) = x - \hat { w } ( x )$ , and the kernels $k ( . ; \Phi _ { x } )$ and $k ( . ; \Phi _ { y } )$ in the sampling step are radial basis function kernels, we recover our warping scheme. The latter can be seen as a specific case of the STN, without the localization step. This result theoretically grounds the use of the STN for Optical Flow in many recent articles Zhu et al. (2017), Yu et al. (2016), Patraucean et al. (2015), Finn et al. (2016): in equation 3, when $D \to 0$ , we recover the brightness constancy constraint equation, used in the latter.
+
+For training, supervision is provided at the output of the warping module. It consists in minimizing the discrepancy between the warped image $\hat { I } _ { t + 1 }$ and the target image $I _ { t + 1 }$ . The loss is measured via a differentiable function and the gradient is back propagated through the warping function in order to adjust the parameters of the convolutional-deconvolutional module generating the vector field. This is detailed in the next section.
+
+# 3.3 LOSS FUNCTION
+
+At each iteration, the model aims at forecasting the next observation, given the previous ones. We evaluate the discrepancy between the warped image $\hat { I } _ { t + 1 }$ and the target image $I _ { t + 1 }$ using the Charbonnier penalty function $\rho ( x ) = ( x + \epsilon ) ^ { \frac { 1 } { \alpha } }$ , where $\epsilon$ and $\alpha$ are parameters to be set. Note that with $\epsilon = 0$ and $\alpha \stackrel { \cdot } { = } \frac { 1 } { 2 }$ , we recover the $\ell _ { 2 }$ loss. The Charbonnier penalty function is known to reduce the influence of outliers compared to an $l _ { 2 }$ norm. We have also tested the Laplacian pyramid loss Ling & Okada (2006), where we enforce convolutions of all deconvolutional layers to be close to down-sampled versions of the target image in the Charbonnier penalty sense, but we have observed an overall decrease in generalization performance.
+
+The proposed NN model has been designed according to the intuition gained from general background knowledge of a physical phenomenon, here advection-diffusion equations. Additional prior knowledge – expressed as partial differential equations, or through constraints – can be easily incorporated in our model, by adding penalty terms in the loss function. As the displacement $w$ is explicitly part of our model, one strength of our model is its capacity to apply some regularization term directly on the motion field. In our experiments, we tested the influence of different terms: divergence $\nabla . w _ { t } ( x ) ^ { 2 }$ , magnitude $\| w _ { t } ( x ) \| ^ { 2 }$ and smoothness $\Vert \nabla w _ { t } ( x ) \Vert ^ { 2 }$ .
+
+$$
+L _ { t } = \sum _ { x \in \Omega } \rho ( \hat { I } _ { t + 1 } ( x ) - I _ { t + 1 } ( x ) ) + \lambda _ { \mathrm { d i v } } ( \nabla . w _ { t } ( x ) ) ^ { 2 } + \lambda _ { \mathrm { m a g n } } \left\| w _ { t } ( x ) \right\| ^ { 2 } + \lambda _ { \mathrm { g r a d } } \left\| \nabla w _ { t } ( x ) \right\| ^ { 2 }
+$$
+
+# 4 EXPERIMENTS
+
+# 4.1 DATASET DESCRIPTION
+
+Since 1982, high resolution SST data has been made available by the NOAA6 weather satellite, Bernstein (1982). Dealing directly with these data requires a lot of preprocessing (e.g. some regions are not available due to clouds, hindering temperature acquisition). In order to avoid such complications which are beyond the scope of this work, we used synthetic but realistic SST data of the Atlantic ocean generated by a sophisticated simulation engine: NEMO (Nucleus for European Modeling of the Ocean) engine 2, Madec (2008). NEMO is a state-of-the-art modelling framework of ocean related engines. It is a primitive equation model adapted to the regional and global ocean circulation problems. Historical data is accumulated in the model to generate a synthesized estimate of the states of the system using data analysis, a specific data assimilation scheme, which means that the data does follow the true temperatures. The resulting dataset is constituted of daily temperature acquisitions of 481 by 781 pixels, from 2006-12-28 to 2017-04-05 (3734 acquisitions).
+
+We extract 64 by 64 pixel sized sub-regions as indicated in figure 4.1. We use data from years 2006 to 2015 for training and validation (94743 training examples), and years 2016 to 2017 for testing. We withhold $20 \%$ of the training data for validation, selected uniformly at random at the beginning of each experiment. For the tests we used sub-regions enumerated 17 to 20 in figure 4.1, where the interactions between hot and cold waters make the dynamics interesting to study. All the regions numbered in figure 4.1, from 2006 to 2015 where used for training 3. Each sequence of images used for training or for evaluation corresponds to a specific numbered sub-region. We make the simplifying hypothesis that the data in a single sub-region contains enough information to forecast the future of the sub-region. As the forecast is for a small temporal horizon we can assume that the influence from outside the region is small enough.
+
+![](images/855f62d169f130a18846bdb7f728e625abf257f5ee4968a7803f13618e9171a8.jpg)  
+Figure 4: Sub regions extracted for the dataset. Test regions are regions 17 to 20.
+
+We normalize the daily SST acquisitions of each sub region using the mean and the standard deviation of all the SST data of the sub-region acquired on the same day of the year for all the years in the training set, i.e. the SST acquisition of sub-region 2 on date September 8th 2017 is standardized using the data of all the September 8th available in the dataset. This removes the seasonal component from SST data.
+
+# 4.2 BASELINE COMPARISON
+
+We compare our model with several baselines. Each model is evaluated with a mean square error metric, forecasting images on a horizon of 6 (we forecast from $I _ { t + 1 }$ to $I _ { t + 6 }$ and then average the MSE). The hyperparameters are tuned using the validation set. Neural network based models are run on a Titan $\mathrm { X p }$ GPU, and runtime is given for comparison purpose.
+
+Concerning the constraints on the vector field $w$ (equation 6. the regularization coefficients selected via validation are $\lambda _ { \mathrm { d i v } } = 1$ , $\lambda _ { \mathrm { { m a g n } } } = - 0 . 0 3$ and $\lambda _ { \mathrm { g r a d } } = 0 . 4$ . The coefficient diffusion $D$ was set to 0.45 by cross validation. We also compare the results with the model without any regularization.
+
+Our reference model for forecasting is Ber´ eziat & Herlin (2015), a numerical assimilation model ´ which relies on data assimilation. In Ber´ eziat & Herlin (2015), the ocean’s dynamics are modeled ´ using shallow water equations Vallis (2017) and the initial conditions, along with other terms, are estimated using assimilation techniques Tremolet (2006). This is a state of the art assimilation model ´ for predicting ocean dynamics, here SST.
+
+The other baselines are 1) an autoregressive convolutional-deconvolutional NN (ACNN), with an architecture similar to our CDNN module, but trained to predict the future image directly, without explicitly representing the motion vector field. Each past observation is used as an input channel (the 4 input images used in the experiments are concatenated), and the output is used as new input for multi step forecasting, 2) a ConvLSTM model Shi et al. (2015), which uses convolutional transitions in the inner LSTM module, and 3) the model in Mathieu et al. (2015) which is a multi-scale ACNN trained as a Generative Adversial Network (GAN). We have used a non-official code for Mathieu et al. (2015), which is made available at https://github.com/dyelax/Adversarial_ Video_Generation. For Ber´ eziat & Herlin (2015), the code has been provided by the authors ´ of the paper. We have implemented the ACNN and ConvLSTM models ourselves. The code for our models, along with these baselines will be made available.
+
+# 4.3 QUANTITATIVE RESULTS
+
+<table><tr><td>Model</td><td>Average Score (MSE)</td><td>Average Time</td></tr><tr><td>Numerical model Béréziat &amp; Herlin (2015)</td><td>1.99</td><td>4.8 s</td></tr><tr><td>ConvLSTM Shi et al. (2015)</td><td>5.76</td><td>0.018 s</td></tr><tr><td>ACNN</td><td>15.84</td><td>0.54 s</td></tr><tr><td>GAN Video Generation (Mathieu et al. (2015))</td><td>4.73</td><td>0.096 s</td></tr><tr><td>Proposed model with regularization</td><td>1.42</td><td>0.040 s</td></tr><tr><td>Proposed model without regularization</td><td>2.01</td><td>0.040 s</td></tr></table>
+
+Table 1: Average score and average time on test data. Average score is calculated using the mean square error metric (MSE), time is in seconds. The regularization coefficients for our model have been set using a validation set with $\lambda _ { \mathrm { d i v } } = 1$ , $\lambda _ { \mathrm { { m a g n } } } = - 0 . 0 3$ and $\lambda _ { \mathrm { g r a d } } = 0 . 4$ .
+
+Quantitatively, our model performs well. The MSE score is better than any of the baselines. The closest NN baseline is Mathieu et al. (2015) which regularizes a regression convolutiondeconvolution model with a GAN. The performance is however clearly below the proposed model and it does not allow to easily incorporate prior constraints inspired from the physics of the phenomenon. ACNN is a direct predictor of the image sequence, implemented via a CDNN module identical to the one used in our model. Its performance is poor. Clearly, a straightforward use of prediction models is not adapted to the complexity of the phenomenon. ConvLSTM performs better: as opposed to the ACNN, it seems to be able to capture a dynamic, although not very accurately. Overall, direct prediction models are not able to capture the complex underlying dynamics and they produce blurry sequences of images. The GAN explicitly forces the network output to eliminate the blurring effect and then makes it able to capture short term dynamics. The state of the art numerical model Ber´ eziat & Herlin (2015), performs well but has a slighthly lower performance than our reg- ´ ularized model, although it incorporates more prior constraints. This shows that pure ML models, when conceived adequately and when trained with enough data, can be competitive with state of the art dedicated models. Regularizing the motion vector $w$ notably increases the performance w.r.t. to the unregularized model. The choice of the constraints (divergence, magnitude and smoothness) inspired here by physical background correspond to relevant priors on the dynamics of the model.
+
+![](images/0538e3fec59746d81f644731393a61c8431a5d123666afbf52804482ea01d9d5.jpg)  
+Figure 5: From top to bottom: target, our model prediction, our model flow, numerical assimilation model , ACNN, ConvLSTM. Data correspond to daily temperatures from January 17 to January 23, 2017
+
+As for the running time, the proposed model is extremely fast, being just above the ConvLSTM model of Shi et al. (2015). The running time of Ber´ eziat & Herlin (2015)’s model is not comparable ´ to the others. It was run on a CPU (no GPU code) when all the others were run on Titan $\mathrm { X p }$ GPU. However, an optimization procedure is required to estimate the motion field, and it is clearly slower than the straightforward NN predictions. Moreover, in order to prevent the numerical scheme from diverging, multiple intermediate forecasts are required.
+
+Besides MSE, we need to analyze the prediction samples qualitatively. Figure 4.3 shows the predictions obtained by the different models. On the top row, the ground truth for a sequence of 4 temperature images corresponding to time $t .$ , $t + 1$ , $t + 3$ and $t + 6$ . The second row corresponds to our regularized model prediction at times $t + 1 , t + 3$ and $t + 6$ (time $t$ corresponds to the last input image, it is repeated on each row). The model seems to conserve temperatures. The prediction is close to the target for $t + 1 , t + 3$ and starts to move away at time $t + 6$ . The third row shows the motion flow estimated by the model. Each color in the flow images corresponds to a motion vector. There is clearly a strong evolving dynamic captured for this sequence. Row 4 is the numerical assimilation model of Ber´ eziat & Herlin (2015). It also clearly captures some dynamics and shows ´ interesting patterns, but it tends to diverge when the prediction horizon increases. The ACNN model (row 5) rapidly produces blurry images; it does not preserve the temperatures and does not seem to capture any dynamics. On row 6 are plotted the predictions of the ConvLSTM model. Temperature is not preserved and although a dynamic is captured, it does not correspond to the target. Overall, the proposed model seems to forecast SST quite accurately, while retrieving a coherent motion vector field.
+
+# 5 RELATED WORK
+
+ML for Physical modeling Close to this work is the field of spatio-temporal statistics. In their reference book Cressie & Wikle (2015) also advocate the use of physical background knowledge to build statistical models. They show how the design of statistical models can be inspired from partial differential equations linked to an observed physical phenomenon. They mainly consider auto-regressive models within a hierarchical Bayesian framework. In Raissi et al. (2017), Archambeau et al. (2007) and Alvarez et al. (2011) the author use PDE-inspired gaussian process to model physical process. Even if the methods and the application are different, the motivation and arguments are similar to the ones developed here.
+
+Another interesting research direction is the use of NNs for reducing the complexity of numerical simulation for physical processes. Generally, in these approaches statistical models are used in place of a computational demanding component of the numerical simulation process. For example in the domain of fluid dynamics, Tompson et al. (2017) and Ladicky et al. (2015) propose to use regressors ´ for simulating fluid and smoke animation. Ladicky et al. (2015) use a random forest to compute ´ particle location and Tompson et al. (2017) use a CNN to approximate part of a numerical PDE scheme. In these approaches, ML is only a component of a numerical simulation scheme whereas we aim at modeling the whole physical process via a Deep Learning approach. Farther to our objective, Rudy et al. (2017) make use of a sparse regression method for discovering the governing partial differential equation(s) of a given system by time series measurements in the spatial domain. Other works have suggested using neural networks for physical process forecast, such as Brajard et al. (2017).
+
+Our work is also related to recent developments in computer vision, in the related but distinct fields of video prediction and motion estimation in videos. Our goal and the domain of application are clearly different from video modeling, but since our solution involves predicting a motion field and the next SST image, the solutions share some similarities. Motion estimation and video predictions by deep architectures have motivated a series of work over the last two years. We briefly review them below and outline the differences.
+
+Optical Flow Optical flow consists in retrieving the apparent motion of objects, surfaces, or particles between two consecutive frames of a video. The extracted motion can be used in many areas such as object detection, object tracking, movement detection, robot navigation and visual odometry. In the vision community, this is considered as a problem by itself and several papers are dedicated to this topic. Classical methods rely on the brightness constancy constrain equation (BCCE) (equation 2), derived from the observation that surfaces usually persist over time and hence the intensity value of a small region remains the same despite its position change Sun et al. (2008). Since using BCCE directly leads to complicated optimizing issues, classic approaches – namely differential methods – approximate BCCE using a first order Taylor expansion and develop variational methods.
+
+As an alternative to these methods, Deep Learning models have been recently proposed for estimating the optical flow between 2 images. Fischer et al. (2015) formulate optical flow as a supervised regression problem, using a CNN to predict motion. Ilg et al. (2016) build on this approach and propose to use an ensemble of these CNN architectures. They assess results on par with state of the art methods for optical flow, while maintaining a small computational overhead. The difficulty here is that these methods require a notable quantity of target data, i.e. optical flow images, while because of the complexity of manually annotating flow images, there are only a few small annotated collections available. Fischer et al. (2015) and Ilg et al. (2016) chose to pretrain their model on a synthetic dataset made of computer animations and their associated motion and show that this information transfers well to real videos. Yu et al. (2016) demonstrate that it is possible to predict the optical flow between two input images in an unsupervised way using a CNN and a Spatial Transformer Network. This is however not extensible for prediction as is done in our setting since this requires the two images $I _ { t }$ and $I _ { t + 1 }$ as input while $I _ { t + 1 }$ is not available at inference time for prediction.
+
+# Video prediction
+
+It is only very recently that video prediction emerged as a task in the Deep Learning community. For this task, people are generally interested at predicting accurately the displacement/ emergence/ disappearing of objects in the video. In our application, the goal is clearly different since we are interested into modeling the whole dynamics behind image changes and not at following moving objects. Let us first introduce some methods that perform prediction by computing optical flow or a similar transformation. Both Patraucean et al. (2015) and Finn et al. (2016) use some form of motion flow estimation. For next frame prediction Patraucean et al. (2015) introduce a STN module at the hidden layer of a LSTM in order do estimate a motion field in this latent space. The resulting image is then decoded in the original image space for prediction. This approach clearly does not allow introducing prior knowledge on the field vector as this has been done in our work. Finn et al. (2016) learn affine transformations on image parts in order to predict object displacement and Van Amersfoort et al. (2017) proposed a similar model.
+
+Let us now consider models that directly attempt to predict the next frame without estimating a motion field. As shown in the experimental section, plain application of autoregressive models produces blurred images. Mathieu et al. (2015), one of our baseline proposed to use different loss functions and a GAN regularization of a CDNN predictor which led to sharper and higher quality predictions. Significant improvements have been obtained with the Video Pixel Network of Kalchbrenner et al. (2016), which is a sophisticated architecture composed of resolution preserving CNN encoders, LSTM and PixelCNN decoders which form a conditional Spatio-temporal video autoencoder with differentiable memory. This model is probably state of the art today for video prediction, They reach a high accuracy on moving MNIST and good performance on a robot video dataset. A drawback is the complexity of the model and the number of parameters: they are using respectively $2 0 \bf { M }$ and $1 \textbf { M }$ frames on these two datasets. We did not test this model since up to our knowledge no code was available.
+
+# 6 CONCLUSION AND FUTURE WORK
+
+The evolution in time of the proposed model is deterministic. Predicting future observations should also deal with the inherent ambiguity and lack of information for the prediction task. A natural future direction would be to incorporate uncertainty in the model’s evolution in the proposed framework. We can extend the proposed model by incorporating a stochastic latent variable in the flow field generation process. A promising direction is the development of generative models which has become popular in Deep Learning, leading to different families of innovative models. For example, the Stochastic Gradient Variational Bayes algorithm (SGVB) Kingma & Welling (2014) provides a framework for learning stochastic latent variables with deep neural networks, and has recently been used by some authors to model time series Karl et al. (2016); Chung et al. (2015); Krishnan et al. (2015). A recent work where both spatial and temporal information are considered is Walker et al. (2016) who model pixel trajectories in a video. As a follow up of our work, we plan to consider such extensions in the future.
+
+The data intensive paradigm offers alternative directions to the classical physical approaches for modeling complex natural processes. Our belief is that cross fertilization of both paradigms is essential for pushing further the frontier of complex data modeling. By using as an example application a relatively complex problem concerning ocean dynamics, we proposed a principled way to design Deep Learning models using inspiration from the physics. The proposed approach can be easily generalized to a class of problems for which the underlying dynamics follow advection-diffusion principles. We have compared the proposed approach to a series of baselines. It is able to reach performance comparable to a state of the art numerical model and clearly outperform alternative NN models used as baselines.
+
+# ACKNOWLEDGMENTS
+
+This work was partially funded by ANR project LOCUST - ANR-15-CE23-0027 and by CLEAR - Center for LEArning & data Retrieval - joint lab. With Thales (www.thalesgroup.com).
+
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+Jonathan Tompson, Kristofer Schlachter, Pablo Sprechmann, and Ken Perlin. Accelerating Eulerian fluid simulation with convolutional networks. In Doina Precup and Yee Whye Teh (eds.), Proceedings of the 34th International Conference on Machine Learning, volume 70 of Proceedings of Machine Learning Research, pp. 3424–3433, International Convention Centre, Sydney, Australia, 06–11 Aug 2017. PMLR. URL http://proceedings.mlr.press/v70/ tompson17a.html.
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+Geoffrey K Vallis. Atmospheric and oceanic fluid dynamics. Cambridge University Press, 2017.
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+Joost Van Amersfoort, Anitha Kannan, Marc’Aurelio Ranzato, Arthur Szlam, Du Tran, and Soumith Chintala. Transformation-Based Models of Video Sequences. In CoRR abs/1701.08435 (2017), pp. 1–11, 2017. URL http://arxiv.org/abs/1701.08435.
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+Jason J. Yu, Adam W. Harley, and Konstantinos G. Derpanis. Back to Basics: Unsupervised Learning of Optical Flow via Brightness Constancy and Motion Smoothness, pp. 3–10. Springer International Publishing, Cham, 2016. ISBN 978-3-319-49409-8. doi: 10.1007/978-3-319-49409-8 1. URL https://doi.org/10.1007/978-3-319-49409-8_1.
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+
+# A PROOF OF THE THEOREM IN SECTION 2 1
+
+Proof. In the following, bold $\mathbf { x }$ and $\mathbf { y }$ will denote vectors of $\mathbb { R } ^ { 2 }$ , while $x$ and $y$ will correspond to the first and second components of $\mathbf { x }$ , respectively. Analogously, $u$ and $v$ will correspond to the components of $w$ . The 2D Fourier Transformation $\mathcal { F }$ of $f : \bar { \mathbb { R } } ^ { 2 }  \bar { \mathbb { R } }$ is defined as
+
+$$
+\begin{array} { l } { \displaystyle \mathcal { F } ( f ) = \int _ { \mathbb { R } ^ { 2 } } f ( \mathbf { x } ) e ^ { - i < \xi , \mathbf { x } > } d \mathbf { x } } \\ { \displaystyle \ = \int _ { \mathbb { R } } \int _ { \mathbb { R } } f ( x , y ) e ^ { - i x \xi _ { 1 } - i y \xi _ { 2 } } d x d y } \end{array}
+$$
+
+We apply the Fourier Transform $\mathcal { F }$ to both sides of 3. As consequence of the linearity of the Fourier transform, we can calculate decompose the Fourier transform of the left hand side in the sum of the transforms of each term. We have three terms: $\begin{array} { r } { \frac { \partial I } { \partial t } , ( w . \nabla ) I } \end{array}$ and $- D \nabla ^ { 2 } I$ .
+
+$$
+\begin{array} { l } { \displaystyle \mathcal { F } ( \frac { \partial I } { \partial t } ) = \int _ { \mathbb { R } ^ { 2 } } \frac { \partial I } { \partial t } e ^ { - i < \mathbf { x } , \xi > } d \mathbf { x } } \\ { \displaystyle \ = \int _ { \mathbb { R } ^ { 2 } } \frac { \partial } { \partial t } ( I e ^ { - i < \mathbf { x } , \xi > } ) d \mathbf { x } } \\ { \displaystyle \ = \frac { \partial } { \partial t } \int _ { \mathbb { R } ^ { 2 } } I e ^ { - i < \mathbf { x } , \xi > } d \mathbf { x } } \\ { \displaystyle \ = \frac { \partial \mathcal { F } ( I ) } { \partial t } } \end{array}
+$$
+
+$$
+\begin{array} { r l } { F ( ( w , \nabla ) I ) = \displaystyle \int _ { \mathbb { R } ^ { 2 } } ( w , \nabla ) I e ^ { - i \omega \cdot \xi } d x } \\ { = \displaystyle \int _ { \mathbb { R } } \int _ { \mathbb { R } } ( w \frac { \partial I } { \partial x } + v \frac { \partial I } { \partial y } ) e ^ { - i x \xi _ { 1 } - \mathrm { i } y \xi _ { 2 } } d x d y } \\ { = \displaystyle \mu \int _ { \mathbb { R } } e ^ { - i \psi \cdot \xi } \int _ { \mathbb { R } } \frac { \partial I } { \partial x } e ^ { - i \xi _ { 1 } } d x d y + v \displaystyle \int _ { \mathbb { R } } e ^ { - i \psi \xi _ { 1 } } \int _ { \mathbb { R } } \frac { \partial I } { \partial y } e ^ { - i y \xi _ { 2 } } d y d x } \\ { = \displaystyle i \xi _ { 1 } w \int _ { \mathbb { R } } e ^ { - i \psi \xi _ { 2 } } \int _ { x } [ e ^ { - i \psi \xi _ { 1 } } d x d y + i \xi _ { 2 } v \int _ { \mathbb { R } } e ^ { - i \psi \xi _ { 1 } } \int _ { \mathbb { R } } \frac { \partial I } { \partial y } e ^ { - i y \xi _ { 2 } } d y d x  } \\ {  = \displaystyle \tilde { U } _ { \mathbb { R } } \int _ { \mathbb { R } } \int _ { \mathbb { R } } e ^ { - i \psi \xi _ { 1 } - \mathrm { i } \psi _ { \xi } } d x d y + v \xi _ { 2 } v \int _ { \mathbb { R } } \int _ { \mathbb { R } } I _ { \mathbb { R } } e ^ { - i \omega \xi _ { 1 } - y \psi } d x d y  } \\ { =  ( \xi _ { 1 } w + i \xi _ { 2 } v ) \int _ { \mathbb { R } } \int _ { \mathbb { R } } \int _ { \mathbb { R } } \int _ { \mathbb { R } } - i \omega \xi _ { 1 } - i \psi \xi _ { 2 } d x d y } \\ { = i \ < \xi , w > F ( I ) } \end{array}
+$$
+
+$$
+\begin{array} { r l } { \mathcal { F } ( - D \nabla ^ { 2 } I ) = - \int _ { \mathbb { R } ^ { 2 } } D \nabla ^ { 2 } I c ^ { - i \nu \xi } s c ^ { - \alpha } k s ^ { \zeta } d x } \\ { = - \int _ { \mathbb { R } ^ { 2 } } \int _ { \mathbb { R } ^ { 0 } } D \big ( \hat { \partial } \hat { \partial } ^ { 2 } I + \hat { \partial } \hat { \partial } ^ { 2 } I \big ) e ^ { - i \alpha \zeta _ { 1 } - i \psi \xi } d x d y } \\ { } & { = - D \int _ { \mathbb { R } ^ { 2 } } e ^ { - i \nu \xi } \int _ { \mathbb { R } ^ { 2 } } \frac { \partial ^ { 2 } I } { \partial x ^ { 2 } } e ^ { - i \alpha \zeta _ { 1 } } d x d y - D \int _ { \mathbb { R } ^ { \epsilon } } e ^ { - i \alpha \zeta _ { 1 } } \int _ { \mathbb { R } } \frac { \partial ^ { 2 } I } { \partial y ^ { 2 } } e ^ { - i \nu \xi \epsilon } d y d x } \\ { } & { = - ( \delta _ { 1 } ) ^ { 2 } D \int _ { \mathbb { R } ^ { 2 } } e ^ { - i \nu \zeta _ { 1 } } \int _ { \mathbb { R } } { F } ^ { - i \nu \zeta _ { 1 } } d x d y d y - ( \delta _ { 2 } ) ^ { 2 } D \int _ { \mathbb { R } ^ { \epsilon } } e ^ { - i \alpha \zeta _ { 1 } } \int _ { \mathbb { R } } { F } ^ { - i \nu \zeta _ { 2 } } d y d x } \\ { } & { = D \mathfrak { L } _ { 1 } ^ { 2 } \int _ { \mathbb { R } ^ { \epsilon } } e ^ { - i \alpha \zeta _ { 2 } } \int _ { \mathbb { R } } { F } ^ { - i \nu \zeta _ { 2 } } d x d y + D \mathfrak { L } _ { 2 } ^ { 2 } \int _ { \mathbb { R } ^ { \epsilon } } e ^ { - i \kappa \zeta _ { 1 } } \int _ { \mathbb { R } } { F } ^ { i \nu \zeta - i \psi \zeta _ { 2 } } d y d x } \\ { } & { = D \mathfrak { L } _ { 2 } ^ { 3 } \int _ { \mathbb { R } ^ { 2 } } \int _ { \mathbb { R } } { F } ^ { - i \nu \zeta _ { 2 } } d x d y + D \mathfrak { L } _ { 2 } ^ { 2 } \int _ { \mathbb { R } } { F } ^ { - i \nu \zeta _ { 1 } } d x d y d x } \\ { } &  = D \mathfrak { L } _ { 1 } ^ { 3 } \int _  \mathbb  \end{array}
+$$
+
+Regrouping all three previously calculated terms, we obtain
+
+$$
+\frac { \partial \mathcal { F } ( I ) } { \partial t } + ( i < \xi , w > + D \left\| \xi \right\| ^ { 2 } ) \mathcal { F } ( I ) = 0
+$$
+
+This is a first order ordinary differential equation of the form $f ^ { \prime } ( t ) + a f ( t ) = 0$ , which admits a known solution $f ( t ) = f ( 0 ) { e ^ { - a t } }$ . Thus, the solution of 11 is
+
+$$
+\begin{array} { r } { \mathcal { F } ( I ) = \mathcal { F } ( I ) _ { 0 } e ^ { - ( i < \xi , w > + D \| \xi \| ^ { 2 } ) t } } \\ { = \mathcal { F } ( I ) _ { 0 } e ^ { - i < \xi , w > t } e ^ { - D t \| \xi \| ^ { 2 } } } \end{array}
+$$
+
+where $\mathcal { F } ( I ) _ { 0 }$ denotes the initial condition of the advection diffusion equation in the frequency domain. In order to obtain a solution of 3 in the spatial domain, we calculate the inverse Fourier Transform ${ \mathcal { F } } ^ { - 1 }$ of 12. The multiplication of two functions in the frequency domain is equivalent to their convolution in the spatial domain, i.e. ${ \mathcal { F } } ( f * g ) = { \mathcal { F } } ( f ) { \mathcal { F } } ( g )$ . Hence, the inverse of both terms $\mathcal { F } ( I ) _ { 0 } e ^ { - i < \xi , w > t }$ and $e ^ { - D t \| \boldsymbol { \xi } \| ^ { 2 } }$ can be calculated separately:
+
+Multiplication by a complex exponential in the frequency domain is equivalent to a shift in the spatial domain : $\bar { e } ^ { - i < \xi , w \bar { > } } \mathcal { F } ( f ( \bar { \mathbf { x } } ) ) = \mathcal { F } ( f ( \mathbf { x } - w ) )$ , for $\boldsymbol { v } \in \mathbb { R } ^ { 2 }$ . Thus, for the first term,
+
+$$
+\mathcal { F } ^ { - 1 } ( \mathcal { F } ( I ) _ { 0 } e ^ { - ( i < \xi , w > ) t } ) = I _ { 0 } ( \mathbf { x } - w )
+$$
+
+For the second term, we use the fact that the Fourier Transform of a Gaussian function also is a Gaussian function, i.e. $\begin{array} { r } { \mathcal { F } \big ( \frac { 1 } { 2 \pi \sigma ^ { 2 } } e ^ { - \frac { 1 } { 2 \sigma ^ { 2 } } \| \mathbf { x } \| ^ { 2 } } \big ) = e ^ { - \frac { 1 } { 2 } \sigma ^ { 2 } \| \boldsymbol { \xi } \| ^ { 2 } } } \end{array}$ . Identifying $\sigma ^ { 2 }$ with $2 D t$ , we have:
+
+$$
+\mathcal { F } ^ { - 1 } ( e ^ { - D t \| \boldsymbol { \xi } \| ^ { 2 } } ) = \frac { 1 } { 4 \pi D t } e ^ { - \frac { 1 } { 4 D t } \| \mathbf { x } \| ^ { 2 } }
+$$
+
+As has been stated above, the solution is a convolution of both previously calculated terms:
+
+$$
+\begin{array} { r } { I ( \mathbf { x } , t ) = \displaystyle \int _ { \mathbb { R } ^ { 2 } } \frac { 1 } { 4 \pi D t } e ^ { - \frac { 1 } { 4 D t } \| \mathbf { y } \| ^ { 2 } } I _ { 0 } ( \mathbf { x } - w - \mathbf { y } ) d y } \\ { = \displaystyle \int _ { \mathbb { R } ^ { 2 } } \frac { 1 } { 4 \pi D t } e ^ { - \frac { 1 } { 4 D t } \| \mathbf { x } - w - \mathbf { y } \| ^ { 2 } } I _ { 0 } ( \mathbf { y } ) d \mathbf { y } } \end{array}
+$$
+
+# B ON THE GENERALIZATION IN SPACE AND TIME
+
+The ability of the model to adapt to other conditions should be evaluated on other regions. This, however, requires a complete study by itself and is beyond the scope of this paper. We, however, present below complementary experiments aimed at assessing the potential of the proposed model for forecasting SST on sequences distant in time and space from the ones used for training.
+
+# B.1 TEMPORAL DIMENSION
+
+In section 4, training has been performed on data from 2006 - 2015 and testing on the period 2016- 2017. In order to provide some indication of the model behavior on more distant time intervals between train and test data, we have performed experiments using the same regions (17 to 20) as in section 4, but using the period 2011 to 2017 for training and period 2006 to 2010 for testing. Figure B.1 shows the MSE curve on this test set, each point corresponding to the mean MSE on predictions performed on 6 days ahead the current date. The most important conclusion is probably that the MSE error remains in the same range for all these years. All the yearly error curve show a clear seasonal phenomenon with a higher prediction error during summer. A similar behavior has been observed when exchanging train and test data.
+
+![](images/528806e279e8e0cafff934b99238f30f79d4eaee57852dbf4a070720a4a6a49c.jpg)  
+Figure 6: Evaluation of our model’s accuracy in time on data from 2006 to 2010 using data from 2011 to 2017 for training. Regions 17 to 20 were used for both periods. Each day, we produce daily forecasts for 6 days ahead and calculate the associated mean square error.
+
+# B.2 SPATIAL DIMENSION
+
+In the experiments, the models have been trained and evaluated on selected regions (numbered 17 to 20 in Figure 4.1), considered as the most interesting for the observed dynamics.
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Test Regions 17&amp;18</td><td rowspan=1 colspan=1>Test Regions 8&amp;9</td></tr><tr><td rowspan=1 colspan=1>Model trained on Regions 17 &amp; 18</td><td rowspan=1 colspan=1>1.43</td><td rowspan=1 colspan=1>1.22</td></tr><tr><td rowspan=1 colspan=1>Model trained on Regions 8&amp;9</td><td rowspan=1 colspan=1>1.90</td><td rowspan=1 colspan=1>1.19</td></tr></table>
+
+Table 2: Evaluation of our model’s spatial generalization ability. We train our model on two distinct regions and calculate the MSE on both regions for each trained model.
+
+We describe below some results providing indications on how the model performs on regions different from the training ones. For these experiments, the model has been trained on regions 17 and 18 in Figure 4.1 and tested on two other regions (regions 8 and 9), and vice versa (trained on 8 and 9 and tested on 17 and 18). The two couples of regions have been selected so as to have different latitude and longitude. The underlying physical processes generating the data are known to be different in these regions: the overall motion in regions 17 and 18 is greater, and the difference between extreme temperature is larger, compared to regions 8 and 9. Experimental conditions are similar to the one described in section 4, i.e. 2006-2015 have been used for training and 2016-1017 for testing.
+
+Results in Table B.2 show that the model generalizes reasonably well to unseen data from distant spatial regions, with a slight decrease in performance when training and test regions do not correspond. The performance loss is 0.47 for regions (17, 18) which show a strong dynamics, whereas it is only 0.03 for regions (8, 9) for which the dynamics are more stable. Most notably, MSE performance depends more on the region itself than on the train/ test conditions. Error is always higher in regions with strong dynamics (17, 18) than on more stable regions (8, 9) whatever the train/ test conditions are. Note that to further improve the results on distant data, it is possible to fine-tune the model using data from the studied regions.
+
+![](images/7fa34cba44f56995c4ccd948262fe818c202d7bb2038a7cc5cbd04fe8d410b0f.jpg)  
+Figure 7: Output for the 6 of May to the 9 of May 2016, Output , From top to bottom: target, our model prediction, our model flow
+
+![](images/4ffe62ca072ede6593044ce1af87b6350592816bc760f6a707c20a78f7a4aa63.jpg)  
+Figure 8: Output for the 6 of January to the 9 of January 2016. From top to bottom: target, our model prediction, our model flow

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+        "text": "DEEP LEARNING FOR PHYSICAL PROCESSES: INCORPORATING PRIOR SCIENTIFIC KNOWLEDGE ",
+        "text_level": 1,
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+        "text": "Emmanuel de Bezenac ´ ∗, Arthur Pajot ∗, Patrick Gallinari emmanuel.de-bezenac, arthur.pajot, patrick.gallinari @lip6.fr Sorbonne Universites, UMR 7606, LIP6, F-75005 Paris, France ´ ",
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+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        "text": "We consider the use of Deep Learning methods for modeling complex phenomena like those occurring in natural physical processes. With the large amount of data gathered on these phenomena the data intensive paradigm could begin to challenge more traditional approaches elaborated over the years in fields like maths or physics. However, despite considerable successes in a variety of application domains, the machine learning field is not yet ready to handle the level of complexity required by such problems. Using an example application, namely Sea Surface Temperature Prediction, we show how general background knowledge gained from the physics could be used as a guideline for designing efficient Deep Learning models. In order to motivate the approach and to assess its generality we demonstrate a formal link between the solution of a class of differential equations underlying a large family of physical phenomena and the proposed model. Experiments and comparison with series of baselines including a state of the art numerical approach is then provided. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "A physical process is a sustained phenomenon marked by gradual changes through a series of states occurring in the physical world. Physicists and environmental scientists attempt to model these processes in a principled way through analytic descriptions of the scientist’s prior knowledge of the underlying processes. Conservation laws, physical principles or phenomenological behaviors are generally formalized using differential equations. This physical paradigm has been, and still is the main framework for modeling complex natural phenomena like e.g. those involved in climate. With the availability of very large datasets captured via different types of sensors, this physical modeling paradigm is being challenged by the statistical Machine Learning (ML) paradigm, which offers a prior-agnostic approach. However, despite impressive successes in a variety of domains as demonstrated by the deployment of Deep Learning methods in fields such as vision, language, speech, etc, the statistical approach is not yet ready to challenge the physical paradigm for modeling complex natural phenomena, or at least it has not demonstrated how to. This is a new challenge for this field and an emerging research direction in the ML community. We believe that knowledge and techniques accumulated for modeling physical processes in well developed fields such as maths or physics could be useful as a guideline to design efficient learning systems and conversely, that the ML paradigm could open new directions for modeling such complex phenomena. In this paper we then raise two issues: 1) are modern ML techniques ready to be used to model complex physical phenomena, and 2) how general knowledge gained from the physical modeling paradigm could help designing efficient ML models. ",
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+        "type": "text",
+        "text": "In this work, we tackle these questions by considering a specific physical modeling problem: forecasting sea surface temperature (SST). SST plays a significant role in analyzing and assessing the dynamics of weather and other biological systems. Accurately modeling and predicting such dynamics is critical in various applications such as weather forecasting, or planning of coastal activities. Since 1982, weather satellites have made huge quantities of very high resolution SST data available Bernstein (1982). Standard physical methods for forecasting SST use coupled ocean-atmosphere prediction systems, based on the Navier Stokes equations. These models rely on multiple physical hypotheses and do not optimally exploit the information available in the data. On the other hand, despite the availability of large amounts of data, direct applications of ML methods do not lead to competitive state of the art results, as will be seen in section 4. ",
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We use SST as a typical and representative problem of intermediate complexity. Our goal is not to offer one more solution to this problem, but to use it as an illustration for advancing on the challenges mentioned above. The way we handle this problem is general enough to be transfered to a more general class of transport problems. ",
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "We propose a Deep Neural Network (NN) model, inspired from general physical motivations which offers a new approach for solving this family of problems. We first motivate our approach by introducing in section 2 the solution of a general class of partial differential equations (PDE) which is a core component of a large family of transport and propagation phenomena in physics. This general solution is used as a guideline for introducing a Deep Learning architecture for SST prediction which is described in section 3. Experiments and comparison with a series of baselines is introduced in section 4. A review of related work is finally presented in section 5. ",
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+        "text": "The main contributions of this work are: 1) an example showing how to incorporate general physical background for designing a NN aimed at modeling a relatively complex prediction task. We believe the approach to be general enough to be used for a family of transport problems obeying general advection-diffusion principles. 2) formal links between our model’s prediction and the solution of a general advection diffusion PDE 3) an unsupervised model for estimating motion fields, given a sequence of images. 4) a proof, on a relatively complex physical modeling problem, that full data intensive approaches based on deep architectures can be competitive with state of the art dedicated numerical methods. ",
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+    {
+        "type": "text",
+        "text": "2 PHYSICAL MOTIVATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Forecasting consists in predicting future temperature maps using past records. Temperatures are acquired via satellite imagery. If we focus on a specific area, we can formulate the problem as prediction of future temperature images of this area using past images. The classical approach to forecasting SST consists in using numerical models representing prior knowledge on the conservation laws and physical principles, which take the form of PDEs. These models are then coupled with SST data using assimilation techniques in order to adjust to initial conditions. It is then integrated forward in time to predict SST evolution. For the sea surface, temperature variation is related to a fluid transport problem. In fluids, transport occurs through the combination of two principles: advection and diffusion. During advection, a fluid transports some conserved quantity $I$ (the temperature for SST) or material via bulk motion, i.e.for small variations $\\Delta x , \\Delta t$ , conservation is expressed as: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fef2111d1356bb277eb5013508398a104158b6c5ee84badd6935dbc2c5a25769.jpg",
+        "text": "$$\nI ( x , t ) = I ( x + \\Delta x , t + \\Delta t )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "applying a first order approximation of the right hand side and moving the resulting terms to the left hand side of equation 1, we obtain the advection equation, known also as the Brightness Constancy Constraint Equation (BCCE): ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d2d51e4573a9d2776c62f0f819bbc52366e42d45e3405117c8ae7c1293fa6c7e.jpg",
+        "text": "$$\n\\frac { \\partial I } { \\partial t } + ( w . \\nabla ) I = 0\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $\\nabla$ denotes the gradient operator, and $w$ the motion vector $\\textstyle { \\frac { \\Delta x } { \\Delta t } }$ . This equation describes the temporal evolution of quantity $I$ for displacement $w$ . Note that this equation is also the basis for many variational methods for Optical Flow. To retrieve the motion, numerical schemes are applied, and the resulting system of equations, along with a an additional constraint on $w$ is solved for $w$ . This motion can then be used to forecast the future value of $I$ . ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a0b2a0a2477ded7e8789c12711e33ec36c7f332daa71fd7fbbd18fe0baeb8cf9.jpg",
+        "text": "$$\n\\frac { \\partial I } { \\partial t } + ( w . \\nabla ) I = D \\nabla ^ { 2 } I\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "$\\nabla ^ { 2 }$ denotes the Laplacian operator and $D$ the diffusion coefficient. Note that when $D \\to 0$ , we recover the advection equation 2. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "This equation describes a large family of physical processes (e.g. fluid dynamics, heat conduction, wind dynamics, etc). Let us now state a result, characterizing the general solutions of equation 3. ",
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+    {
+        "type": "text",
+        "text": "Theorem 1. 1 For any initial condition $I _ { 0 } \\in L ^ { 1 } ( \\mathbb { R } ^ { 2 } )$ with $I _ { 0 } ( \\pm \\infty ) = 0$ , there exists a unique global solution $I ( x , t )$ to the advection-diffusion equation 3: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/51cdaa73a43906f93e0b953047dee2e1621ccdb9be84b097c1adb26f7ec0d9b2.jpg",
+        "text": "$$\nI ( x , t ) = \\int _ { \\mathbb { R } ^ { 2 } } k ( x - w , y ) I _ { 0 } ( y ) d y\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { k ( u , v ) = \\frac { 1 } { 4 \\pi D t } e ^ { - \\frac { 1 } { 4 D t } \\left. u - v \\right. ^ { 2 } } } \\end{array}$ is a radial basis function kernel, or alternatively, a 2 dimensional Gaussian probability density with mean $u$ and variance $2 D t$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Equation 4 provides a principled way to calculate $I ( x , t )$ for any time $t$ using the initial condition $I _ { 0 }$ , provided the motion $w$ and the diffusion coefficient $D$ are known. It states that quantity $I ( x , t )$ can be computed from the initial condition $I _ { 0 }$ via a convolution with a Gaussian probability density function. In other words, if $I$ was used as a model for the evolution of the SST and the surface’s underlying advecting mechanisms were known, future surface temperatures could be predicted from previous ones. Unfortunately neither the initial conditions, the motion vector nor the diffusion coefficient are known. They have to be estimated from the data. Inspired from the general form of solution 4, we propose a ML method, expressed as a Deep Learning architecture for predicting SST. This model will learn to predict a motion field analog to the $w$ in equation 4, which will be used to predict future images. ",
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+    {
+        "type": "text",
+        "text": "3 MODEL ",
+        "text_level": 1,
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+    },
+    {
+        "type": "image",
+        "img_path": "images/723d416121047ed73a5ae5137105467c9d588884d82dff94268dd9f826d12a3b.jpg",
+        "image_caption": [
+            "Figure 1: Motion is estimated from the input images $( I _ { t - k - 1 : t } )$ with a convolutional neural network (top left CDNN component). A warping scheme then displaces the last input image along this motion estimate to produce the future image. The error signal is calculated using the target future image $I _ { t + 1 }$ , and is backprogated through the warping scheme to correct the CDNN. To produce multiple time-step forecasts, the predicted image is fed back in the CDNN in an autoregressive manner. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The model consists of two main components, as illustrated in Figure 1. One predicts the motion field from a sequence of past input images, this is convolutional-deconvolutional (CDNN) module on the top of figure 1, and the other warps the last input image using the motion field from the first component, in order to produce an image forecast. The entire system is trained in an end-to-end fashion, using only the supervision from the target SST image. By doing so, we are able to produce an interpretable latent state which corresponds in our problem to the velocity field advecting the temperatures. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Let us first introduce some notations. Each SST image $I _ { t }$ is acquired on a bounded rectangle of $\\mathbb { R } ^ { 2 }$ , named $\\Omega$ . We denote $I _ { t } ( x )$ and $w _ { t } ( x )$ the sea surface temperature and the two-dimensional motion vector at time $t \\in \\mathbb { R }$ at position $x \\in \\Omega$ . $I _ { t } : \\Omega \\to \\mathbb { R }$ and $\\dot { w } _ { t } : \\Omega \\to \\mathbb { R } ^ { 2 }$ represent the temperatures and the motion vector field at time $t$ defined on $\\Omega$ . When time $t$ and position $x$ are available from the context, we will drop the subscript $t$ from $w _ { t } ( x )$ and $I _ { t } ( x )$ , along with $x$ for clarity. Given a sequence of $k$ consecutive SST images $\\{ I _ { t - k - 1 } , . . . , I _ { t } \\}$ (also denoted as $I _ { t - k - 1 : t } )$ , our goal is to predict the next image $I _ { t + 1 }$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/fdfbc089060a5d4b341bb93d8bfc0394e30c6e4e3b5ae9494dd7e8d78b641924.jpg",
+        "image_caption": [
+            "Figure 2: Architecture of the CDNN motion estimation component. For the estimated motion flow $\\hat { w } _ { t }$ , colours correspond to the flow orientation and colour intensity to the flow intensity "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As indicated in section 2, provided the underlying motion field is known, one can compute SST forecasts. Let us introduce how the motion field is estimated in our architecture. We are looking for a vector field $w$ which when applied to the geometric space $\\Omega$ renders $I _ { t }$ close to $I _ { t + 1 }$ , i.e. $I _ { t + 1 } ( \\bar { x } ) \\simeq$ $I _ { t } ( x + w ( x ) )$ , $\\forall x \\in \\Omega$ . If $I _ { t + 1 }$ were known, we could estimate $w$ , but $I _ { t + 1 }$ is precisely what we are looking for. Instead, we choose to use a convolutional-deconvolutional architecture to predict a motion vector for each pixel. As shown in figure 2, this network makes use of skip connections He et al. (2015), allowing fine grained information from the first layers to flow through in a more direct manner. We use a Batch Normalization layer between each convolution, and Leaky $R e L U$ (with parameter value set to 0.1) non-linearities between convolutions and transposed-convolutions. We used $k = 4$ concatenated images $I _ { t - k - 1 : t }$ as input for training. We have selected this architecture experimentally, testing different state-of-the-art convolution-deconvolution network architectures. Let $\\hat { w } \\in \\mathbb { R } ^ { 2 \\times \\mathbf { \\bar { W } } \\times H }$ be the output of the network, where $W$ and $H$ are respectively the width and height of the images, and $\\mathit { \\Omega } ^ { , }$ corresponds to the two components of the flow at each point of the motion field. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Generally, and this is the case for our problem, we do not have a direct supervision on the motion vector field, since the target motion is usually not available. Using the warping scheme introduced below, we will nonetheless be able to (weakly) supervise $w$ , based on the discrepancy of the warped version of the $I _ { t }$ image and the target image $I _ { t + 1 }$ . ",
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+    {
+        "type": "text",
+        "text": "3.2 WARPING SCHEME ",
+        "text_level": 1,
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+        "page_idx": 3
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+    {
+        "type": "image",
+        "img_path": "images/f26b7c0fe8bb8e32a7af7bd237402c2fba74592354a184042a168128f6b08f8e.jpg",
+        "image_caption": [
+            "Figure 3: Warping scheme. To calculate the pixel value for time $t + 1$ at position $x$ , we first compute its previous position at time $t$ , i.e. $x - w$ . We then center a Gaussian in that position in order to obtain a weight value for each pixel in $I _ { t }$ based on its distance with $x - w$ , and compute a weighted average of the pixel values of $I _ { t }$ . This weighted average will correspond to the new pixel value at $x$ in $I _ { t + 1 }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Discretizing the solution of the advection-diffusion equation in section 2 by replacing the integral with a sum, and setting image $I _ { t }$ as the initial condition, we obtain a method to calculate the future image, based on the motion field estimate $\\hat { w }$ . The latter is used as a warping scheme: ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e8a196d1bf52ed7f432394343c8de7e606cd2b23990b4363b08b0d38f6372cc2.jpg",
+        "text": "$$\n\\hat { I } _ { t + 1 } ( x ) = \\sum _ { y \\in \\Omega } k ( x - \\hat { w } ( x ) , y ) I _ { t } ( y )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { k ( x - \\underset { . } { \\hat { w } } , y ) = \\underset { . } { \\frac { 1 } { 4 \\pi D \\Delta t } } e ^ { - \\frac { 1 } { 4 D \\Delta t } \\| x - \\hat { w } - y \\| ^ { 2 } } } \\end{array}$ is a radial basis function kernel, as in equation 4, parameterized by the diffusion coefficient $D$ and the time step value $\\Delta t$ between $t$ and $t + 1$ and $\\hat { w }$ is the estimated value of the vector flow $w$ . To calculate the temperature for time $t + 1$ at position $x$ , we compute the scalar product between $k ( x - { \\hat { w } } , . )$ , a Gaussian centered in $x - \\hat { w }$ , and the previous image $I _ { t }$ . Simply put, it is a weighted average of the temperatures $I _ { t }$ , where the weight values are larger when the pixel’s positions that are closer to $x - \\hat { w }$ . Informally, $x - \\hat { w }$ corresponds to the pixel’s previous position at time $t$ . See figure 3. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As seen by the relation with the solution of the advection-diffusion equation, the proposed warping mechanism is then clearly adapted to the modeling of phenomena governed by the advectiondiffusion equation. SST forecasting is a particular case, but the proposed scheme can be used for any problems in which advection and diffusion are occurring. Moreover, this warping scheme is entirely differentiable, allowing backpropagation of the error signal to the motion fireld estimating module. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This warping mechanism has been inspired by the Spatial Transformer Network (STN) Jaderberg et al. (2015), originally designed to be incorporated as a layer in a convolutional neural network architecture in order to gain invariance under geometric transformations. Using the notations in Jaderberg et al. (2015), when the inverse geometric transformation $\\mathcal { T } _ { \\theta }$ of the grid generator step is set to $\\mathcal { T } _ { \\theta } ( x ) = x - \\hat { w } ( x )$ , and the kernels $k ( . ; \\Phi _ { x } )$ and $k ( . ; \\Phi _ { y } )$ in the sampling step are radial basis function kernels, we recover our warping scheme. The latter can be seen as a specific case of the STN, without the localization step. This result theoretically grounds the use of the STN for Optical Flow in many recent articles Zhu et al. (2017), Yu et al. (2016), Patraucean et al. (2015), Finn et al. (2016): in equation 3, when $D \\to 0$ , we recover the brightness constancy constraint equation, used in the latter. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For training, supervision is provided at the output of the warping module. It consists in minimizing the discrepancy between the warped image $\\hat { I } _ { t + 1 }$ and the target image $I _ { t + 1 }$ . The loss is measured via a differentiable function and the gradient is back propagated through the warping function in order to adjust the parameters of the convolutional-deconvolutional module generating the vector field. This is detailed in the next section. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 LOSS FUNCTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "At each iteration, the model aims at forecasting the next observation, given the previous ones. We evaluate the discrepancy between the warped image $\\hat { I } _ { t + 1 }$ and the target image $I _ { t + 1 }$ using the Charbonnier penalty function $\\rho ( x ) = ( x + \\epsilon ) ^ { \\frac { 1 } { \\alpha } }$ , where $\\epsilon$ and $\\alpha$ are parameters to be set. Note that with $\\epsilon = 0$ and $\\alpha \\stackrel { \\cdot } { = } \\frac { 1 } { 2 }$ , we recover the $\\ell _ { 2 }$ loss. The Charbonnier penalty function is known to reduce the influence of outliers compared to an $l _ { 2 }$ norm. We have also tested the Laplacian pyramid loss Ling & Okada (2006), where we enforce convolutions of all deconvolutional layers to be close to down-sampled versions of the target image in the Charbonnier penalty sense, but we have observed an overall decrease in generalization performance. ",
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+        "type": "text",
+        "text": "The proposed NN model has been designed according to the intuition gained from general background knowledge of a physical phenomenon, here advection-diffusion equations. Additional prior knowledge – expressed as partial differential equations, or through constraints – can be easily incorporated in our model, by adding penalty terms in the loss function. As the displacement $w$ is explicitly part of our model, one strength of our model is its capacity to apply some regularization term directly on the motion field. In our experiments, we tested the influence of different terms: divergence $\\nabla . w _ { t } ( x ) ^ { 2 }$ , magnitude $\\| w _ { t } ( x ) \\| ^ { 2 }$ and smoothness $\\Vert \\nabla w _ { t } ( x ) \\Vert ^ { 2 }$ . ",
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+        "img_path": "images/18fe2dff46e590585c081bd45c5186c206d8fe936d24ac0df7343cba03be85cd.jpg",
+        "text": "$$\nL _ { t } = \\sum _ { x \\in \\Omega } \\rho ( \\hat { I } _ { t + 1 } ( x ) - I _ { t + 1 } ( x ) ) + \\lambda _ { \\mathrm { d i v } } ( \\nabla . w _ { t } ( x ) ) ^ { 2 } + \\lambda _ { \\mathrm { m a g n } } \\left\\| w _ { t } ( x ) \\right\\| ^ { 2 } + \\lambda _ { \\mathrm { g r a d } } \\left\\| \\nabla w _ { t } ( x ) \\right\\| ^ { 2 }\n$$",
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+        "text": "4 EXPERIMENTS ",
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+        "text": "4.1 DATASET DESCRIPTION ",
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+        "text": "Since 1982, high resolution SST data has been made available by the NOAA6 weather satellite, Bernstein (1982). Dealing directly with these data requires a lot of preprocessing (e.g. some regions are not available due to clouds, hindering temperature acquisition). In order to avoid such complications which are beyond the scope of this work, we used synthetic but realistic SST data of the Atlantic ocean generated by a sophisticated simulation engine: NEMO (Nucleus for European Modeling of the Ocean) engine 2, Madec (2008). NEMO is a state-of-the-art modelling framework of ocean related engines. It is a primitive equation model adapted to the regional and global ocean circulation problems. Historical data is accumulated in the model to generate a synthesized estimate of the states of the system using data analysis, a specific data assimilation scheme, which means that the data does follow the true temperatures. The resulting dataset is constituted of daily temperature acquisitions of 481 by 781 pixels, from 2006-12-28 to 2017-04-05 (3734 acquisitions). ",
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+        "text": "We extract 64 by 64 pixel sized sub-regions as indicated in figure 4.1. We use data from years 2006 to 2015 for training and validation (94743 training examples), and years 2016 to 2017 for testing. We withhold $20 \\%$ of the training data for validation, selected uniformly at random at the beginning of each experiment. For the tests we used sub-regions enumerated 17 to 20 in figure 4.1, where the interactions between hot and cold waters make the dynamics interesting to study. All the regions numbered in figure 4.1, from 2006 to 2015 where used for training 3. Each sequence of images used for training or for evaluation corresponds to a specific numbered sub-region. We make the simplifying hypothesis that the data in a single sub-region contains enough information to forecast the future of the sub-region. As the forecast is for a small temporal horizon we can assume that the influence from outside the region is small enough. ",
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+        "image_caption": [
+            "Figure 4: Sub regions extracted for the dataset. Test regions are regions 17 to 20. "
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+        "text": "We normalize the daily SST acquisitions of each sub region using the mean and the standard deviation of all the SST data of the sub-region acquired on the same day of the year for all the years in the training set, i.e. the SST acquisition of sub-region 2 on date September 8th 2017 is standardized using the data of all the September 8th available in the dataset. This removes the seasonal component from SST data. ",
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+        "text": "4.2 BASELINE COMPARISON",
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+        "text": "We compare our model with several baselines. Each model is evaluated with a mean square error metric, forecasting images on a horizon of 6 (we forecast from $I _ { t + 1 }$ to $I _ { t + 6 }$ and then average the MSE). The hyperparameters are tuned using the validation set. Neural network based models are run on a Titan $\\mathrm { X p }$ GPU, and runtime is given for comparison purpose. ",
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+        "text": "Concerning the constraints on the vector field $w$ (equation 6. the regularization coefficients selected via validation are $\\lambda _ { \\mathrm { d i v } } = 1$ , $\\lambda _ { \\mathrm { { m a g n } } } = - 0 . 0 3$ and $\\lambda _ { \\mathrm { g r a d } } = 0 . 4$ . The coefficient diffusion $D$ was set to 0.45 by cross validation. We also compare the results with the model without any regularization. ",
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+        "text": "Our reference model for forecasting is Ber´ eziat & Herlin (2015), a numerical assimilation model ´ which relies on data assimilation. In Ber´ eziat & Herlin (2015), the ocean’s dynamics are modeled ´ using shallow water equations Vallis (2017) and the initial conditions, along with other terms, are estimated using assimilation techniques Tremolet (2006). This is a state of the art assimilation model ´ for predicting ocean dynamics, here SST. ",
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+        "text": "The other baselines are 1) an autoregressive convolutional-deconvolutional NN (ACNN), with an architecture similar to our CDNN module, but trained to predict the future image directly, without explicitly representing the motion vector field. Each past observation is used as an input channel (the 4 input images used in the experiments are concatenated), and the output is used as new input for multi step forecasting, 2) a ConvLSTM model Shi et al. (2015), which uses convolutional transitions in the inner LSTM module, and 3) the model in Mathieu et al. (2015) which is a multi-scale ACNN trained as a Generative Adversial Network (GAN). We have used a non-official code for Mathieu et al. (2015), which is made available at https://github.com/dyelax/Adversarial_ Video_Generation. For Ber´ eziat & Herlin (2015), the code has been provided by the authors ´ of the paper. We have implemented the ACNN and ConvLSTM models ourselves. The code for our models, along with these baselines will be made available. ",
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+        "text": "4.3 QUANTITATIVE RESULTS ",
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+        "table_body": "<table><tr><td>Model</td><td>Average Score (MSE)</td><td>Average Time</td></tr><tr><td>Numerical model Béréziat &amp; Herlin (2015)</td><td>1.99</td><td>4.8 s</td></tr><tr><td>ConvLSTM Shi et al. (2015)</td><td>5.76</td><td>0.018 s</td></tr><tr><td>ACNN</td><td>15.84</td><td>0.54 s</td></tr><tr><td>GAN Video Generation (Mathieu et al. (2015))</td><td>4.73</td><td>0.096 s</td></tr><tr><td>Proposed model with regularization</td><td>1.42</td><td>0.040 s</td></tr><tr><td>Proposed model without regularization</td><td>2.01</td><td>0.040 s</td></tr></table>",
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+        "type": "text",
+        "text": "Table 1: Average score and average time on test data. Average score is calculated using the mean square error metric (MSE), time is in seconds. The regularization coefficients for our model have been set using a validation set with $\\lambda _ { \\mathrm { d i v } } = 1$ , $\\lambda _ { \\mathrm { { m a g n } } } = - 0 . 0 3$ and $\\lambda _ { \\mathrm { g r a d } } = 0 . 4$ . ",
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+    {
+        "type": "text",
+        "text": "Quantitatively, our model performs well. The MSE score is better than any of the baselines. The closest NN baseline is Mathieu et al. (2015) which regularizes a regression convolutiondeconvolution model with a GAN. The performance is however clearly below the proposed model and it does not allow to easily incorporate prior constraints inspired from the physics of the phenomenon. ACNN is a direct predictor of the image sequence, implemented via a CDNN module identical to the one used in our model. Its performance is poor. Clearly, a straightforward use of prediction models is not adapted to the complexity of the phenomenon. ConvLSTM performs better: as opposed to the ACNN, it seems to be able to capture a dynamic, although not very accurately. Overall, direct prediction models are not able to capture the complex underlying dynamics and they produce blurry sequences of images. The GAN explicitly forces the network output to eliminate the blurring effect and then makes it able to capture short term dynamics. The state of the art numerical model Ber´ eziat & Herlin (2015), performs well but has a slighthly lower performance than our reg- ´ ularized model, although it incorporates more prior constraints. This shows that pure ML models, when conceived adequately and when trained with enough data, can be competitive with state of the art dedicated models. Regularizing the motion vector $w$ notably increases the performance w.r.t. to the unregularized model. The choice of the constraints (divergence, magnitude and smoothness) inspired here by physical background correspond to relevant priors on the dynamics of the model. ",
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+        "img_path": "images/0538e3fec59746d81f644731393a61c8431a5d123666afbf52804482ea01d9d5.jpg",
+        "image_caption": [
+            "Figure 5: From top to bottom: target, our model prediction, our model flow, numerical assimilation model , ACNN, ConvLSTM. Data correspond to daily temperatures from January 17 to January 23, 2017 "
+        ],
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+    {
+        "type": "text",
+        "text": "As for the running time, the proposed model is extremely fast, being just above the ConvLSTM model of Shi et al. (2015). The running time of Ber´ eziat & Herlin (2015)’s model is not comparable ´ to the others. It was run on a CPU (no GPU code) when all the others were run on Titan $\\mathrm { X p }$ GPU. However, an optimization procedure is required to estimate the motion field, and it is clearly slower than the straightforward NN predictions. Moreover, in order to prevent the numerical scheme from diverging, multiple intermediate forecasts are required. ",
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+    {
+        "type": "text",
+        "text": "Besides MSE, we need to analyze the prediction samples qualitatively. Figure 4.3 shows the predictions obtained by the different models. On the top row, the ground truth for a sequence of 4 temperature images corresponding to time $t .$ , $t + 1$ , $t + 3$ and $t + 6$ . The second row corresponds to our regularized model prediction at times $t + 1 , t + 3$ and $t + 6$ (time $t$ corresponds to the last input image, it is repeated on each row). The model seems to conserve temperatures. The prediction is close to the target for $t + 1 , t + 3$ and starts to move away at time $t + 6$ . The third row shows the motion flow estimated by the model. Each color in the flow images corresponds to a motion vector. There is clearly a strong evolving dynamic captured for this sequence. Row 4 is the numerical assimilation model of Ber´ eziat & Herlin (2015). It also clearly captures some dynamics and shows ´ interesting patterns, but it tends to diverge when the prediction horizon increases. The ACNN model (row 5) rapidly produces blurry images; it does not preserve the temperatures and does not seem to capture any dynamics. On row 6 are plotted the predictions of the ConvLSTM model. Temperature is not preserved and although a dynamic is captured, it does not correspond to the target. Overall, the proposed model seems to forecast SST quite accurately, while retrieving a coherent motion vector field. ",
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+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "ML for Physical modeling Close to this work is the field of spatio-temporal statistics. In their reference book Cressie & Wikle (2015) also advocate the use of physical background knowledge to build statistical models. They show how the design of statistical models can be inspired from partial differential equations linked to an observed physical phenomenon. They mainly consider auto-regressive models within a hierarchical Bayesian framework. In Raissi et al. (2017), Archambeau et al. (2007) and Alvarez et al. (2011) the author use PDE-inspired gaussian process to model physical process. Even if the methods and the application are different, the motivation and arguments are similar to the ones developed here. ",
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+    {
+        "type": "text",
+        "text": "Another interesting research direction is the use of NNs for reducing the complexity of numerical simulation for physical processes. Generally, in these approaches statistical models are used in place of a computational demanding component of the numerical simulation process. For example in the domain of fluid dynamics, Tompson et al. (2017) and Ladicky et al. (2015) propose to use regressors ´ for simulating fluid and smoke animation. Ladicky et al. (2015) use a random forest to compute ´ particle location and Tompson et al. (2017) use a CNN to approximate part of a numerical PDE scheme. In these approaches, ML is only a component of a numerical simulation scheme whereas we aim at modeling the whole physical process via a Deep Learning approach. Farther to our objective, Rudy et al. (2017) make use of a sparse regression method for discovering the governing partial differential equation(s) of a given system by time series measurements in the spatial domain. Other works have suggested using neural networks for physical process forecast, such as Brajard et al. (2017). ",
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+    {
+        "type": "text",
+        "text": "Our work is also related to recent developments in computer vision, in the related but distinct fields of video prediction and motion estimation in videos. Our goal and the domain of application are clearly different from video modeling, but since our solution involves predicting a motion field and the next SST image, the solutions share some similarities. Motion estimation and video predictions by deep architectures have motivated a series of work over the last two years. We briefly review them below and outline the differences. ",
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+    {
+        "type": "text",
+        "text": "Optical Flow Optical flow consists in retrieving the apparent motion of objects, surfaces, or particles between two consecutive frames of a video. The extracted motion can be used in many areas such as object detection, object tracking, movement detection, robot navigation and visual odometry. In the vision community, this is considered as a problem by itself and several papers are dedicated to this topic. Classical methods rely on the brightness constancy constrain equation (BCCE) (equation 2), derived from the observation that surfaces usually persist over time and hence the intensity value of a small region remains the same despite its position change Sun et al. (2008). Since using BCCE directly leads to complicated optimizing issues, classic approaches – namely differential methods – approximate BCCE using a first order Taylor expansion and develop variational methods. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As an alternative to these methods, Deep Learning models have been recently proposed for estimating the optical flow between 2 images. Fischer et al. (2015) formulate optical flow as a supervised regression problem, using a CNN to predict motion. Ilg et al. (2016) build on this approach and propose to use an ensemble of these CNN architectures. They assess results on par with state of the art methods for optical flow, while maintaining a small computational overhead. The difficulty here is that these methods require a notable quantity of target data, i.e. optical flow images, while because of the complexity of manually annotating flow images, there are only a few small annotated collections available. Fischer et al. (2015) and Ilg et al. (2016) chose to pretrain their model on a synthetic dataset made of computer animations and their associated motion and show that this information transfers well to real videos. Yu et al. (2016) demonstrate that it is possible to predict the optical flow between two input images in an unsupervised way using a CNN and a Spatial Transformer Network. This is however not extensible for prediction as is done in our setting since this requires the two images $I _ { t }$ and $I _ { t + 1 }$ as input while $I _ { t + 1 }$ is not available at inference time for prediction. ",
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+    {
+        "type": "text",
+        "text": "Video prediction ",
+        "text_level": 1,
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+        "page_idx": 9
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+    {
+        "type": "text",
+        "text": "It is only very recently that video prediction emerged as a task in the Deep Learning community. For this task, people are generally interested at predicting accurately the displacement/ emergence/ disappearing of objects in the video. In our application, the goal is clearly different since we are interested into modeling the whole dynamics behind image changes and not at following moving objects. Let us first introduce some methods that perform prediction by computing optical flow or a similar transformation. Both Patraucean et al. (2015) and Finn et al. (2016) use some form of motion flow estimation. For next frame prediction Patraucean et al. (2015) introduce a STN module at the hidden layer of a LSTM in order do estimate a motion field in this latent space. The resulting image is then decoded in the original image space for prediction. This approach clearly does not allow introducing prior knowledge on the field vector as this has been done in our work. Finn et al. (2016) learn affine transformations on image parts in order to predict object displacement and Van Amersfoort et al. (2017) proposed a similar model. ",
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Let us now consider models that directly attempt to predict the next frame without estimating a motion field. As shown in the experimental section, plain application of autoregressive models produces blurred images. Mathieu et al. (2015), one of our baseline proposed to use different loss functions and a GAN regularization of a CDNN predictor which led to sharper and higher quality predictions. Significant improvements have been obtained with the Video Pixel Network of Kalchbrenner et al. (2016), which is a sophisticated architecture composed of resolution preserving CNN encoders, LSTM and PixelCNN decoders which form a conditional Spatio-temporal video autoencoder with differentiable memory. This model is probably state of the art today for video prediction, They reach a high accuracy on moving MNIST and good performance on a robot video dataset. A drawback is the complexity of the model and the number of parameters: they are using respectively $2 0 \\bf { M }$ and $1 \\textbf { M }$ frames on these two datasets. We did not test this model since up to our knowledge no code was available. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION AND FUTURE WORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The evolution in time of the proposed model is deterministic. Predicting future observations should also deal with the inherent ambiguity and lack of information for the prediction task. A natural future direction would be to incorporate uncertainty in the model’s evolution in the proposed framework. We can extend the proposed model by incorporating a stochastic latent variable in the flow field generation process. A promising direction is the development of generative models which has become popular in Deep Learning, leading to different families of innovative models. For example, the Stochastic Gradient Variational Bayes algorithm (SGVB) Kingma & Welling (2014) provides a framework for learning stochastic latent variables with deep neural networks, and has recently been used by some authors to model time series Karl et al. (2016); Chung et al. (2015); Krishnan et al. (2015). A recent work where both spatial and temporal information are considered is Walker et al. (2016) who model pixel trajectories in a video. As a follow up of our work, we plan to consider such extensions in the future. ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "The data intensive paradigm offers alternative directions to the classical physical approaches for modeling complex natural processes. Our belief is that cross fertilization of both paradigms is essential for pushing further the frontier of complex data modeling. By using as an example application a relatively complex problem concerning ocean dynamics, we proposed a principled way to design Deep Learning models using inspiration from the physics. The proposed approach can be easily generalized to a class of problems for which the underlying dynamics follow advection-diffusion principles. We have compared the proposed approach to a series of baselines. It is able to reach performance comparable to a state of the art numerical model and clearly outperform alternative NN models used as baselines. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "This work was partially funded by ANR project LOCUST - ANR-15-CE23-0027 and by CLEAR - Center for LEArning & data Retrieval - joint lab. With Thales (www.thalesgroup.com). ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            409,
+            285,
+            422
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Mauricio A Alvarez, David Luengo, and Neil D Lawrence. Linear latent force models using gaussian processes. 2011. ",
+        "bbox": [
+            174,
+            431,
+            823,
+            459
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Cedric Archambeau, Dan Cornford, Manfred Opper, and John Shawe-Taylor. Gaussian process approximations of stochastic differential equations. In Gaussian Processes in Practice, pp. 1–16, 2007. ",
+        "bbox": [
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+            468,
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+            510
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Dominique Ber´ eziat and Isabelle Herlin. ´ Coupling Dynamic Equations and Satellite Images for Modelling Ocean Surface Circulation, pp. 191–205. Springer International Publishing, Cham, 2015. ISBN 978-3-319-25117-2. doi: 10.1007/978-3-319-25117-2 12. URL https://doi. org/10.1007/978-3-319-25117-2_12. ",
+        "bbox": [
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+        "page_idx": 10
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+    {
+        "type": "text",
+        "text": "A PROOF OF THE THEOREM IN SECTION 2 1 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            550,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Proof. In the following, bold $\\mathbf { x }$ and $\\mathbf { y }$ will denote vectors of $\\mathbb { R } ^ { 2 }$ , while $x$ and $y$ will correspond to the first and second components of $\\mathbf { x }$ , respectively. Analogously, $u$ and $v$ will correspond to the components of $w$ . The 2D Fourier Transformation $\\mathcal { F }$ of $f : \\bar { \\mathbb { R } } ^ { 2 }  \\bar { \\mathbb { R } }$ is defined as ",
+        "bbox": [
+            174,
+            132,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c1e10b449d1712428e04327e795e62e9ccc969d4924535b3d40dd65f446d9ea5.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathcal { F } ( f ) = \\int _ { \\mathbb { R } ^ { 2 } } f ( \\mathbf { x } ) e ^ { - i < \\xi , \\mathbf { x } > } d \\mathbf { x } } \\\\ { \\displaystyle \\ = \\int _ { \\mathbb { R } } \\int _ { \\mathbb { R } } f ( x , y ) e ^ { - i x \\xi _ { 1 } - i y \\xi _ { 2 } } d x d y } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            184,
+            630,
+            253
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We apply the Fourier Transform $\\mathcal { F }$ to both sides of 3. As consequence of the linearity of the Fourier transform, we can calculate decompose the Fourier transform of the left hand side in the sum of the transforms of each term. We have three terms: $\\begin{array} { r } { \\frac { \\partial I } { \\partial t } , ( w . \\nabla ) I } \\end{array}$ and $- D \\nabla ^ { 2 } I$ . ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/be4ad8f62a7fdc91a94efa4a55bad61766cf617c58ac3902370065f63ef5b851.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathcal { F } ( \\frac { \\partial I } { \\partial t } ) = \\int _ { \\mathbb { R } ^ { 2 } } \\frac { \\partial I } { \\partial t } e ^ { - i < \\mathbf { x } , \\xi > } d \\mathbf { x } } \\\\ { \\displaystyle \\ = \\int _ { \\mathbb { R } ^ { 2 } } \\frac { \\partial } { \\partial t } ( I e ^ { - i < \\mathbf { x } , \\xi > } ) d \\mathbf { x } } \\\\ { \\displaystyle \\ = \\frac { \\partial } { \\partial t } \\int _ { \\mathbb { R } ^ { 2 } } I e ^ { - i < \\mathbf { x } , \\xi > } d \\mathbf { x } } \\\\ { \\displaystyle \\ = \\frac { \\partial \\mathcal { F } ( I ) } { \\partial t } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            387,
+            327,
+            611,
+            463
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/47f3c8b54c83d55ee10071d59cf6281dd4c9688e184a0820a4ed385206cdad5e.jpg",
+        "text": "$$\n\\begin{array} { r l } { F ( ( w , \\nabla ) I ) = \\displaystyle \\int _ { \\mathbb { R } ^ { 2 } } ( w , \\nabla ) I e ^ { - i \\omega \\cdot \\xi } d x } \\\\ { = \\displaystyle \\int _ { \\mathbb { R } } \\int _ { \\mathbb { R } } ( w \\frac { \\partial I } { \\partial x } + v \\frac { \\partial I } { \\partial y } ) e ^ { - i x \\xi _ { 1 } - \\mathrm { i } y \\xi _ { 2 } } d x d y } \\\\ { = \\displaystyle \\mu \\int _ { \\mathbb { R } } e ^ { - i \\psi \\cdot \\xi } \\int _ { \\mathbb { R } } \\frac { \\partial I } { \\partial x } e ^ { - i \\xi _ { 1 } } d x d y + v \\displaystyle \\int _ { \\mathbb { R } } e ^ { - i \\psi \\xi _ { 1 } } \\int _ { \\mathbb { R } } \\frac { \\partial I } { \\partial y } e ^ { - i y \\xi _ { 2 } } d y d x } \\\\ { = \\displaystyle i \\xi _ { 1 } w \\int _ { \\mathbb { R } } e ^ { - i \\psi \\xi _ { 2 } } \\int _ { x } [ e ^ { - i \\psi \\xi _ { 1 } } d x d y + i \\xi _ { 2 } v \\int _ { \\mathbb { R } } e ^ { - i \\psi \\xi _ { 1 } } \\int _ { \\mathbb { R } } \\frac { \\partial I } { \\partial y } e ^ { - i y \\xi _ { 2 } } d y d x  } \\\\ {  = \\displaystyle \\tilde { U } _ { \\mathbb { R } } \\int _ { \\mathbb { R } } \\int _ { \\mathbb { R } } e ^ { - i \\psi \\xi _ { 1 } - \\mathrm { i } \\psi _ { \\xi } } d x d y + v \\xi _ { 2 } v \\int _ { \\mathbb { R } } \\int _ { \\mathbb { R } } I _ { \\mathbb { R } } e ^ { - i \\omega \\xi _ { 1 } - y \\psi } d x d y  } \\\\ { =  ( \\xi _ { 1 } w + i \\xi _ { 2 } v ) \\int _ { \\mathbb { R } } \\int _ { \\mathbb { R } } \\int _ { \\mathbb { R } } \\int _ { \\mathbb { R } } - i \\omega \\xi _ { 1 } - i \\psi \\xi _ { 2 } d x d y } \\\\ { = i \\ < \\xi , w > F ( I ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            227,
+            470,
+            772,
+            693
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2a0c10eacce80df1f6be7842e12f14367bf1e7e2970fd9bccb55a14fcd0a4e3d.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\mathcal { F } ( - D \\nabla ^ { 2 } I ) = - \\int _ { \\mathbb { R } ^ { 2 } } D \\nabla ^ { 2 } I c ^ { - i \\nu \\xi } s c ^ { - \\alpha } k s ^ { \\zeta } d x } \\\\ { = - \\int _ { \\mathbb { R } ^ { 2 } } \\int _ { \\mathbb { R } ^ { 0 } } D \\big ( \\hat { \\partial } \\hat { \\partial } ^ { 2 } I + \\hat { \\partial } \\hat { \\partial } ^ { 2 } I \\big ) e ^ { - i \\alpha \\zeta _ { 1 } - i \\psi \\xi } d x d y } \\\\ { } & { = - D \\int _ { \\mathbb { R } ^ { 2 } } e ^ { - i \\nu \\xi } \\int _ { \\mathbb { R } ^ { 2 } } \\frac { \\partial ^ { 2 } I } { \\partial x ^ { 2 } } e ^ { - i \\alpha \\zeta _ { 1 } } d x d y - D \\int _ { \\mathbb { R } ^ { \\epsilon } } e ^ { - i \\alpha \\zeta _ { 1 } } \\int _ { \\mathbb { R } } \\frac { \\partial ^ { 2 } I } { \\partial y ^ { 2 } } e ^ { - i \\nu \\xi \\epsilon } d y d x } \\\\ { } & { = - ( \\delta _ { 1 } ) ^ { 2 } D \\int _ { \\mathbb { R } ^ { 2 } } e ^ { - i \\nu \\zeta _ { 1 } } \\int _ { \\mathbb { R } } { F } ^ { - i \\nu \\zeta _ { 1 } } d x d y d y - ( \\delta _ { 2 } ) ^ { 2 } D \\int _ { \\mathbb { R } ^ { \\epsilon } } e ^ { - i \\alpha \\zeta _ { 1 } } \\int _ { \\mathbb { R } } { F } ^ { - i \\nu \\zeta _ { 2 } } d y d x } \\\\ { } & { = D \\mathfrak { L } _ { 1 } ^ { 2 } \\int _ { \\mathbb { R } ^ { \\epsilon } } e ^ { - i \\alpha \\zeta _ { 2 } } \\int _ { \\mathbb { R } } { F } ^ { - i \\nu \\zeta _ { 2 } } d x d y + D \\mathfrak { L } _ { 2 } ^ { 2 } \\int _ { \\mathbb { R } ^ { \\epsilon } } e ^ { - i \\kappa \\zeta _ { 1 } } \\int _ { \\mathbb { R } } { F } ^ { i \\nu \\zeta - i \\psi \\zeta _ { 2 } } d y d x } \\\\ { } & { = D \\mathfrak { L } _ { 2 } ^ { 3 } \\int _ { \\mathbb { R } ^ { 2 } } \\int _ { \\mathbb { R } } { F } ^ { - i \\nu \\zeta _ { 2 } } d x d y + D \\mathfrak { L } _ { 2 } ^ { 2 } \\int _ { \\mathbb { R } } { F } ^ { - i \\nu \\zeta _ { 1 } } d x d y d x } \\\\ { } &  = D \\mathfrak { L } _ { 1 } ^ { 3 } \\int _  \\mathbb  \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            186,
+            703,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Regrouping all three previously calculated terms, we obtain ",
+        "bbox": [
+            173,
+            103,
+            566,
+            119
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7a38b2c228e4547f383b8a4150ef651a91d402f6a0ce96178c725dd86f28a186.jpg",
+        "text": "$$\n\\frac { \\partial \\mathcal { F } ( I ) } { \\partial t } + ( i < \\xi , w > + D \\left\\| \\xi \\right\\| ^ { 2 } ) \\mathcal { F } ( I ) = 0\n$$",
+        "text_format": "latex",
+        "bbox": [
+            356,
+            135,
+            643,
+            166
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "This is a first order ordinary differential equation of the form $f ^ { \\prime } ( t ) + a f ( t ) = 0$ , which admits a known solution $f ( t ) = f ( 0 ) { e ^ { - a t } }$ . Thus, the solution of 11 is ",
+        "bbox": [
+            171,
+            175,
+            826,
+            205
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/85a5a0136fedc29c2976cd292af0cf1329cef1f0003a13060b18ce57d89ee58a.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { F } ( I ) = \\mathcal { F } ( I ) _ { 0 } e ^ { - ( i < \\xi , w > + D \\| \\xi \\| ^ { 2 } ) t } } \\\\ { = \\mathcal { F } ( I ) _ { 0 } e ^ { - i < \\xi , w > t } e ^ { - D t \\| \\xi \\| ^ { 2 } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            382,
+            229,
+            616,
+            275
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { F } ( I ) _ { 0 }$ denotes the initial condition of the advection diffusion equation in the frequency domain. In order to obtain a solution of 3 in the spatial domain, we calculate the inverse Fourier Transform ${ \\mathcal { F } } ^ { - 1 }$ of 12. The multiplication of two functions in the frequency domain is equivalent to their convolution in the spatial domain, i.e. ${ \\mathcal { F } } ( f * g ) = { \\mathcal { F } } ( f ) { \\mathcal { F } } ( g )$ . Hence, the inverse of both terms $\\mathcal { F } ( I ) _ { 0 } e ^ { - i < \\xi , w > t }$ and $e ^ { - D t \\| \\boldsymbol { \\xi } \\| ^ { 2 } }$ can be calculated separately: ",
+        "bbox": [
+            173,
+            286,
+            825,
+            362
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Multiplication by a complex exponential in the frequency domain is equivalent to a shift in the spatial domain : $\\bar { e } ^ { - i < \\xi , w \\bar { > } } \\mathcal { F } ( f ( \\bar { \\mathbf { x } } ) ) = \\mathcal { F } ( f ( \\mathbf { x } - w ) )$ , for $\\boldsymbol { v } \\in \\mathbb { R } ^ { 2 }$ . Thus, for the first term, ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ba531a1f044abcd42e6f2843fb8749e3c545fc64cfa2cb6c218f204d0f5a8720.jpg",
+        "text": "$$\n\\mathcal { F } ^ { - 1 } ( \\mathcal { F } ( I ) _ { 0 } e ^ { - ( i < \\xi , w > ) t } ) = I _ { 0 } ( \\mathbf { x } - w )\n$$",
+        "text_format": "latex",
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+        "page_idx": 14
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+    {
+        "type": "text",
+        "text": "For the second term, we use the fact that the Fourier Transform of a Gaussian function also is a Gaussian function, i.e. $\\begin{array} { r } { \\mathcal { F } \\big ( \\frac { 1 } { 2 \\pi \\sigma ^ { 2 } } e ^ { - \\frac { 1 } { 2 \\sigma ^ { 2 } } \\| \\mathbf { x } \\| ^ { 2 } } \\big ) = e ^ { - \\frac { 1 } { 2 } \\sigma ^ { 2 } \\| \\boldsymbol { \\xi } \\| ^ { 2 } } } \\end{array}$ . Identifying $\\sigma ^ { 2 }$ with $2 D t$ , we have: ",
+        "bbox": [
+            173,
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+            830,
+            489
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+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/39d72d35abb3d10beb5e25ff6d0314e9a54221ca15fade3e62b0b71d3c4d672d.jpg",
+        "text": "$$\n\\mathcal { F } ^ { - 1 } ( e ^ { - D t \\| \\boldsymbol { \\xi } \\| ^ { 2 } } ) = \\frac { 1 } { 4 \\pi D t } e ^ { - \\frac { 1 } { 4 D t } \\| \\mathbf { x } \\| ^ { 2 } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            379,
+            503,
+            619,
+            535
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "As has been stated above, the solution is a convolution of both previously calculated terms: ",
+        "bbox": [
+            169,
+            542,
+            769,
+            559
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d5181f94aeaa40708964d718f50eb9c04c1671bbb1d1b3db6da24454c5dc2246.jpg",
+        "text": "$$\n\\begin{array} { r } { I ( \\mathbf { x } , t ) = \\displaystyle \\int _ { \\mathbb { R } ^ { 2 } } \\frac { 1 } { 4 \\pi D t } e ^ { - \\frac { 1 } { 4 D t } \\| \\mathbf { y } \\| ^ { 2 } } I _ { 0 } ( \\mathbf { x } - w - \\mathbf { y } ) d y } \\\\ { = \\displaystyle \\int _ { \\mathbb { R } ^ { 2 } } \\frac { 1 } { 4 \\pi D t } e ^ { - \\frac { 1 } { 4 D t } \\| \\mathbf { x } - w - \\mathbf { y } \\| ^ { 2 } } I _ { 0 } ( \\mathbf { y } ) d \\mathbf { y } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            583,
+            661,
+            652
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "B ON THE GENERALIZATION IN SPACE AND TIME ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            604,
+            118
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "The ability of the model to adapt to other conditions should be evaluated on other regions. This, however, requires a complete study by itself and is beyond the scope of this paper. We, however, present below complementary experiments aimed at assessing the potential of the proposed model for forecasting SST on sequences distant in time and space from the ones used for training. ",
+        "bbox": [
+            174,
+            132,
+            825,
+            189
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "B.1 TEMPORAL DIMENSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            205,
+            383,
+            219
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In section 4, training has been performed on data from 2006 - 2015 and testing on the period 2016- 2017. In order to provide some indication of the model behavior on more distant time intervals between train and test data, we have performed experiments using the same regions (17 to 20) as in section 4, but using the period 2011 to 2017 for training and period 2006 to 2010 for testing. Figure B.1 shows the MSE curve on this test set, each point corresponding to the mean MSE on predictions performed on 6 days ahead the current date. The most important conclusion is probably that the MSE error remains in the same range for all these years. All the yearly error curve show a clear seasonal phenomenon with a higher prediction error during summer. A similar behavior has been observed when exchanging train and test data. ",
+        "bbox": [
+            173,
+            231,
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+            357
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/528806e279e8e0cafff934b99238f30f79d4eaee57852dbf4a070720a4a6a49c.jpg",
+        "image_caption": [
+            "Figure 6: Evaluation of our model’s accuracy in time on data from 2006 to 2010 using data from 2011 to 2017 for training. Regions 17 to 20 were used for both periods. Each day, we produce daily forecasts for 6 days ahead and calculate the associated mean square error. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            204,
+            372,
+            787,
+            595
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "B.2 SPATIAL DIMENSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            676,
+            362,
+            690
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In the experiments, the models have been trained and evaluated on selected regions (numbered 17 to 20 in Figure 4.1), considered as the most interesting for the observed dynamics. ",
+        "bbox": [
+            173,
+            702,
+            823,
+            731
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/0ab62c56b4419a4d94dd17969b76ac3bab26b6aa9a1c61cb38d05e3ef91e0077.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Test Regions 17&amp;18</td><td rowspan=1 colspan=1>Test Regions 8&amp;9</td></tr><tr><td rowspan=1 colspan=1>Model trained on Regions 17 &amp; 18</td><td rowspan=1 colspan=1>1.43</td><td rowspan=1 colspan=1>1.22</td></tr><tr><td rowspan=1 colspan=1>Model trained on Regions 8&amp;9</td><td rowspan=1 colspan=1>1.90</td><td rowspan=1 colspan=1>1.19</td></tr></table>",
+        "bbox": [
+            222,
+            742,
+            776,
+            787
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Evaluation of our model’s spatial generalization ability. We train our model on two distinct regions and calculate the MSE on both regions for each trained model. ",
+        "bbox": [
+            173,
+            797,
+            823,
+            827
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We describe below some results providing indications on how the model performs on regions different from the training ones. For these experiments, the model has been trained on regions 17 and 18 in Figure 4.1 and tested on two other regions (regions 8 and 9), and vice versa (trained on 8 and 9 and tested on 17 and 18). The two couples of regions have been selected so as to have different latitude and longitude. The underlying physical processes generating the data are known to be different in these regions: the overall motion in regions 17 and 18 is greater, and the difference between extreme temperature is larger, compared to regions 8 and 9. Experimental conditions are similar to the one described in section 4, i.e. 2006-2015 have been used for training and 2016-1017 for testing. ",
+        "bbox": [
+            174,
+            839,
+            825,
+            924
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            821,
+            132
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Results in Table B.2 show that the model generalizes reasonably well to unseen data from distant spatial regions, with a slight decrease in performance when training and test regions do not correspond. The performance loss is 0.47 for regions (17, 18) which show a strong dynamics, whereas it is only 0.03 for regions (8, 9) for which the dynamics are more stable. Most notably, MSE performance depends more on the region itself than on the train/ test conditions. Error is always higher in regions with strong dynamics (17, 18) than on more stable regions (8, 9) whatever the train/ test conditions are. Note that to further improve the results on distant data, it is possible to fine-tune the model using data from the studied regions. ",
+        "bbox": [
+            174,
+            138,
+            825,
+            251
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/7fa34cba44f56995c4ccd948262fe818c202d7bb2038a7cc5cbd04fe8d410b0f.jpg",
+        "image_caption": [
+            "Figure 7: Output for the 6 of May to the 9 of May 2016, Output , From top to bottom: target, our model prediction, our model flow "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            251,
+            154,
+            774,
+            445
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/4ffe62ca072ede6593044ce1af87b6350592816bc760f6a707c20a78f7a4aa63.jpg",
+        "image_caption": [
+            "Figure 8: Output for the 6 of January to the 9 of January 2016. From top to bottom: target, our model prediction, our model flow "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            263,
+            556,
+            766,
+            842
+        ],
+        "page_idx": 17
+    }
+]

parse/train/LSFCEb3GYU7/LSFCEb3GYU7.md

@@ -0,0 +1,509 @@
+# EMERGENT SYMBOLS THROUGH BINDING IN EXTERNAL MEMORY
+
+Taylor W. Webb   
+University of California Los Angeles Los Angeles, CA   
+taylor.w.webb@gmail.com   
+Ishan Sinha, Jonathan D. Cohen   
+Princeton University   
+Princeton, NJ
+
+# ABSTRACT
+
+A key aspect of human intelligence is the ability to infer abstract rules directly from high-dimensional sensory data, and to do so given only a limited amount of training experience. Deep neural network algorithms have proven to be a powerful tool for learning directly from high-dimensional data, but currently lack this capacity for data-efficient induction of abstract rules, leading some to argue that symbol-processing mechanisms will be necessary to account for this capacity. In this work, we take a step toward bridging this gap by introducing the Emergent Symbol Binding Network (ESBN), a recurrent network augmented with an external memory that enables a form of variable-binding and indirection. This binding mechanism allows symbol-like representations to emerge through the learning process without the need to explicitly incorporate symbol-processing machinery, enabling the ESBN to learn rules in a manner that is abstracted away from the particular entities to which those rules apply. Across a series of tasks, we show that this architecture displays nearly perfect generalization of learned rules to novel entities given only a limited number of training examples, and outperforms a number of other competitive neural network architectures.
+
+# 1 INTRODUCTION
+
+Human intelligence is characterized by a remarkable capacity to detect the presence of simple, abstract rules that govern high-dimensional sensory data, such as images or sounds, and then apply these to novel data. This capacity has been extensively studied by psychologists in both the visual domain, in tasks such as Raven’s Progressive Matrices (Raven & Court, 1938), and the auditory domain, in tasks that employ novel, artificial languages (Marcus et al., 1999).
+
+In recent years, deep neural network algorithms have reemerged as a powerful tool for learning directly from high-dimensional data, though many studies have now demonstrated that these models suffer from similar limitations as those faced by the earlier generation of neural networks: requiring enormous amounts of training data and tending to generalize poorly outside the distribution of those training data (Lake & Baroni, 2018; Barrett et al., 2018). This stands in sharp contrast to the ability of human learners to infer abstract structure from a limited number of training examples and then systematically generalize that structure to problems involving novel entities.
+
+It has long been argued that the human ability to generalize in this manner depends crucially on a capacity for variable-binding, that is, the ability to represent a problem in terms of abstract symbollike variables that are bound to concrete entities (Holyoak & Hummel, 2000; Marcus, 2001). This in turn can be broken down into two components: 1) a mechanism for indirection, the ability to bind two representations together and then use one representation to refer to and retrieve the other (Kriete et al., 2013), and 2) a representational scheme whereby one of the bound representations codes for abstract variables, and the other codes for the values of those variables.
+
+In this work, we present a novel architecture designed around the goal of having a capacity for abstract variable-binding. This is accomplished through two important design considerations. First, the architecture possesses an explicit mechanism for indirection, in the form of a two-column external memory. Second, the architecture is separated into two information-processing streams, one that maintains learned embeddings of concrete entities (in our case, images), and one in which a recurrent controller learns to represent and operate over task-relevant variables. These two streams only interact in the form of bindings in the external memory, allowing the controller to learn to perform tasks in a manner that is abstracted away from the particular entities involved. We refer to this architecture as the Emergent Symbol Binding Network (ESBN), due to the fact that this arrangement allows abstract, symbol-like representations to emerge during the learning process, without the need to incorporate symbolic machinery.
+
+We evaluate this architecture on a suite of tasks involving relationships among images that are governed by abstract rules. Across these tasks, we show that the ESBN is capable of learning abstract rules from a limited number of training examples and systematically generalizing these rules to novel entities. By contrast, the other architectures that we evaluate are capable of learning these rules in some cases, but fail to generalize them successfully when trained on a limited number of problems involving a limited number of entities. We conclude from these results that a capacity for variable-binding is a necessary component for human-like abstraction and generalization, and that the ESBN is a promising candidate for how to incorporate such a capacity into neural network algorithms.
+
+# 2 TASKS
+
+![](images/dd626285c79ee67e710196f3856936479741bdc56e6485c4e26bff446cfd34c7.jpg)  
+Figure 1: Abstract rule learning tasks. Each task involves generalizing rules to objects not seen during training. (a) Same/different discrimination task. (b) Relational match-to-sample task (answer is 2). (c) Distribution-of-three task (answer is 2). (d) Identity rules task (ABA pattern, answer is 1).
+
+We consider a series of tasks, each involving the application of an abstract rule to a set of images. For all tasks, we employ the same set of $n = 1 0 0$ images, in which each image is a distinct Unicode character (the specific characters used are shown in A.7). We construct training sets in which $m$ images are withheld (where $0 \leq m \leq n - o .$ , and $o$ is the minimum number of images necessary to create a problem in a given task) consisting of problems that employ only the remaining $( n -$ $m$ ) images, and then test on problems that employ only the $m$ withheld images, thus requiring generalization to novel entities. In the easiest generalization regime $\mathbf { \bar { \rho } } m = 0$ ) the test set contains problems composed of the same entities as observed during training (though the exact order of these entities differs). In the most extreme generalization regime, we evaluate models that have only been trained on the minimum number of entities for a given task, and then must generalize what they learn to the majority of the $n$ images in the complete set. This regime poses an extremely challenging test of the ability to learn to perform these tasks from limited training experience, in a manner that is abstracted away from the specific entities observed during training.
+
+The first task that we study is a same/different discrimination task (Figure 1a). In this task, two images are presented, and the task is to determine whether they are the same or different. Though this task may appear quite simple, it has been shown that the ability to generalize this simple rule to novel entities is actually a significant challenge for deep neural networks (Kim et al., 2018), a pattern that we also observe in our results.
+
+The second task that we consider is a relational match-to-sample (RMTS) task (Figure 1b), essentially a higher-order version of a same/different task. In this task, a source pair of objects is compared to two target pairs. The task is to identify the target pair with the same relation as the source pair; e.g., if the source pair contains two of the same object, to identify the target pair that contains two of the same object. It was initially believed that the ability to perform this task is not unique to humans (Premack, 1983), but it has now been shown that this ability depends on a visual entropy confound that arises from using large arrays of objects rather than simple pairs (Fagot et al., 2001). When the task is presented in a manner that does not allow this confound to be exploited (as is the case in our experiments), the ability to perform the task with novel entities appears to be unique to humans, and therefore is a good test of the human ability for abstract rule learning.
+
+Next we consider a task based on Raven’s Progressive Matrices (RPM; Raven & Court (1938)). RPM is a commonly used visual problem-solving task, and is one of the most widely used tests of fluid intelligence (Snow et al., 1984), the ability to reason and make inferences in a novel domain (as opposed to crystallized intelligence, the ability to solve familiar tasks). In this task, a $3 \times 3$ array of figural elements is presented, in which the elements are governed by a simple rule, or set of rules, with the lower right element of the array left blank. The task is to infer the rule that governs the elements in the array, and then use that rule to select from among 8 candidate completions. Many of the rules that govern RPM problems are relations involving sets. One such rule is sometimes referred to as distribution-of-three (Carpenter et al., 1990), according to which the same set of three elements (e.g. a triangle, square, and circle) will appear in each row, though the order doesn’t matter. The task in this case is simply to identify the set, determining which element is missing from the final row, and locating this element among the choices.
+
+Though multiple RPM-inspired datasets have recently been proposed (Barrett et al., 2018; Zhang et al., 2019), in this work we choose to strip away unnecessary complexity, focusing on $2 \times 3$ arrays governed by a single rule (Figure 1c), in order to focus specifically on the capacity for generalization of an abstract rule to novel entities. We find that, even in this simplified setting, this form of generalization is extremely challenging.
+
+The final task that we consider is a visual version of the identity rules task studied by Marcus et al. (1999). In this task, an abstract pattern (e.g. ABA or ABB) must be inferred from a sequence of elements. For instance, in the original study, the following sequence ‘ga ni ga, li na li, wo fe wo’ is governed by an ABA rule, whereas the sequence ‘ga ni ni, li na na, wo fe fe’ is governed by an ABB rule. This study played an important role in debates concerning the presence of algebraic rulelike processes in human cognition, because it demonstrated that even 7-month-old human infants are capable of detecting this abstract regularity and generalizing it to novel entities, whereas neural networks tend to overfit to the specific entities involved and fail to generalize the rule.
+
+In our implementation, we use visual images rather than sounds, and present the task as a $2 \times 3$ array (Figure 1d). In this task, each problem is governed by either an ABA, ABB, or AAA rule. The task is to determine which of these patterns is present in the first row, and then to apply that pattern by selecting an element from a set of 4 choices to complete the second row.
+
+For all four tasks, we consider generalization regimes in which some number of images $( m \in$ $\{ 0 , 5 0 , 8 5 , 9 5 \}$ out of $n = 1 0 0$ ) are withheld from training. For the same/different discrimination task, on which only two images are necessary to construct a problem, we also consider the case in which $m = 9 8$ (such that the training set consists of problems involving only $n - m = 2$ images, the minimum number necessary to construct the task).
+
+In most settings, we construct training sets consisting of $1 0 ^ { 4 }$ problems. This is a tiny fraction of all possible problems (on the order of $\mathrm { { \bar { 1 0 } ^ { 9 } } }$ when the multiple choice options are considered)1 Thus, even in the easiest generalization regime $( m = 0$ ) this is an extremely small amount of training data relative to the size of the task space. In the most extreme regimes, in which $m \geq 9 5$ , it is only possible to construct a few hundred problems, resulting in even more limited training experience.
+
+# 3 APPROACH
+
+For each task, we treat the problem as a sequence of images $\pmb { x } _ { t = 1 } . . . \pmb { x } _ { t = T }$ , with an associated target $\textbf {  { y } }$ . In the same/different discrimination task, there are $T \ = \ 2$ images, and $\textbf {  { y } }$ is a binary target indicating whether the images are the same or different. In the RMTS task, there are $T = 6$ images, consisting of the source pair followed by two target pairs, and $\textbf {  { y } }$ is a binary target indicating which target pair matches the source pair. In both the distribution-of-three task and the identity rules task, there are $T = 9$ images, consisting of the three entries in the first row, the two non-empty entries in the second row, and the four multiple-choice options, and $\textbf {  { y } }$ is a four-way classification target, indicating which of the multiple-choice options is correct.
+
+All images are $3 2 \times 3 2$ grayscale images containing a single Unicode character. For each problem, we first process each image independently by a shared encoder $f _ { e }$ , generating image embeddings $z _ { t = 1 } , . . . z _ { t = T }$ , and then pass these embeddings to a sequential model component $f _ { s }$ that generates a response (either through a sigmoid output layer for tasks with a binary target, or a softmax layer for tasks with a four-way classification target). The sequential component is either the ESBN or one of a number of alternative architectures described below. We use the same encoder architecture $f _ { e }$ (detailed in A.3) for all models. All components are trained end-to-end, including the encoder.
+
+# 3.1 TEMPORAL CONTEXT NORMALIZATION
+
+We use temporal context normalization (TCN), recently shown to improve out-of-distribution generalization in relational reasoning tasks (Webb et al., 2020). TCN is similar to batch normalization, but, instead of normalizing over the batch dimension, normalizes over a task-relevant temporal window. This has the effect of preserving information about the relations between the entities present within this window (e.g. the size of those entities relative to one another), resulting in better generalization of learned relations to novel contexts (i.e. out-of-distribution).
+
+We found that TCN significantly improved generalization for all of the models on all of the tasks studied in the present work2. Therefore, the primary results we report all incorporate this technique ( A.5.1 includes a comparison of the performance of all models on all tasks with and without TCN). Specifically, we applied TCN to the embeddings $z _ { t = 1 } , . . . z _ { t = T }$ extracted by the encoder. Webb et al. (2020) also reported that it is sometimes useful to apply TCN separately to different components of a sequence. We found that this was the case for the RMTS task that we studied, in which we found it useful to apply TCN separately to the embeddings for the source pair and each target pair. For all of the other tasks that we studied, TCN was applied over the entire sequence for each problem.
+
+# 3.2 EMERGENT SYMBOL BINDING NETWORK
+
+![](images/476acc9f1cb7fbaf9e923b8b24ee79c4de4f1eee886b14ed4777c92f158a40c5.jpg)  
+Figure 2: Emergent Symbol Binding Network. $f _ { s }$ consists of an LSTM controller plus output layers for $\hat { \pmb { y } }$ , $k _ { w }$ , and $g$ (not shown). $f _ { e }$ is a multilayer feedforward encoder that translates an image $_ { \textbf { \em x } }$ into a low-dimensional embedding $_ { z }$ . These two pathways only interact indirectly via a key/value memory.
+
+The ESBN (Figure 2; Algorithm 1) uses an LSTM controller $( f _ { s } )$ with a differentiable external memory that is explicitly separated into keys $( M _ { k } )$ and values $( M _ { v } )$ . At each time step $t$ , a key/value pair is written to memory. The keys written to memory, $k _ { w _ { t } }$ , are generated by an output layer from the LSTM controller, and the values are the individual input embeddings, ${ \boldsymbol { z } } _ { t }$ , of the input sequence, unmodified by the LSTM. Our hypothesis was that factoring the model into two separate information processing streams would allow the LSTM to learn how to represent abstract variables in the keys it generates, which could then be explicitly bound to associated values (image embeddings) learned by the separate encoder network $( f _ { e } )$ , allowing the ESBN to employ a form of indirection.
+
+To retrieve keys from memory, similarity scores are computed by comparing (via a dot product) the image embedding ${ \boldsymbol { z } } _ { t }$ to all of the values in memory $M _ { v _ { t - 1 } }$ . These similarity scores are passed through a softmax nonlinearity to generate weights ${ \pmb w } _ { k _ { t } }$ , and passed through a sigmoid nonlinearity (with learned gain and bias parameters, $\gamma$ and $\beta$ ) to generate confidence values $c _ { k _ { t } }$ (one weight and confidence value per entry in memory). The weights are used to compute 1) a weighted sum of all keys in memory $M _ { k _ { t - 1 } }$ , and 2) a weighted sum of all associated confidence values $c _ { k _ { t } }$ . Finally, the retrieved key and associated confidence value are concatenated and multiplied by a learned sigmoidal gate $g _ { t }$ to form $\boldsymbol { k } _ { \boldsymbol { r } _ { t } }$ , the input to the LSTM controller at the next time step.
+
+<table><tr><td>Algorithm 1: Emergent Symbol Binding Network. （ll) indicates the concatenation of a vector and a scalar, forming a vector with one additional dimension.{,} indicates the concatenation of a matrix and a vector,forming a matrix with one additional row. o() is the logistic sigmoid function.</td></tr><tr><td>ht=0←0; Mkt=0←{}; Mut=o←{; for t in1...Tdo Zt←fe(xt)；</td></tr><tr><td>yt，gt，kwt，ht ←fs(ht-1,krt-1); if t is 1 then krt←0; else</td></tr><tr><td>Wkt ← softmax(Mut-1·zt); Ckt← σ(γ(Mvt-1·zt)+β);</td></tr><tr><td>t-1 krt←gt M Wkt(i)(Mkt-1(i)llckt (i)) ; i=1</td></tr></table>
+
+# 3.3 ALTERNATIVE ARCHITECTURES
+
+The simplest alternative architecture that we consider is an LSTM (Hochreiter & Schmidhuber, 1997) without external memory. We pass the low-dimensional embeddings $z _ { t = 1 } , . . . z _ { t = T }$ directly to the LSTM, and generate a prediction $\hat { \pmb { y } }$ by passing the final hidden state through an output layer.
+
+Next we consider two alternative external memory architectures: the Neural Turing Machine (NTM; Graves et al. (2014)) and Metalearned Neural Memory (MNM; Munkhdalai et al. (2019)). This comparison allows us to determine to what extent our results depend on the specific details of the ESBN’s external memory, and, in particular, the separation between its two informationprocessing pathways vs. the mere presence of an external memory. Our NTM implementation consists of an LSTM controller (which takes image embeddings as input, and generates a prediction $\hat { \textbf { \textit { y } } }$ as output) that interacts with an external memory using both content-based and location-based read/write mechanisms. Our MNM implementation employs the publicly available code from the original paper, modified so as to employ the same encoder architecture and TCN procedure as the other architectures that we test. Just as with the ESBN, we allow both of these architectures an extra time step to process the information retrieved from memory following the final input.
+
+We also consider the Relation Net (RN; Santoro et al. (2017)), an architecture that has proven to be an effective approach for a wide range of relational reasoning tasks. In our implementation, we treat the low-dimensional image embeddings as individual ‘objects’ in the RN framework, using a shared MLP to process all pair-wise combinations of these embeddings, summing the outputs from this MLP, and then passing them to another MLP that generates the prediction $\hat { \boldsymbol y }$ .
+
+We also compare our model against the Transformer (Vaswani et al., 2017), an architecture originally developed in the domain of natural language processing, but has proven to be effective for a wide range of sequential data, and demonstrated a capacity for some degree of extrapolation (Saxton et al., 2019). After applying the transformer architecture to the sequence of image embeddings (allowing self-attention between these embeddings), we compute an average of the (transformed) embeddings, and then pass this to a small MLP that then generates the task output $\hat { \pmb { y } }$ .
+
+Finally, we consider the PrediNet (Shanahan et al., 2019). PrediNet was designed with the goal of being ‘explicitly relational,’ and has been shown to be effective at generalizing learned relations to novel entities. We apply the PrediNet’s multi-head attention over the 1D temporal sequence of image embeddings (as opposed to applying attention over a 2D image, as in the original work), and then pass the output of the PrediNet module to a small MLP that generates $\hat { \boldsymbol y }$ .
+
+![](images/4779e2f7fc5d68739eae072614bbe070f56d2a84d5b7a937fd32aecfc5d1091a.jpg)  
+Figure 3: Results for all four tasks with $m$ objects withheld (out of $n = 1 0 0$ ) during training. Results reflect test accuracy averaged over 10 trained networks ( $\pm$ the standard error of the mean).
+
+Figure 3 shows the generalization results for all four tasks. Our primary finding is that the ESBN displayed nearly perfect generalization $( \geq 9 5 \% )$ of the learned rule in all four tasks, even when trained on a very limited number of problems (just hundreds of problems, in the case of the most extreme generalization regimes) involving a limited number of entities (as few as just two entities, in the case of the same/different task), and tested on completely novel entities. Some of the alternative architectures that we evaluated showed a surprising capability to generalize to novel entities in some tasks as seen, for instance, in the generalization results for the Transformer and RN on the same/different and RMTS tasks (though we note that all architectures incorporate TCN, without which generalization is significantly worse, as shown in A.5.1). Nevertheless, none of these alternative architectures were able to generalize what they learned in the most extreme generalization regimes, whereas the ESBN performed comparably well across all regimes.
+
+Notably, the RN performed very poorly on the distribution-of-three and identity rules tasks, even in the easiest regime $( m = 0$ ). We speculate that this results from the fact that the RN is biased toward pair-wise relations, whereas these tasks are both based on a ternary relation. It is possible to represent this ternary relation as a combination of pair-wise relations, but doing so requires a more complex strategy and therefore likely more training data. We include results in A.5.2 demonstrating that the RN is capable of successfully generalizing in this task (though not in the most extreme regimes) when trained on an order of magnitude more data $1 0 ^ { 5 }$ instead of $1 0 ^ { 4 }$ examples). We also present results for the Temporal Relation Network (Zhou et al., 2018), an RN variant that incorporates ternary relations via subsampling, though we find that this doesn’t help as much as increasing the amount of training data.
+
+![](images/a290863da2658abe383e52e522aff79c3ca9a9e6836ac6dd48ca5ac230ec8b77.jpg)  
+Figure 4: Training accuracy time courses for all models on the $m = 0$ regime. Each time course reflects an average over 10 trained networks. Error bars reflect the standard error of the mean.
+
+In addition to requiring a very small amount of training data and generalizing systematically to novel entities, the ESBN also requires very little training time. Figure 4 shows training accuracy time courses for the RMTS, distribution-of-three, and identity rules tasks for all models 3. The ESBN converged to nearly perfect training accuracy within 100 to 200 training updates on all four tasks, whereas the other models required thousands, or even tens of thousands of training updates to reach convergence4.
+
+We also performed some experiments to better understand how the ESBN operates, and why it was so effective. First, we tested whether the systematic generalization exhibited by the ESBN was dependent on the use of convolutional layers in the encoder, which naturally confer a significant degree of generalization in tasks that involve shape recognition. We found that the ESBN generalized to novel entities comparably well when using either an MLP encoder or a random projection (see A.5.4 for details), suggesting that the ESBN is capable of generalizing learned rules to any arbitrary set of entities, regardless of how those entities are encoded. For comparison, we also performed the same experiments with the Transformer (the best performing alternative architecture on our tasks) and found that, by contrast, its performance was significantly impaired by the use of a random projection instead of a convolutional encoder.
+
+Second, we performed an ablation experiment on the confidence value appended to retrieved memories. We found that ablation of these confidence values impaired the ESBN’s performance in both the same/different and RMTS tasks, but not the distribution-of-three or identity rules tasks (see A.5.5 for details). A likely reason for this result is that the distribution-of-three and identity rules tasks only require retrieval of the best match from memory, whereas the same/different and RMTS tasks require a sense of how good of a match that memory is, which is exactly the information that the confidence value conveys. This dissociation mirrors the distinction sometimes made in cognitive psychology between recollection and familiarity (Yonelinas, 2001).
+
+Third, we performed an analysis of the key representations learned by the controller. We hypothesized that the controller would learn to represent abstract variables in the keys that it writes to memory, and that these representations therefore shouldn’t vary based on the values to which they are bound. This analysis revealed a high degree of overlap between the keys written during training and test (involving entirely different entities), suggesting that this was indeed the case (see A.6 for details). This ability to arbitrarily bind values to variables, without affecting the representations of those variables, is a key property of symbol-processing systems, and is likely the basis of the strong systematic generalization exhibited by the ESBN.
+
+# 5 RELATED WORK
+
+There have been a number of proposals for augmenting neural networks with an external memory. An influential early line of work, Complementary Learning Systems (McClelland et al., 1995), proposed that neural systems benefit from having components that learn on different time scales, and argues that this combination allows neural networks both to learn general, abstract structure (using standard learning algorithms) and to rapidly encode arbitrary new items (using an external memory). In recent years, there have been a number of proposals for how to implement the latter efficiently, including Fast Weights (Ba et al., 2016a), the NTM and closely related Differentiable Neural Computer (Graves et al., 2016), and the Differentiable Neural Dictionary (DND; Pritzel et al. (2017)). Our external memory approach is most closely related to the DND, which also involves a two-column key/value memory. Variations on key/value memory have also been employed in other more recently proposed approaches, such as the Memory Recall Agent (Fortunato et al., 2019) and the Dual-Coding Episodic Memory (Hill et al., 2020), where it afforded various benefits in terms of generalization. One critical difference between our model and this previous work is that the ESBN’s controller is forced to interact with perceptual inputs only indirectly through its memory, a design decision that we argue is crucial to its ability to systematically generalize what it learns.
+
+It is worth noting that architectures such as Fast Weights and the NTM are, in principle, capable of implementing variable-binding, though it is a separate question whether such a strategy will result from learning in any particular task. Along these lines, a recent study from Chen et al. (2019) found that both of these architectures are capable of generalizing learned structure to novel entities when allowed a sufficiently dense sampling of the space of potential objects (the ‘objects’ in their study were randomly sampled 50-dimensional vectors). This contrasts with our findings, in which the NTM performed poorly when trained on far fewer samples from a much higher-dimensional space (in the $m = 9 5$ regime). This suggests that indirection and variable-binding, though possible in principle for architectures such as the NTM, do not emerge in practice when given only a limited amount of training experience, whereas this capacity is explicitly built into the ESBN.
+
+At a high level, the idea of factoring a model into two distinct information processing streams, one that codes abstract task-relevant variables or roles and one that codes concrete entities, has been explored before. Kriete et al. (2013) proposed the PBWM Indirection model, in which one population of neurons acted as a pointer to another population of neurons by gating its activity, and showed that this model enabled a significant degree of generalization to novel role/filler bindings. Whittington et al. (2019) proposed the Tolman-Eichenbaum machine, a model that is capable of learning abstract relational structure (such as 2D spatial maps), and showed that this model captured a number of phenomena relating to grid cells and place cells. Russin et al. (2019) proposed Syntactic Attention, an architecture involving separate pathways for processing syntax vs. semantics, and showed that this approach was capable of a significant degree of compositional generalization on the challenging SCAN dataset. Relative to this previous work, our central contribution is the development of a simple model that can learn abstract rules directly from high-dimensional data (images), exploiting this same high-level idea to enable nearly perfect generalization of those rules to novel entities.
+
+There has been extensive modeling work focusing on some of the tasks that we study. The recent development of two datasets modeled after Raven’s Progressive Matrices, Procedurally Generated Matrices (Barrett et al., 2018), and RAVEN (Zhang et al., 2019), has spurred the development of models that are capable of solving RPM-like problems (Jahrens & Martinetz, 2020; Wu et al., 2020). However, these models typically require very large training sets (on the order of $1 0 ^ { 6 }$ training examples), and largely fail to generalize outside of the specific conditions under which they are trained, whereas the ESBN exhibits the ability to learn rapidly and generalize out-of-distribution.
+
+There have also been a number of models proposed to account for the human ability to rapidly learn identity rules (Alhama & Zuidema, 2019). Though some of these models achieved significant generalization of learned identity rules to novel entities, they did so mostly through the inclusion of highly task-specific mechanisms. By contrast, our aim in the present work was to present a general approach that could be applied to a wider range of tasks.
+
+Finally, there have been a number of recent proposals for so-called ‘neurosymbolic’ models, incorporating elements from both the neural network and symbolic modeling frameworks (Mao et al., 2019; Nye et al., 2020). Though we have emphasized the notion of ‘emergent symbols’ in the present work, we stress that this is quite distinct from neurosymbolic modeling efforts since we do not explicitly incorporate any symbolic machinery into the ESBN. Instead, our approach was to show how the functional equivalent of symbols can emerge in a neural network model with an appropriate architecture and binding mechanism.
+
+# 6 DISCUSSION
+
+# 6.1 LIMITATIONS AND FUTURE WORK
+
+One open question is whether the strict division between the two information processing streams in the ESBN is necessary, and whether it limits the sorts of relations and rules that it can learn. In future work, it may be desirable to soften this division, for instance by encouraging it in a regularization term, rather than strictly enforcing it architecturally.
+
+A second limitation is that the tasks we study are not as complex as other similar tasks that have recently been studied, such as the two recently proposed RPM-like benchmarks (Barrett et al., 2018; Zhang et al., 2019). In the present work, we intentionally stripped away some of this complexity in order to make progress on the issue of out-of-distribution generalization. Extending the ESBN to more complex tasks will likely require the incorporation of visual attention mechanisms to enable selective sequential processing of individual elements within a scene. There are many recently proposed approaches for doing this (Gregor et al., 2015; Locatello et al., 2020). In future work, we look forward to extending the ESBN in this manner and testing it on more complex tasks.
+
+# 6.2 RELATION TO WORK IN NEUROSCIENCE
+
+It is worth considering how the present work relates to pre-existing theories of how the brain might implement variable-binding. Classic proposals for variable-binding in neural systems emphasize dynamic binding of representations, either by computing the tensor product between those representations (Smolensky, 1990), or by establishing synchronous activation between two pools of units (Hummel & Holyoak, 1997). An alternative proposal is that variable-binding is accomplished via semi-permanent synaptic changes in the hippocampus, relying on the same mechanism that plays a central role in episodic memory (Cer & O’Reilly, 2006). This approach relies on contextual information and retrieval processes to prevent potential interference between conflicting memories, rather than explicit unbinding mechanisms. Our model is more in line with the latter account, since it does not possess an unbinding mechanism. As such, our model can be seen as part of a recent trend toward the reinterpretation of putatively working memory functions in terms of episodic memory (Beukers et al., 2020).
+
+# 7 CONCLUSION
+
+In this work, we have presented a model of abstract rule learning based on a novel architecture, the ESBN, and shown that this model is capable of rapidly learning abstract rules directly from images given only a small amount of training experience, and then successfully generalizing those rules to novel entities. Key to the model’s performance is its separation into two streams that only interact through indirection, allowing the ESBN to learn tasks in a manner that is abstracted away from the specific entities involved, and resulting in the emergence of symbol-like representations. We believe that these results suggest that such a variable-binding capacity is an essential ingredient for achieving human-like abstraction and generalization, and hope that the ESBN will be a useful tool for doing so.
+
+# ACKNOWLEDGMENTS
+
+We would like to thank Zachary Dulberg, Steven Frankland, Randall O’Reilly, Alexander Petrov, and Simon Segert for their helpful feedback and discussions.
+
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+
+# A APPENDIX
+
+# A.1 CODE AVAILABILITY
+
+All code, including code for dataset generation, model implementation, training, and evaluation, is available on GitHub.
+
+# A.2 DATASET GENERATION
+
+In this section, we provide details on the dataset generation process for all tasks. In all of our simulations, a dataset was generated from scratch (according to the procedures described below) at the beginning of each training run, such that different runs involved different datasets, though the statistics were the same across these datasets. We did this to prevent the possibility that our results would reflect biases present in a particular dataset.
+
+Table 1: Training and test set sizes for the same/different discrimination task.   
+
+<table><tr><td></td><td>m=0</td><td>m= 50</td><td>m= 85</td><td>m= 95</td><td>m=98</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Training</td><td>18,810</td><td>4,900</td><td>420</td><td>40</td><td>4</td></tr><tr><td>Test</td><td>990</td><td>4,900</td><td>10,000</td><td>10,000</td><td>10,000</td></tr></table>
+
+# A.2.1 SAME/DIFFERENT DISCRIMINATION
+
+Given $n = 1 0 0$ total images, with $m = 0$ withheld during training, there are $n ^ { 2 } = 1 0 ^ { 4 }$ possible same/different problems. To prevent the potential for networks to be biased by the fact that the overwhelming majority of these are ‘different’ problems, we created balanced datasets by including duplicates of the ‘same’ problems. Specifically, we randomly sampled (with replacement) $n ( n - 1 )$ of the $n$ unique ‘same’ trials and combined them with the $n ( n - 1 )$ unique ‘different’ trials, resulting in $2 n ( n - 1 ) = 1 9$ , 800 total problems. We reserved 990 of these problems for test, yielding training sets including 18, 810 problems (ensuring that duplicates of the same problem did not appear in both the training and test sets).
+
+We followed a similar procedure for the other regimes, generating balanced datasets by duplicating the ‘same’ problems when necessary. These datasets incorporated either all of the problems that resulted from this procedure (given the $n { - } m$ images available for training, or the $m$ images available for test), or $1 0 , 0 0 0$ problems, whichever was smaller. The exact size of each of these datasets is shown in Table 1.
+
+# A.2.2 RMTS
+
+Table 2: Training and test set sizes for the RMTS task.   
+
+<table><tr><td></td><td>m=0</td><td>m= 50</td><td>m=85</td><td>m= 95</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Training</td><td>10,000</td><td>10,000</td><td>10,000</td><td>480</td></tr><tr><td>Test</td><td>10.000</td><td>10,000</td><td>10,000</td><td>10,000</td></tr></table>
+
+For the RMTS task, we constructed balanced training and test sets ensuring that there were an equal number of problems with a ‘same’ vs. ‘different’ source pair. These datasets contained either 10, 000 problems, or the minimum number of problems possible in a given regime, whichever was smaller (Table 2). For most regimes, 10, 000 problems constitutes a tiny fraction of the full space of possible problems (ranging from $1 0 ^ { 9 }$ for the $m = 0$ regime to $1 0 ^ { 5 }$ for the training set in the $m = 8 5$ regime), and thus there was no need to duplicate problems to achieve balanced datasets. For the $m = 9 5$ regime, there are only 480 possible training problems, which happen to include the same number of ‘same’ and ‘different’ trial types.
+
+# A.2.3 DISTRIBUTION-OF-THREE
+
+Table 3: Training and test set sizes for the distribution-of-three task.   
+
+<table><tr><td></td><td>m=0</td><td>m= 50</td><td>m=85</td><td>m= 95</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Training</td><td>10,000</td><td>10,000</td><td>10,000</td><td>360</td></tr><tr><td>Test</td><td>10,000</td><td>10,000</td><td>10,000</td><td>10,000</td></tr></table>
+
+For the distribution-of-three task, we generated problems by randomly selecting three of the available images in a given regime (either $n - m$ images during training, or $m$ images during test), and then randomly sampling two permutations of those images for the two rows (allowing the possibility that the same permutation appears in both rows) of the $2 \times 3$ matrix. We then randomly selected a fourth image to appear with the other three as possible answers, and randomly permuted these four answer choices. When taking into account the identity of this fourth image, and the permutation of the answer choices, the number of unique distribution-of-three problems in the $m = 0$ regime is on the order of $1 0 ^ { 1 0 }$ .
+
+For most regimes, we randomly created training and test sets consisting of 10, 000 randomly sampled problems. For the $m = 9 5$ regime, the training set consisted of 360 problems (the total number of unique problems possible in this regime when not considering the identity and order of the answer choices, which were randomly selected).
+
+# A.2.4 IDENTITY RULES
+
+Table 4: Training and test set sizes for the identity rules task.   
+
+<table><tr><td></td><td>m=0</td><td>m= 50</td><td>m=85</td><td>m=95</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Training</td><td>10,000</td><td>10,000</td><td>10,000</td><td>8,640</td></tr><tr><td>Test</td><td>10,000</td><td>10,000</td><td>10,000</td><td>10,000</td></tr></table>
+
+For the identity rules task, we constructed datasets with an approximately balanced (through uniform random sampling) number of ABA, ABB, and AAA problems. These datasets consisted of either $1 0 , 0 0 0$ problems, or the minimum number of possible problems in a given regime, whichever was smaller (Table 4). For the training set in the $m = 9 5$ regime, datasets consisting of 8, 640 problems were constructed from the 7, 200 possible unique problems in this regime, by duplicating the AAA problems to match the number of ABA/ABB problems. For all other datasets, $1 0 , 0 0 0$ problems constituted a small fraction of the total number of possible problems (ranging from $1 0 ^ { \bar { 9 } }$ for the $m = 0$ regime to $1 0 ^ { 6 }$ for the training set in the $m = 8 5$ regime), and no duplication was necessary to achieve balanced problem types.
+
+# A.3 IMPLEMENTATION DETAILS FOR ALL MODELS
+
+# A.3.1 ENCODER
+
+We used the same feedforward encoder architecture to process each of the images in a sequence $\pmb { x } _ { t = 1 } . . . \pmb { x } _ { t = T }$ , generating low-dimensional embeddings $z _ { t = 1 } , . . . z _ { t = T }$ that were then passed to the core sequential component of each model (either the ESBN or one of the alternative architectures)5. This encoder consisted of three convolutional layers, each with 32 channels, a $4 \times 4$ kernel, and a stride of 2, followed by two fully-connected layers with 256 units and 128 units respectively. All layers used ReLU nonlinearities. All weights were initialized using a Kaiming normal distribution (He et al., 2015), and all biases were initialized to 0.
+
+# A.3.2 TASK OUTPUT LAYER
+
+All models had an output layer for generating $\hat { \boldsymbol y }$ . The number of units and the nonlinearity depended on the task. For the same/different and RMTS tasks, the output layer had 1 unit and a sigmoid nonlinearity (producing a number between 0 and 1 to code for ‘same’ vs. ‘different’, or pair 1 vs. pair 2). For the distribution-of-three and identity rules tasks, the output layer had 4 units and a softmax nonlinearity (to select 1 of the 4 answer choices). The weights of the output layer were initialized using an Xavier normal distribution (Glorot & Bengio, 2010), and the biases were initialized to 0.
+
+# A.3.3 ESBN
+
+The details of the ESBN’s operations are given in Algorithm 1. The LSTM controller had 1 layer with 512 units, and employed the standard tanh nonlinearities and sigmoidal gates. The controller also had output layers for $k _ { w }$ (256 units with a ReLU nonlinearity), $g$ (1 unit with a sigmoid nonlinearity), and $\hat { \boldsymbol y }$ . The input to the controller at each time step was $k _ { r }$ , the key retrieved from memory at the previous time step (along with the associated confidence value, $c _ { k } .$ ). At the beginning of each sequence, $k _ { r }$ and the controller’s hidden state $^ { h }$ were initialized to 0.
+
+After processing a full sequence, the ESBN was allowed an additional time step for the controller to process the retrieved key associated with the final input. After this additional time step, the final hidden state of the LSTM was passed through the task output layer to generate the prediction $\hat { \pmb { y } }$ . We note that it is also possible to retrieve values from memory (from $M _ { v }$ ) using a similar procedure and then decode these values to make predictions in image space (Sinha et al., 2020), but in the present work we focus only on the classification component of the model.
+
+The input weights for the LSTM controller were initialized using an Xavier normal distribution with a gain value of $5 / 3$ . The weights for the LSTM’s gates, as well as the weights for the output layer for the gate $g$ , were initialized with an Xavier normal distribution (with a gain of 1). The weights for the output layer that produced $k _ { w }$ were initialized using a Kaiming normal distribution. All biases were initialized to 0. The parameters $\gamma$ and $\beta$ were initialized to 1 and 0 respectively.
+
+# A.3.4 LSTM
+
+The LSTM architecture had 1 layer with 512 units. Image embeddings were passed to the LSTM as a sequence, after which $\hat { \pmb { y } }$ was generated through a task output layer. The LSTM’s hidden state was initialized to 0 at the beginning of each sequence. The LSTM’s weights were initialized using the same scheme as the LSTM controller in the ESBN (using an Xavier normal distribution, with a gain of $5 / 3$ for the input weights and 1 for the gates), and biases were initialized to 0.
+
+# A.3.5 NTM
+
+The NTM had an LSTM controller (1 layer with 512 units). The LSTM’s hidden state was initialized to 0 at the beginning of each sequence, and the LSTM’s parameters were initialized in the same way as the LSTM architecture and the controller for the ESBN. The NTM had one write head and one read head. The read head had the following output layers:
+
+1. read key: 256 units, tanh nonlinearity, weights initialized using an Xavier normal distribution with a gain of $5 / 3$ .   
+2. key strength: 1 unit, softplus nonlinearity, weights initialized using a Kaiming normal distribution.   
+3. interpolation gate: 1 unit, sigmoid nonlinearity, weights initialized using an Xavier normal distribution with a gain of 1.   
+4. shift weights: 3 units (corresponding to the allowable shifts $- 1 , 0$ , and 1), softmax nonlinearity, weights initialized using an Xavier normal distribution with a gain of 1.
+
+The write head had the following output layers:
+
+1. erase vector: 256 units, sigmoid nonlinearity, weights initialized using an Xavier normal distribution with a gain of 1.   
+2. add vector: 256 units, tanh nonlinearity, weights initialized using an Xavier normal distribution with a gain of $5 / 3$ .   
+3. write key: 256 units, tanh nonlinearity, weights initialized using an Xavier normal distribution with a gain of $5 / 3$ .   
+4. key strength: 1 unit, softplus nonlinearity, weights initialized using a Kaiming normal distribution.   
+5. interpolation gate: 1 unit, sigmoid nonlinearity, weights initialized using an Xavier normal distribution with a gain of 1.   
+6. shift weights: 3 units (corresponding to the allowable shifts $- 1 , 0$ , and 1), softmax nonlinearity, weights initialized using an Xavier normal distribution with a gain of 1.
+
+All biases for these output layers were initialized to 0. The NTM used these outputs to interact with its external memory, employing all of the location- and content-based mechanisms described in the original work (Graves et al., 2014). Cosine similarity was used as a similarity measure for the content-based mechanisms. The memory matrix had 10 rows of size 256. The initial state of the memory at the beginning of each sequence was learned. The learned initial state was initialized (at the beginning of training) using an Xavier normal distribution.
+
+The input to the LSTM controller at each time step consisted of the image embedding corresponding to that time step and the read vector from the previous time step. At the beginning of each sequence, the read vector, read weights, and write weights were initialized to 0. Just as with the ESBN, the NTM was allowed an additional time step to process the read vector retrieved from memory after observing the final image embedding, after which $\hat { \pmb { y } }$ was generated through an output layer from the LSTM controller.
+
+# A.3.6 MNM
+
+We implemented the MNM using publicly available code released with the original publication (Munkhdalai et al., 2019). Specifically, we used the version of MNM that employs a learned local update (‘MNM-p’ in the original paper). Before passing the images in our tasks to the MNM model, we applied the same encoder and TCN procedure used for the other architectures that we tested. Other than this modification, the original implementation, including all architectural hyperparameters, was unmodified.
+
+# A.3.7 RN
+
+The RN implementation consisted of two MLPs. The first MLP (used to process all pair-wise combinations of image embeddings) had a hidden layer of size 512 and an output layer of size 256. The outputs from the first MLP were summed, and then passed to the second MLP, which had a hidden layer of size 256 and an output layer for generating $\hat { y }$ . All layers (except the output layer) used ReLU nonlinearities. All weights were initialized using a Kaiming normal distribution (except the output layer, which was initialized according to the description in A.3.2), and all biases were initialized to 0. Before passing the image embeddings to the first MLP, they were appended with a tag (an integer from 0 to $T - 1$ ) indicating their position in the input sequence.
+
+# A.3.8 TRN
+
+The TRN employs two key design decisions intended to prevent the combinatorial explosion that would naturally result from the inclusion of n-ary relations:
+
+1. Only considering temporally ordered, non-redundant sets (whereas the original RN considers all possible pairs of objects, including both permutations of the same pair, and the pair of each object with itself).   
+2. Subsampling from these sets.
+
+We found that it was computationally feasible to implement a TRN with ternary relations in our tasks by only using (1), without the need to subsample. Thus, our implementation considers all temporally ordered, non-redundant sets of two and three.
+
+Each pair of image embeddings was processed by an MLP with a hidden layer of size 512 and an output layer of size 256. The outputs of this MLP for all pairs were then summed, and processed by an additional fully-connected layer with 256 units, yielding a single vector representing all pairs.
+
+Each set of three image embeddings was processed by a separate MLP with the same hyperparameters (hidden layer of 512 units, output layer of 256 units). The outputs of this MLP for all sets of three were then summed, and processed by a separate fully-connected layer with 256 units, yielding a single vector representing all sets of three.
+
+These two vectors, representing all pairs and sets of three, were then summed and passed to an additional fully-connected layer with 256 units, and then to the output layer to generate $\hat { \pmb { y } }$ . All layers (except for the output layer) used ReLU nonlinearities. All weights in these layers were initialized using a Kaiming normal distribution, and all biases were initialized to 0.
+
+Just as with the RN, we append the image embeddings with a tag indicating their position in the sequence before passing them to the first MLP in the TRN.
+
+# A.3.9 TRANSFORMER
+
+The Transformer implementation consisted of a single Transformer encoder layer. We also experimented with 2- and 3-layer Transformers but these did not generalize as well as the 1-layer Transformer in the tasks that we studied. Positional encoding (as described by Vaswani et al. (2017)) was applied to the sequence of image embeddings, which were then passed to the Transformer layer. The self-attention layer had 8 heads. The MLP had a single hidden layer with 512 units, and used ReLU nonlinearities. Residual connections and layer normalization (Ba et al., 2016b) were used following both the self-attention layer and the MLP. The self-attention weights (for generating the keys, queries, and values) were initialized using an Xavier normal distribution. The MLP weights were initialized using a Kaiming normal distribution.
+
+After applying the Transformer layer, the (transformed) embeddings were averaged and passed to an output MLP. The output MLP had a single hidden layer with 256 units, and an output layer for generating $\hat { \pmb { y } }$ . The hidden layer used ReLU nonlinearities, and the weights were initialized using a Kaiming normal distribution. All biases were initialized to 0.
+
+# A.3.10 PREDINET
+
+The PrediNet implementation was as close as possible to the model described in the original work (Shanahan et al., 2019), except that the multi-head attention was applied over the 1D temporal sequence of image embeddings, rather than over a 2D feature map (since there was no spatial component to the tasks that we studied). Before being passed to the PrediNet module, the image embeddings were appended with a tag (an integer from 0 to $T - 1 \dot s$ ) indicating their temporal position. The PrediNet module used keys of size 16, 32 heads, and 16 relations. All weights in the PrediNet module were initialized using an Xavier normal distribution.
+
+The output of all PrediNet heads was concatenated and passed to an output MLP. This MLP had a single hidden layer with 8 units, and an output layer for generating $\hat { \boldsymbol y }$ . The hidden layer used ReLU nonlinearities, and the weights were initialized using a Kaiming normal distribution. All biases were initialized to 0.
+
+# A.4 TRAINING DETAILS
+
+Table 5: Learning rates for all models trained without TCN.   
+
+<table><tr><td></td><td>Same/different</td><td>RMTS</td><td>Distribution-of-threeIdentity rules</td><td></td></tr><tr><td colspan="5"></td></tr><tr><td>ESBN</td><td>5e-5</td><td>5e-5</td><td>5e-5</td><td>5e-5</td></tr><tr><td>Transformer</td><td>5e-4</td><td>5e-4</td><td>5e-4</td><td>5e-4</td></tr><tr><td>NTM</td><td>5e-4</td><td>5e-4</td><td>5e-4</td><td>5e-4</td></tr><tr><td>MNM</td><td>5e-4</td><td>5e-4</td><td>5e-4</td><td>5e-4</td></tr><tr><td>LSTM</td><td>5e-4</td><td>5e-4</td><td>5e-4</td><td>5e-4</td></tr><tr><td>PrediNet</td><td>5e-4</td><td>5e-4</td><td>5e-5</td><td>5e-5</td></tr><tr><td>RN</td><td>5e-4</td><td>5e-5</td><td>5e-4</td><td>5e-4</td></tr></table>
+
+All models were trained with a batch size of 32 using the ADAM optimizer (Kingma & Ba, 2014). The learning rate for all models trained with TCN was $5 e ^ { - } 4$ . Some of the models failed to converge when trained without TCN, requiring a smaller learning rate of $5 e ^ { - } 5$ . The learning rates used for all models when trained without TCN are shown in Table 5.
+
+Because different generalization regimes (different values for $m$ ) involved different training set sizes, and therefore involved fewer training updates per epoch, the number of training epochs required to reliably achieve convergence varied based on the regime. The default number of training epochs for all tasks and regimes is shown in Table 6.
+
+Some models required additional training on some tasks to reach convergence. The PrediNet and the RN required longer training on the distribution-of-three task (Table 7), and the PrediNet, RN, and Transformer required longer training on the identity rules task (Table 8).
+
+Table 6: Default number of training epochs for all tasks and regimes.   
+
+<table><tr><td></td><td>m=0</td><td>m=50 m=85m=95</td><td></td><td></td><td>m=98</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Same/different</td><td>50</td><td>50</td><td>50</td><td>100</td><td>100</td></tr><tr><td>RMTS</td><td>50</td><td>50</td><td>50</td><td>200</td><td>1</td></tr><tr><td>Distribution-of-three</td><td>50</td><td>50</td><td>50</td><td>150</td><td></td></tr><tr><td>Identity rules</td><td>50</td><td>50</td><td>50</td><td>50</td><td>1</td></tr></table>
+
+$$
+m = 0 \quad m = 5 0 \quad m = 8 5 \quad m = 9 5
+$$
+
+Table 7: Number of training epochs for the PrediNet and RN on the distribution-of-three task.   
+
+<table><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>PrediNet</td><td>100</td><td>100</td><td>100</td><td>150</td></tr><tr><td>RN</td><td>150</td><td>150</td><td>150</td><td>800</td></tr></table>
+
+Table 8: Number of training epochs for the PrediNet, RN, and Transformer on the identity rules task.   
+
+<table><tr><td></td><td></td></tr><tr><td>100 100</td><td>100</td></tr></table>
+
+$$
+m = 0 \quad m = 5 0 \quad m = 8 5 \quad m = 9 5
+$$
+
+When training the RN on larger datasets for the distribution-of-three and identity rules tasks, the same learning rate and number of training epochs as used when training on smaller datasets was sufficient to reach convergence.
+
+# A.5 SUPPLEMENTARY RESULTS
+
+# A.5.1 RESULTS WITH AND WITHOUT TCN
+
+Tables 9 - 12 show the results for all models trained both with and without TCN. With the exception of the PrediNet on the same/different task, every model benefited on every task from the incorporation of TCN, in many cases substantially. Results for models trained with TCN (indicated by $^ \bullet +$ TCN’) correspond to the results presented in Figure 3 (except for the results of the PrediNet on the same/different task, for which the version of the model trained without TCN is plotted in Figure 3).
+
+We note that, even with a lower learning rate of $5 e ^ { - } 5$ , some models failed to converge without TCN, such as the ESBN on the same/different task, or the RN on the RMTS task. It is possible that some of these models might have performed better if we had optimized them further by training for longer or trying different learning rates, but we opted not to do that since TCN was so effective across all of the models and tasks that we studied.
+
+Table 9: Results for same/different task. Results reflect test accuracy averaged over 10 trained networks $\pm$ the standard error of the mean).   
+
+<table><tr><td></td><td>m=0</td><td>m= 50</td><td>m=85</td><td>m= 95</td><td>m=98</td></tr><tr><td colspan="6"></td></tr><tr><td>ESBN+TCN</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td></tr><tr><td>ESBN</td><td>50.0 ± 0.02</td><td>50.0 ± 0.0</td><td>50.1 ± 0.1</td><td>49.8 ± 0.2</td><td>50.1 ± 0.1</td></tr><tr><td>Transformer+ TCN</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>72.3 ± 5.2</td></tr><tr><td>Transformer</td><td>100.0 ± 0.0</td><td>99.9 ± 0.02</td><td>95.4 ± 0.6</td><td>73.7 ± 1.8</td><td>56.1 ± 1.3</td></tr><tr><td>NTM+ TCN</td><td>100.0 ± 0.0</td><td>99.99 ± 0.0</td><td>94.9 ± 0.6</td><td>66.7 ± 2.5</td><td>53.3 ± 1.4</td></tr><tr><td>NTM</td><td>99.0 ± 0.9</td><td>98.6 ± 0.3</td><td>84.9 ± 2.4</td><td>57.0 ± 2.2</td><td>52.5 ± 0.9</td></tr><tr><td>MNM+ TCN</td><td>100.0 ± 0.0</td><td>99.95 ± 0.03</td><td>97.8 ± 0.4</td><td>72.0 ± 2.4</td><td>52.3 ± 0.5</td></tr><tr><td>MNM</td><td>98.9 ± 0.1</td><td>95.1 ± 1.8</td><td>88.6 ±1.1</td><td>59.1 ± 1.6</td><td>51.7 ± 0.7</td></tr><tr><td>LSTM+TCN</td><td>100.0 ± 0.0</td><td>99.97 ± 0.01</td><td>96.9 ± 0.3</td><td>69.4 ± 1.5</td><td>54.8 ± 1.1</td></tr><tr><td>LSTM</td><td>88.2 ±3.2</td><td>97.0 ± 0.5</td><td>85.5 ± 2.4</td><td>61.8 ± 1.7</td><td>56.5 ± 1.6</td></tr><tr><td>PrediNet+ TCN</td><td>100.0 ± 0.0</td><td>99.7 ± 0.1</td><td>96.0 ± 1.3</td><td>67.2 ± 2.9</td><td>61.6 ± 2.3</td></tr><tr><td>PrediNet</td><td>100.0 ± 0.0</td><td>99.9 ± 0.03</td><td>97.0 ± 0.4</td><td>90.0 ± 1.6</td><td>68.5 ± 2.8</td></tr><tr><td>RN + TCN</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>99.9 ± 0.04</td><td>66.8 ± 6.6</td></tr><tr><td>RN</td><td>99.98 ± 0.02</td><td>98.5 ± 0.4</td><td>53.2 ±1.4</td><td>50.5 ± 0.2</td><td>52.3 ± 0.7</td></tr></table>
+
+Table 10: Results for relational match-to-sample task. Results reflect test accuracy averaged over 10 trained networks ( $\pm$ the standard error of the mean).   
+
+<table><tr><td></td><td>m=0</td><td>m= 50</td><td>m= 85</td><td>m= 95</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ESBN+TCN</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>95.0 ± 0.7</td></tr><tr><td>ESBN</td><td>86.4 ± 6.1</td><td>69.4 ± 6.5</td><td>50.0 ± 0.1</td><td>51.0 ± 0.5</td></tr><tr><td>Transformer</td><td>100.0 ± 0.0</td><td>99.98 ± 0.01</td><td>99.1 ± 0.4</td><td>79.8 ± 2.5</td></tr><tr><td>Transformer</td><td>99.4 ± 0.1</td><td>96.8 ± 0.7</td><td>86.4 ± 1.9</td><td>49.9 ± 0.2</td></tr><tr><td>NTM+TCN</td><td>100.0 ± 0.0</td><td>99.97 ± 0.01</td><td>96.8 ± 0.5</td><td>80.1 ± 2.3</td></tr><tr><td>NTM</td><td>99.5 ± 0.1</td><td>92.5 ± 4.7</td><td>81.2 ± 1.5</td><td>50.1 ± 0.2</td></tr><tr><td>MNM+TCN</td><td>99.99 ± 0.0</td><td>99.9 ± 0.03</td><td>98.7 ± 0.3</td><td>50.0 ± 0.2</td></tr><tr><td>MNM</td><td>74.6 ± 7.6</td><td>63.6 ± 5.7</td><td>78.3 ± 3.7</td><td>50.0 ± 0.2</td></tr><tr><td>LSTM+ TCN</td><td>99.99 ± 0.0</td><td>99.8 ± 0.03</td><td>94.9 ± 1.3</td><td>60.7 ± 3.7</td></tr><tr><td>LSTM</td><td>99.1 ± 0.3</td><td>90.2 ± 2.0</td><td>80.9 ± 1.1</td><td>50.2 ± 0.1</td></tr><tr><td>PrediNet+TCN</td><td>99.7 ± 0.1</td><td>99.6 ± 0.1</td><td>94.6 ± 2.2</td><td>68.4 ± 2.7</td></tr><tr><td>PrediNet</td><td>54.9 ± 4.7</td><td>50.1 ± 0.2</td><td>65.9 ± 3.7</td><td>49.7 ± 0.2</td></tr><tr><td>RN+TCN</td><td>100.0 ± 0.0</td><td>99.99 ± 0.0</td><td>99.5 ± 0.3</td><td>79.6 ± 2.1</td></tr><tr><td>RN</td><td>50.1 ± 0.2</td><td>49.9 ± 0.2</td><td>50.2 ± 0.2</td><td>50.0 ± 0.1</td></tr></table>
+
+Table 11: Results for distribution-of-three task. Results reflect test accuracy averaged over 10 trained networks $\pm$ the standard error of the mean).   
+
+<table><tr><td></td><td>m=0</td><td>m= 50</td><td>m= 85 m= 95</td></tr><tr><td colspan="4"></td></tr><tr><td>ESBN+TCN</td><td>98.7 ± 0.4</td><td>99.0 ± 0.3</td><td>99.5 ± 0.2 99.7 ± 0.1</td></tr><tr><td>ESBN</td><td>99.98 ± 0.0</td><td>97.4 ± 0.2</td><td>92.4 ± 1.1 62.0 ± 4.0</td></tr><tr><td>Transformer+ TCN</td><td>88.7 ± 2.6</td><td>95.0 ± 1.2</td><td>92.7 ± 1.5 32.1 ± 1.0</td></tr><tr><td>Transformer</td><td>62.1 ± 3.3</td><td>68.6 ± 3.6 72.6 ± 4.4</td><td>28.0 ± 0.8</td></tr><tr><td>NTM+TCN</td><td>95.5 ± 0.4</td><td>95.2 ± 0.4</td><td>94.3 ± 0.8 34.0 ± 0.5</td></tr><tr><td>NTM</td><td>92.9 ± 0.5</td><td>87.1 ± 1.4</td><td>78.2 ± 1.4 26.7 ± 0.3</td></tr><tr><td>MNM+ TCN</td><td>94.7 ± 0.3</td><td>93.6 ± 0.4 90.6 ± 0.7</td><td>32.2 ± 0.6</td></tr><tr><td>MNM</td><td>58.5 ± 8.9</td><td>68.7 ± 6.2 48.4 ± 5.5</td><td>25.6 ± 0.3</td></tr><tr><td>LSTM+TCN</td><td>96.0 ± 0.6</td><td>94.8 ± 0.5</td><td>92.9 ± 0.8 34.8 ± 0.8</td></tr><tr><td>LSTM</td><td>91.3 ± 0.6</td><td>85.3 ± 1.5</td><td>71.6 ± 4.3 27.5 ± 0.3</td></tr><tr><td>PrediNet+TCN</td><td>95.2 ± 0.3</td><td>94.6 ± 0.4</td><td>93.3 ± 0.9 27.8 ± 0.5</td></tr><tr><td>PrediNet</td><td>75.1 ± 3.0</td><td>65.7 ± 7.4</td><td>78.0 ± 6.0 25.7 ± 0.1</td></tr><tr><td>RN+TCN</td><td>35.6 ± 3.0</td><td>50.6 ± 7.6</td><td>72.2 ± 6.8 26.5 ± 0.3</td></tr><tr><td>RN</td><td>25.1 ± 0.1</td><td>24.9 ± 0.1 25.7 ± 0.3</td><td>25.2 ± 0.1</td></tr></table>
+
+Table 12: Results for identity rules task. Results reflect test accuracy averaged over 10 trained networks $\pm$ the standard error of the mean).   
+
+<table><tr><td></td><td>m=0</td><td>m= 50</td><td>m=85</td><td>m= 95</td></tr><tr><td colspan="5"></td></tr><tr><td>ESBN+TCN</td><td>99.6 ± 0.2</td><td>99.6 ± 0.1</td><td>99.9 ± 0.04</td><td>99.2 ± 0.4</td></tr><tr><td>ESBN</td><td>100.0 ± 0.0</td><td>99.4 ± 0.1</td><td>97.8 ± 0.2</td><td>95.2 ± 0.4</td></tr><tr><td>Transformer+ TCN</td><td>98.3 ± 0.7</td><td>97.1 ± 1.0</td><td>92.0 ±1.7</td><td>67.1 ± 2.4</td></tr><tr><td>Transformer</td><td>75.5 ± 4.1</td><td>71.6 ± 5.1</td><td>85.4 ± 4.6</td><td>38.6 ± 2.2</td></tr><tr><td>NTM+TCN</td><td>98.2 ± 0.6</td><td>97.8 ± 0.5</td><td>93.9 ± 0.6</td><td>64.9 ± 1.2</td></tr><tr><td>NTM</td><td>94.6 ± 0.3</td><td>90.1 ± 0.8</td><td>82.2 ± 1.2</td><td>25.0 ± 0.1</td></tr><tr><td>MNM+TCN</td><td>95.2 ± 0.4</td><td>93.8 ± 0.4</td><td>90.8 ± 0.5</td><td>61.5 ± 1.5</td></tr><tr><td>MNM</td><td>70.9 ± 10.2</td><td>69.5 ± 9.7</td><td>49.8 ± 8.4</td><td>24.9 ± 0.2</td></tr><tr><td>LSTM+TCN</td><td>98.9 ± 0.1</td><td>97.7 ± 0.3</td><td>92.1 ± 0.7</td><td>62.5 ± 1.1</td></tr><tr><td>LSTM</td><td>93.8 ± 0.5</td><td>89.3 ± 0.6</td><td>73.7 ± 5.7</td><td>24.8 ± 0.1</td></tr><tr><td>PrediNet+ TCN</td><td>93.0 ± 0.8</td><td>92.8 ± 0.7</td><td>89.8 ± 0.8</td><td>59.9 ± 2.6</td></tr><tr><td>PrediNet</td><td>40.8 ± 0.4</td><td>40.5 ± 1.9</td><td>40.3 ± 2.2</td><td>32.2 ± 0.6</td></tr><tr><td>RN+ TCN</td><td>41.5 ± 6.7</td><td>40.2 ± 1.0</td><td>48.7 ± 2.0</td><td>41.4 ± 2.0</td></tr><tr><td>RN</td><td>41.1 ± 7.2</td><td>37.3 ± 3.4</td><td>31.6 ± 2.8</td><td>25.4 ± 0.4</td></tr></table>
+
+# A.5.2 PERFORMANCE OF RN ON TERNARY RELATIONS
+
+Table 13 shows the results for the RN (w/ TCN) on the distribution-of-three and identity rules tasks when trained on larger training sets $1 0 ^ { 5 }$ instead of $1 0 ^ { 4 }$ training examples). These results show that with more training data, the RN, which is biased toward processing pair-wise relations, is able to learn these tasks (which are based on ternary relations) in a manner that enables some degree of generalization. Note that these results do not include the $m = 9 5$ regime, because there are not enough images in that regime to create larger training sets than were originally used.
+
+Table 13: Results for the RN on the distribution-of-three and identity rules tasks when trained on a larger training set. Results reflect test accuracy averaged over 10 trained networks ( $\pm$ the standard error of the mean).   
+
+<table><tr><td></td><td>m=0</td><td>m= 50</td><td>m=85</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td>Distribution-of-three</td><td>84.5 ± 10.0</td><td>84.6 ± 9.9</td><td>72.4 ± 9.7</td></tr><tr><td>Identity rules</td><td>89.2 ± 5.0</td><td>99.7 ± 0.1</td><td>86.6 ± 4.1</td></tr></table>
+
+We also tested the TRN, which incorporates ternary relations through subsampling, on these tasks (with the standard training set size of $1 0 ^ { 4 }$ training examples). Table 14 shows the results. This yielded a slight improvement over the RN (when trained on $1 0 ^ { 4 }$ training examples), though not as much of an improvement as resulted from training the RN with a larger training set. This result may seem surprising given that the TRN explicitly incorporates ternary relations. We note two possible explanations for this result:
+
+1. The systematic comparison of every pair of objects, including permutations and comparisons of each object with itself, allows the RN to take advantage of a very powerful form of data augmentation, enforcing a certain degree of systematicity in the relations that it learns. By only considering temporally ordered and non-redundant sets, the TRN is not able to take advantage of this to the same extent, and therefore might not learn relations that generalize as well.   
+2. The distribution-of-three and identity rules tasks both involve not only ternary sets, but the higher-order comparison of multiple pairs of ternary sets (the first row vs. the combination of the second row with each candidate answer). One could presumably engineer a solution to this problem within the RN framework, but we take it as a strength of the ESBN that no such special engineering is necessary in this case.
+
+Table 14: Results for the TRN on the distribution-of-three and identity rules tasks. Results reflect test accuracy averaged over 10 trained networks $\pm$ the standard error of the mean).   
+A.5.3 TRAINING TIME COURSES FOR SAME/DIFFERENT TASK   
+
+<table><tr><td></td><td>m=0</td><td>m= 50</td><td>m= 85</td><td>m = 95</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Distribution-of-three</td><td></td><td></td><td>60.2±5.4 77.5±5.7 88.7±0.8</td><td>327.8±0.5</td></tr><tr><td>Identity rules</td><td></td><td>40.3 ± 2.0 43.6± 2.0</td><td>52.8 ± 2.7</td><td>44.9 ± 1.0</td></tr></table>
+
+Figure 5 shows the training time courses for all models on the same/different task. Unlike the other three tasks we studied (for which training time courses are shown in Figure 4), all models were able to learn this task within a few hundred training updates (though all models except the ESBN failed to generalize in the most extreme regime).
+
+# A.5.4 ALTERNATIVE ENCODER ARCHITECTURES
+
+In order to determine whether the systematic generalization exhibited by the ESBN depended to some extent on the convolutional layers in its encoder, we performed experiments with two alternative encoder architectures: a multilayer perceptron (MLP) encoder, and a random projection.
+
+![](images/aaa7fb6a3176c2b300c4bdc9fc10c870841ebe3d2619a27cc2e4ec70c2d07eb7.jpg)  
+Figure 5: Training accuracy time courses on $m = 0$ regime of the same/different task. Each time course reflects an average over 10 trained networks. Error bars reflect the standard error of the mean.
+
+![](images/c366872be7b0df16e430db884b2883ef8a8ba7fd1f3bd25e27dc986eb9ad5467.jpg)  
+Figure 6: Results for all four tasks with convolutional (conv), multilayer perceptron (MLP), or random (rand) encoders. Results reflect test accuracy averaged over 10 trained networks ( $\pm$ the standard error of the mean).
+
+The MLP encoder consisted of 3 fully-connected layers, with 512, 256, and 128 units, each of which used ReLU nonlinearities. All weights were initialized using a Kaiming normal distribution, and all biases were set to 0.
+
+The random projection encoder involved only a single, untrained, fully-connected layer that projected from the flattened image to 128 units, followed by a ReLU nonlinearity. Weights were sampled from a Kaiming normal distribution, and biases were set to 0.
+
+Figure 6 and Tables 15 - 18 show the results for these experiments, along with the original version of the model (with a convolutional encoder) for comparison. To enable a fair comparison with the original model, all experiments employed TCN. The results show that the ESBN performed comparably well with all three of the encoder architectures. This was confirmed by performing paired t-tests on the average test accuracy in each task/generalization condition (each combination of task and value of $m$ ) for the MLP vs. convolutional encoder $t = - 1 . 7$ , $p = 0 . 1$ ) and for the random vs. convolutional encoder $t = 1 . 6$ , $p = 0 . 1 3$ ).
+
+For comparison, we also performed experiments with these alternative encoders in the Transformer architecture. These experiments revealed that, in contrast with the ESBN, the Transformer’s performance was significantly impaired by the use of a random vs. convolutional encoder $( t = - 4 . 0$ , $p = 0 . 0 0 1 )$ , though it appeared to perform comparably well with an MLP vs. convolutional encoder $t = - 1 . 6$ , $p = 0 . 1 4 )$ ).
+
+Table 15: Results for same/different task with convolutional (conv), multilayer perceptron (MLP), or random (rand) encoders. Results reflect test accuracy averaged over 10 trained networks $\pm$ the standard error of the mean).   
+
+<table><tr><td></td><td>m=0</td><td>m= 50</td><td>m=85</td><td>m=95</td><td>m=98</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ESBN (conv)</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td></tr><tr><td>ESBN (MLP)</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td></tr><tr><td>ESBN (rand)</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td></tr><tr><td>Transformer (conv)</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>72.3 ± 5.2</td></tr><tr><td>Transformer (MLP)</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>77.6 ± 4.6</td></tr><tr><td>Transformer (rand)</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>50.6 ± 0.3</td></tr></table>
+
+Table 16: Results for relational match-to-sample task with convolutional (conv), multilayer perceptron (MLP), or random (rand) encoders. Results reflect test accuracy averaged over 10 trained networks $\pm$ the standard error of the mean).   
+
+<table><tr><td></td><td>m=0</td><td>m= 50</td><td>m=85</td><td>m= 95</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ESBN (conv)</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>95.0 ± 0.7</td></tr><tr><td>ESBN (MLP)</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>97.2 ± 0.2</td></tr><tr><td>ESBN (rand)</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>93.8 ± 0.4</td></tr><tr><td>Transformer (conv)</td><td>100.0 ± 0.0</td><td>99.98 ± 0.01</td><td>99.1 ± 0.4</td><td>79.8 ± 2.5</td></tr><tr><td>Transformer (MLP)</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>99.9 ± 0.1</td><td>73.6 ± 5.3</td></tr><tr><td>Transformer (rand)</td><td>99.99 ± 0.01</td><td>99.9 ± 0.04</td><td>95.4 ± 3.2</td><td>46.8 ± 1.7</td></tr></table>
+
+Table 17: Results for distribution-of-three task with convolutional (conv), multilayer perceptron (MLP), or random (rand) encoders. Results reflect test accuracy averaged over 10 trained networks ( $\pm$ the standard error of the mean).   
+
+<table><tr><td></td><td>m=0</td><td>m= 50</td><td>m=85</td><td>m= 95</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ESBN (conv)</td><td>98.7 ± 0.4</td><td>99.0 ± 0.3</td><td>99.5 ± 0.2</td><td>99.7 ± 0.1</td></tr><tr><td>ESBN (MLP)</td><td>99.0 ± 0.1</td><td>98.4 ± 0.3</td><td>98.0 ± 0.3</td><td>95.9 ± 0.5</td></tr><tr><td>ESBN (rand)</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td></tr><tr><td>Transformer (conv)</td><td>88.7 ± 2.6</td><td>95.0 ± 1.2</td><td>92.7 ± 1.5</td><td>32.1 ± 1.0</td></tr><tr><td>Transformer (MLP)</td><td>92.7 ± 2.1</td><td>93.3 ± 1.5</td><td>92.1 ± 0.8</td><td>35.3 ± 1.2</td></tr><tr><td>Transformer (rand)</td><td>66.0 ± 6.2</td><td>80.8 ± 2.5</td><td>60.9 ± 2.4</td><td>26.9 ± 0.4</td></tr></table>
+
+Table 18: Results for identity rules task with convolutional (conv), multilayer perceptron (MLP), or random (rand) encoders. Results reflect test accuracy averaged over 10 trained networks $\pm$ the standard error of the mean).   
+
+<table><tr><td></td><td>m=0</td><td>m= 50</td><td>m= 85</td><td>m= 95</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ESBN (conv.)</td><td>99.6 ± 0.2</td><td>99.6 ± 0.1</td><td>99.9 ± 0.04</td><td>99.2 ± 0.4</td></tr><tr><td>ESBN (MLP)</td><td>99.3 ± 0.2</td><td>98.6 ± 0.3</td><td>97.7 ± 0.4</td><td>95.5 ± 1.0</td></tr><tr><td>ESBN (random)</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td></tr><tr><td>Transformer (conv.)</td><td>98.3 ± 0.7</td><td>97.1 ± 1.0</td><td>92.0 ±1.7</td><td>67.1 ± 2.4</td></tr><tr><td>Transformer (MLP)</td><td>85.5 ± 4.0</td><td>84.8 ± 3.8</td><td>86.4 ± 2.3</td><td>59.8 ± 1.5</td></tr><tr><td>Transformer (random)</td><td>47.8 ±1.1</td><td>51.4 ± 1.9</td><td>48.6 ±1.1</td><td>27.5 ± 0.6</td></tr></table>
+
+# A.5.5 CONFIDENCE ABLATION EXPERIMENT
+
+In order to determine the importance of the confidence values appended to retrieved memories, we tested a version of the ESBN without these confidence values. These results are shown in Table 19 and Figure 7. The ablation of confidence values prevented the ESBN from being able to perform the same/different task at all, and resulted in much slower training on the RMTS task. By contrast, ablation of confidence values did not affect performance, either in terms of generalization or training time, for the distribution-of-three or identity rules tasks. This can be explained by the fact that these tasks only require the retrieval of the best match from memory, whereas the same/different and RMTS tasks require the model to know how good of a match the best match is, which is precisely the information conveyed by confidence values.
+
+Table 19: Results for the confidence ablation experiment. Results reflect test accuracy averaged over 10 trained networks $\pm$ the standard error of the mean).   
+
+<table><tr><td></td><td>m=0</td><td>m= 50</td><td>m= 85</td><td>m= 95</td><td>m= 98</td></tr><tr><td colspan="6"></td></tr><tr><td>Same/different</td><td>50.0 ± 0.02</td><td>50.0 ± 0.0</td><td>50.0 ± 0.05</td><td>49.8 ± 0.1</td><td>50.0 ± 0.1</td></tr><tr><td>RMTS</td><td>99.95 ± 0.01</td><td>99.9 ± 0.02</td><td>99.9 ± 0.02</td><td>96.0 ± 0.6</td><td></td></tr><tr><td>Distribution-of-three</td><td>99.2 ± 0.2</td><td>99.0 ± 0.3</td><td>99.5 ± 0.3</td><td>99.8 ± 0.1</td><td></td></tr><tr><td>Identity rules</td><td>99.6 ± 0.1</td><td>99.6 ± 0.2</td><td>99.8 ± 0.1</td><td>99.2 ± 0.2</td><td></td></tr></table>
+
+![](images/4f3140e98ad78df5a251cc44698547758bc2b3dc25ada989b5a077dd22f67752.jpg)  
+Figure 7: Training accuracy time courses for the ESBN model without confidence values on the $m = 0$ regime, shown with the time courses for all other models for comparison. Each time course reflects an average over 10 trained networks. Error bars reflect the standard error of the mean.
+
+It is also worth noting one potential alternative to an explicit, inbuilt confidence value. In our implementation, the ESBN’s memory is empty at the beginning of each sequence that it processes. However, when multiple entries are present in memory, as will generally be the case in realistic, temporally extended settings, the presentation of a previously unseen item will result in the retrieval of a mixture of (weakly matched) memories. This mixed representation can therefore serve as a reliable cue for the degree to which the current percept matches a stored memory, obviating the need for an explicit confidence value. To demonstrate this, we implemented a version of the ESBN that begins each sequence with a single, learned key/value entry stored in memory (initialized to 0 at the beginning of training). Table 20 shows that this approach allows the ESBN to learn and perfectly generalize on the same/different task. Figure 8 shows that this approach allows the ESBN to retain the short training time of the original model on the RMTS task.
+
+Table 20: Results on the same/different task for the ESBN model with a learned default memory instead of confidence values. Results reflect test accuracy averaged over 10 trained networks ( $\pm$ the standard error of the mean).
+
+![](images/c31ae8d2cb9633c10939d6f607f5e27e2afb3fd53d987d9a72940c252af1aef8.jpg)  
+Figure 8: Training accuracy time courses on $m = 0$ regime of the RMTS task for the ESBN model with a learned default memory instead of confidence values. Each time course reflects an average over 10 trained networks. Error bars reflect the standard error of the mean.
+
+![](images/ca9d748d252c788908ecf5de7247e9e4f232304204241105a959dc8695758a69.jpg)  
+Figure 9: Representations learned by ESBN (projected along first two principal components). (a) Keys written to memory during time steps 1-9 (training set). (b) Keys written to memory during time steps 1-3 (training set vs. test set). (c) Keys retrieved from memory following second appearance of objects that first appeared during time steps 1-3 (training set vs. test set).
+
+To better understand how the ESBN works, we performed an analysis of the representations that it learned on the distribution-of-three task. Specifically, we performed an analysis of a network trained on the most difficult generalization regime $( m = 9 5 $ ), by performing principal component analysis (PCA) on all key vectors written to and retrieved from memory for both the training and test sets, and visualizing these vectors along the first two principal components.
+
+First, we looked at the keys that were written to memory $( k _ { w } )$ . We found that the keys for the first three time steps were tightly clustered, whereas the keys for the subsequent time steps (4-9) were more diffuse (Figures 9a and 9b). This makes sense because, in the distribution-of-three task, the ESBN only needs to be able to reliably retrieve what it wrote during the first three time steps (when the objects in the first row were presented). For time steps 4-9, the only important consideration is that the keys written to memory not overlap with those written during the first three time steps, which also appears to be the case.
+
+Second, we compared the keys written to memory for the first three time steps in the training vs. test sets (Figure 9b). This revealed that, for a given time step, the keys written to memory in the training vs. test sets were remarkably similar (so much so that they are completely overlapping for time steps 1 and 2).
+
+Third, we looked at the keys that were retrieved from memory following the second appearance of the objects that appeared on time steps 1-3. We found that 1) these closely matched the distribution of keys written to memory during time steps 1-3, and 2) these were highly overlapping for the training vs. test sets (Figure 9c).
+
+Taken together, these results help to explain why the ESBN was so successful in this generalization regime, despite the very small degree of overlap between the distribution of training and test images. Because the ESBN’s controller was relatively isolated from the part of the model that deals with image embeddings, it was able to learn to encode abstract symbol-like representations (such as ‘first image’, ‘second image’, and ‘third image’), that did not depend on the identity of the images. Then, when queried with an image, was able to successfully retrieve the image’s corresponding abstract encoding, even when that image was quite different than those observed during training. That is, the model learned representations to use as keys that could be used for binding and indirection in the same way that symbols are used in traditional computational architectures.
+
+# A.7 UNICODE CHARACTERS
+
+Figure 10 shows all 100 images that were used to construct the abstract rule learning tasks.
+
+![](images/887b1ae25cca3a239fbf94d507ce08a8916ce532f8964e7f75552108e7422ff2.jpg)  
+Figure 10

parse/train/MjNFN44NbZm/MjNFN44NbZm.md

@@ -0,0 +1,334 @@
+# Policy Finetuning: Bridging Sample-Efficient Offline and Online Reinforcement Learning
+
+Tengyang Xie UIUC tx10@illinois.edu
+
+Nan Jiang UIUC nanjiang@illinois.edu
+
+Huan Wang Salesforce Research huan.wang@salesforce.com
+
+Caiming Xiong Salesforce Research cxiong@salesforce.com
+
+Yu Bai Salesforce Research yu.bai@salesforce.com
+
+# Abstract
+
+Recent theoretical work studies sample-efficient reinforcement learning (RL) extensively in two settings: learning interactively in the environment (online RL), or learning from an offline dataset (offline RL). However, existing algorithms and theories for learning near-optimal policies in these two settings are rather different and disconnected. Towards bridging this gap, this paper initiates the theoretical study of policy finetuning, that is, online RL where the learner has additional access to a “reference policy” $\mu$ close to the optimal policy $\pi _ { \star }$ in a certain sense. We consider the policy finetuning problem in episodic Markov Decision Processes (MDPs) with $S$ states, $A$ actions, and horizon length $H$ . We first design a sharp offline reduction algorithm—which simply executes $\mu$ and runs offline policy optimization on the collected dataset—that finds an $\varepsilon$ near-optimal policy within $\bar { \tilde { O } } ( \bar { H ^ { 3 } } S C ^ { \star } / \varepsilon ^ { 2 } )$ episodes, where $C ^ { \star }$ is the single-policy concentrability coefficient between $\mu$ and $\pi _ { \star }$ . This offline result is the first that matches the sample complexity lower bound in this setting, and resolves a recent open question in offline RL. We then establish an $\Omega ( H ^ { 3 } S \bar { \operatorname* { m i n } } \{ C ^ { \star } , A \} / \varepsilon ^ { 2 } )$ sample complexity lower bound for any policy finetuning algorithm, including those that can adaptively explore the environment. This implies that—perhaps surprisingly—the optimal policy finetuning algorithm is either offline reduction or a purely online RL algorithm that does not use $\mu$ . Finally, we design a new hybrid offline/online algorithm for policy finetuning that achieves better sample complexity than both vanilla offline reduction and purely online RL algorithms, in a relaxed setting where $\mu$ only satisfies concentrability partially up to a certain time step. Overall, our results offer a quantitative understanding on the benefit of a good reference policy, and make a step towards bridging offline and online RL.
+
+# 1 Introduction
+
+Reinforcement learning (RL)—where agents learn to play sequentially in an environment to maximize a cumulative reward function—has achieved great recent success in many artificial intelligence challenges such as video games playing [38, 52], large-scale strategy games (e.g. GO) [44, 45], robotic manipulation [3, 32], behavior learning in social scenarios [8], and more. In many such challenging domains, achieving human-like or superhuman performance requires training the RL agent with millions of samples (steps of acting or game playing) or more. Understanding and improving the sample efficiency of RL algorithms has been a central topic of research.
+
+Sample-efficient RL has been studied in a rich body of theoretical work in two main settings: online $R L$ , in which the learner has interactive access to the environment and can execute any policy; and offline $R L$ , in which the learner only has access to an “offline” dataset collected by executing some (one or many) policies within the environment, and is not allowed to further access the environment. These two settings share some common learning goals such as the sample complexity (number of episodes of playing) for finding the optimal policy. However, existing algorithms and theories in the online and offline setting seem rather different and disconnected—In online RL, state-of-the-art sample-efficient algorithms typically explore the entire environment, e.g. by using optimism to encourage visitation to unseen states and actions [9, 27, 19, 41, 21, 5, 22, 12, 23, 53]. In contrast, offline RL does not allow interactive exploration, and sample-efficient policy optimization algorithms typically focus on optimizing an unbiased (or downward biased) estimator of the value function [39, 48, 4, 40, 10, 56, 35, 58, 25, 42]. It is therefore of interest to ask whether these two types of algorithms and theories can be connected in any way.
+
+Further, on the empirical end, insights and patterns from offline RL often help as well in designing online RL algorithms and improving the sample efficiency in the real world. For example, there are online RL algorithms that alternate between data collection steps using a fixed policy, and policy improvement steps by learning on the collected dataset [20]. The replay buffer in value-based algorithms can also be seen as a local form of offline (off-policy) policy optimization and are often be used in conjunction with optimistic exploration techniques [38, 18, 49]. The prevalence of these algorithms also offers practical motivations for us to look for a more unified understanding of online and offline RL in theory. These reasonings motivate us to ask the following question:
+
+# Can we bridge sample-efficient offline and online RL from a theoretical perspective?
+
+This paper proposes policy finetuning, a new RL setting that investigates the benefit of a good initial policy in reinforcement learning, and encapsulates challenges of both online and offline RL. In the policy finetuning problem, the learner is given interactive access to the environment and asked to learn a near-optimal policy, but in addition has access to a reference policy $\mu$ that is good in certain aspects. This setting offers great flexibility for the algorithm design: For example, the algorithm is allowed to either simply collect data from $\mu$ and run any offline policy optimization algorithm on the collected dataset. It is also allowed to play any other policy interactively, including those that adaptively explores the environment. The policy finetuning problem offers a common playground for both offline and online types of algorithms, and has a unified performance metric (sample complexity for finding the near-optimal policy) for comparing their performance.
+
+We study the policy finetuning problem theoretically in finite-horizon Markov Decision Processes (MDPs) with $H$ time steps, $S$ states, and $A$ actions. We summarize our contributions as follows.
+
+• We begin by considering offline reduction algorithms which simply collect data using the reference policy $\mu$ and run an offline policy optimization algorithm on the collected dataset. This setting equivalent to offline RL with behavior policy $\mu$ , and thus our result translates to a same result for offline RL as well. We design an algorithm PEVI-ADV that is able to find an $\varepsilon$ -optimal policy (for small $\varepsilon$ ) within $\tilde { O } ( H ^ { 3 } S \bar { C } ^ { \star } / \varepsilon ^ { 2 } )$ episodes of play, where $C ^ { \star }$ is the single-policy concentrability coefficient between $\mu$ and some optimal policy $\pi _ { \star }$ (Section 3). This improves over the best existing offline result by an $H ^ { 2 }$ factor in the same setting and matches the lower bound (up to log factors), thereby resolving the recent open question of [42] on tight offline RL under single-policy concentrability.   
+• Under the same assumption on $\mu$ , we establish an $\Omega ( H ^ { 3 } S \operatorname* { m i n } { \{ C ^ { \star } , A \} } / \varepsilon ^ { 2 } )$ sample complexity lower bound for any policy finetuning algorithm, including those that adaptively explores the environment (Section 4). This implies that the optimal policy finetuning algorithm is either offline reduction via PEVI-ADV, or a “purely” online RL algorithm from scratch (such as UCBVI), depending on whether $C ^ { \star } \leq A$ . This comes rather surprising, as it rules out possibilities of combining online exploration and knowledge of $\mu$ to further improve the sample complexity over the aforementioned two baselines.   
+• Finally, we consider policy finetuning in a more challenging setting where $\mu$ only satisfies concentrability up to a certain time step. We design a “hybrid offline/online” algorithm HOOVI that combines online exploration and offline data collection, and show that it achieves better sample complexity than both vanilla offline reduction and purely online algorithms in certain cases (Section 5). This gives a positive example on when such hybrid algorithm designs are beneficial.
+
+# 1.1 Related work
+
+Sample-efficient online RL There is a long line of work on establishing provably sample-efficient online RL algorithms. A major portion of these works is concerned with the tabular setting with finitely many states and actions [9, 27, 19, 5, 11, 2, 22, 63]. For episodic MDPs with inhomogeneous transition functions with $S$ states, and $A$ actions, and horizon length $H$ , the optimal sample complexity for finding the $\varepsilon$ near-optimal policy is $\widetilde { \cal O } ( H ^ { 3 } S A / \varepsilon ^ { 2 } )$ , achieved by various algorithms such as UCBVI of Azar et al. [5] and UCB-Advantage of Zhang et al. [63]. Our paper adapts the reference-advantage decomposition technique of Zhang et al. [63] to designing sharp offline algorithms. Online RL with with large state/action spaces are also studied by using function approximation in conjunction with structural assumptions on the MDP [23, 61, 62, 1, 41, 21, 47, 53, 57, 14, 24].
+
+Offline RL Offline/batch RL studies the case where the agent only has access to an offline dataset obtained by executing a behavior policy in the environment. Sample-efficient learning results in offline RL typically work by assuming either sup-concentrability assumptions [39, 48, 4, 40, 15, 51, 10, 56]) or lower bounded exploration constants [58, 59] to ensure the sufficient coverage of offline data over all (relevant) states and actions. However, such strong coverage assumptions can often fail to hold in practice [16]. More recent works address this by using either policy constraint/regularization [16, 35, 29, 55], or the pessimism principle to optimize conservatively on the offline data [30, 60, 28, 25, 59, 42]. The policy-constraint/regularization-based approaches prevent the policy to visit states and actions that has no or low coverage from the offline data. Our proposed offline RL algorithm PEVI-ADV (Algorithm 1) is inspired by the pessimistic value iteration algorithms of [25, 42] and achieves an improved sample complexity over these work under the same single-policy concentrability assumption on the behavior policy.
+
+Bridging online and offline RL Kalashnikov et al. [26] observed empirically that the performance of policies trained purely from offline data can be improved considerably by a small amount of additional online fine-tuning. A recent line of work studied low switching cost RL [6, 63, 17, 54]— which forbits online RL algorithms from switching its policy too often—as an interpolation between the online and offline settings. The same problem is also studied empirically as deployment-efficient RL [36, 46]. While we also attempt to bridge online and offline RL, our work differs from this line in that our policy finetuning setting allows a direct comparison between “fully offline” and “fully online” algorithms, whereas the low switching cost setting prohibits fully online algorithms.
+
+# 2 Preliminaries
+
+Markov Decision Processes In this paper, we consider episodic Markov decision processes (MDPs) with time-inhomogeneous transitions, specified by $M = ( \mathcal { S } , \mathcal { A } , H , \mathbb { P } , r )$ , where $s$ is the state space, $\mathcal { A }$ is the action space, $H$ is the horizon length, $\mathbb { P } = \{ \mathbb { P } _ { h } \} _ { h = 1 } ^ { H }$ where $\mathbb { P } _ { h } ( \cdot | s , a ) \in \Delta _ { S }$ is the transition probabilities at step $h$ , and $r = \{ r _ { h } : \mathcal { S } \times \mathcal { A } \to [ 0 , 1 ] \} _ { h = 1 } ^ { H }$ =1  are the deterministic1 reward functions at time step $h \in [ H ]$ . Without loss of generality, we assume that the initial state $s _ { 1 }$ is deterministic2.
+
+Policies, value functions, visitation distributions A policy $\pi = \{ \pi _ { h } ( \cdot | s ) \} _ { h \in [ H ] , s \in { \mathcal S } }$ consists of distributions $\pi _ { h } ( \cdot | s ) \in \Delta _ { { \cal A } }$ . We use $\mathbb { E } _ { \pi } [ \cdot ]$ to denote the expectation with respect to the random trajectory induced by $\pi$ in the MDP $M$ , that is, $\left( s _ { 1 } , a _ { 1 } , r _ { 1 } , s _ { 2 } , a _ { 2 } , r _ { 2 } , \ldots , s _ { H } , a _ { H } , r _ { H } \right)$ , where $a _ { h } =$ $\pi _ { h } ( s _ { h } )$ , $r _ { h } = r _ { h } ( s _ { h } , a _ { h } )$ , $s _ { h + 1 } \sim \mathbb { P } _ { h } ( \cdot | s _ { h } , a _ { h } )$ . For each policy $\pi$ , let $V _ { h } ^ { \pi } : { \cal S }  \mathbb { R }$ and $Q _ { h } ^ { \pi }$ : $S \times \mathcal { A }  \mathbb { R }$ denote its value functions and Q functions at each time step $h \in [ H ]$ , that is,
+
+$$
+V _ { h } ^ { \pi } ( s ) : = \mathbb { E } _ { \pi } \bigg [ \sum _ { h ^ { \prime } = h } ^ { H } r _ { h ^ { \prime } } ( s _ { h ^ { \prime } } , a _ { h ^ { \prime } } ) \bigg | s _ { h } = s \bigg ] , Q _ { h } ^ { \pi } ( s , a ) : = \mathbb { E } _ { \pi } \bigg [ \sum _ { h ^ { \prime } = h } ^ { H } r _ { h ^ { \prime } } ( s _ { h ^ { \prime } } , a _ { h ^ { \prime } } ) \bigg | s _ { h } = s , a _ { h } = a \bigg ] .
+$$
+
+The operators $\mathbb { P } _ { h }$ and $\mathbb { V } _ { h }$ are defined as $[ \mathbb { P } _ { h } V _ { h + 1 } ] ( s , a ) : = \mathbb { E } [ V _ { h + 1 } ( s ^ { \prime } ) | s _ { h } = s , a _ { h } = a ]$ and $[ \mathbb { V } _ { h } V _ { h + 1 } ] ( s , a ) : = \mathrm { V a r } [ V _ { h + 1 } ( s ^ { \prime } ) | s _ { h } = s , a _ { h } = a ]$ for any value function $V _ { h + 1 }$ at time step $h + 1$
+
+We also use $\widehat { \mathbb { P } } _ { h }$ and $\widehat { V } _ { h }$ to denote empirical versions of these operators building on estimated models (which will be clear in the context).
+
+We use $\pi _ { \star } : = \arg \operatorname* { m a x } _ { \pi } V _ { 1 } ^ { \pi } ( s _ { 1 } )$ to denote any optimal policy, and $V _ { h } ^ { \star } : = V _ { h } ^ { \pi _ { \star } }$ and $Q _ { h } ^ { \star } : = Q _ { h } ^ { \pi _ { \star } }$ to denote the value function and $\mathrm { Q }$ function of $\pi ^ { \star }$ at all $h \in [ H ]$ . Throughout this paper, our learning goal is to find an near-optimal policy $\widehat { \pi }$ such that ${ \cal V } _ { 1 } ^ { \star } ( s _ { 1 } ) \dot { - } \dot { \cal V } _ { 1 } ^ { \widehat \pi } ( s _ { 1 } ) \le \varepsilon$ .
+
+Finally, we let $d _ { h } ^ { \pi }$ denote the state(-action) visitation distributions of $\pi$ at time step $h \in [ H ]$ :
+
+$$
+\begin{array} { r } { d _ { h } ^ { \pi } ( s ) : = \mathbb { P } ( s _ { h } = s | \pi ) , \mathrm { ~ a n d ~ } d _ { h } ^ { \pi } ( s , a ) : = \mathbb { P } ( s _ { h } = s , a _ { h } = a | \pi ) . } \end{array}
+$$
+
+Miscellaneous We use standard $O ( \cdot )$ and $\Omega ( \cdot )$ notation: $A = O ( B )$ is defined as $A \leq C B$ for some absolute constant $C > 0$ (and similarly for $\Omega$ ). The tilded notation $A = { \widetilde { O } } ( B )$ denotes $A \leq C L \cdot B$ where $L$ is a poly-logarithmic factor of problem parameters.
+
+# 2.1 Policy Finetuning
+
+We now introduce the setting of policy finetuning. A policy finetuning problem consists of an MDP $M$ and a reference policy $\mu$ . During the learning stage, the learner can perform the following two types of moves:
+
+(a) Play an episode in the MDP $M$ using any policy (i.e. learner has online interactive access to $M$ ). (b) Access the values of the reference policy $\mu _ { h } ( a | s )$ for all $( h , s , a )$ . For example, the learner can use it to sample actions $a \sim \mu _ { h } ( \cdot | s )$ for any $h , s$ for arbitrarily many times during learning.
+
+The goal of the learner is to output $\varepsilon$ near-optimal policy $\widehat { \pi }$ within as few episodes of play (within the MDP) as possible.
+
+A unique feature about the policy finetuning setting is that it allows both online interactive plays via any online RL algorithm (not necessarily using $\mu$ ), as well as offline reduction which simply collects data by executing the reference policy $\mu$ and do anything with the collected dataset. In particular, this means that any algorithm for offline policy optimization (based on offline datasets) also gives an algorithm for policy finetuning via this offline reduction. Therefore, policy finetuning offers a common playground for both online and offline type algorithms with a unified learning goal.
+
+Assumption on reference policy Throughout most of this paper (except for Section 5), we consider the following assumption on the reference policy $\mu$ .
+
+Assumption A (Single-policy concentrability). The reference policy $\mu$ satisfies that
+
+$$
+\operatorname* { m a x } _ { h \in [ H ] , ( s , a ) \in S \times A } \frac { d _ { h } ^ { \pi _ { \star } } ( s , a ) } { d _ { h } ^ { \mu } ( s , a ) } \leq C ^ { \star }
+$$
+
+(with the convention $0 / 0 = 0 ,$ ) for some deterministic optimal policy $\pi _ { \star }$ and constant $C ^ { \star } \geq 1$
+
+The single-policy concentrability characterizes the distance between the visitation distributions of the reference policy $\mu$ and some optimal policy $\pi ^ { \star }$ . This assumption is considered in the recent work of Rashidinejad et al. [42] on offline RL and is more relaxed than previously assumed concentrability assumptions which typically requires the supremum concentrability against all possible $\pi$ ’s to be bounded [10]. We consider this assumption as it both allows efficient offline RL algorithms [42], and is perhaps also a sensible measure of quality for the reference policy in policy finetuning.
+
+# 3 Sharp offline learning via reference-advantage decomposition
+
+We begin by investigating the sharpest sample complexity for policy finetuning via the offline reduction approach. This requires us to design sharp offline RL algorithms that run on the dataset $\mathcal { D }$ collected by executing $\mu$ . We emphasize that this is both an interesting offline RL question on its own right, and also important for our later discussions on lower bounds and other algorithms for policy finetuning, as the sharpest sample complexity via offline reduction provides a solid baseline.
+
+Warm-up: VI-LCB As a warm-up, we first show that a finite-horizon variant of the VILCB (Value Iteration with Lower Confidence Bounds) algorithm of Rashidinejad et al. [42] achieves sample complexity $\widetilde { O } ( H ^ { 5 } S C ^ { \star } / \varepsilon ^ { 2 } )$ for finding an $\varepsilon$ near-optimal policy. This result is similar to the $\widetilde { O } ( S C ^ { \star } / ( 1 - \gamma ) ^ { 5 } \varepsilon ^ { 2 } )$ guarantee3 for the original VI-LCB in infinite-horizon discounted MDPs [42, Theorem 6]. The main ingredients of our VI-LCB algorithm is a pessimistic value iteration procedure in which we perform value iteration on the empirical model estimated from the dataset $\mathcal { D }$ , along with a negative Hoeffding bonus term to impose pessimism. Due to space constraints, the algorithm description (Algorithm 3) and the proof of Theorem 1 are deferred to Appendix B.
+
+Theorem 1 (VI-LCB for finite-horizon MDPs). Suppose the reference policy $\mu$ satisfies the singlepolicy concentrability (Assumption $A$ ). Then with probability at least $1 - \delta$ , VI-LCB (Algorithm 3) outputs a policy $\widehat { \pi }$ and value estimate $\widehat { V }$ such that
+
+$$
+\begin{array} { r l } & { ) \operatorname* { m a x } _ { h \in [ H ] } \sum _ { s \in { \mathcal { S } } } d _ { h } ^ { \pi _ { \star } } ( s ) ( V _ { h } ^ { \star } ( s ) - \widehat { V } _ { h } ( s ) ) \leq \varepsilon , } \\ & { } \\ & { V _ { 1 } ^ { \star } ( s _ { 1 } ) - V _ { 1 } ^ { \widehat { \pi } } ( s _ { 1 } ) \leq \varepsilon , } \end{array}
+$$
+
+within $n = \widetilde O \bigl ( H ^ { 5 } S C ^ { \star } / \varepsilon ^ { 2 } \bigr )$ episodes.
+
+Theorem 1 serves two main purposes. First, the $\widetilde { O } ( H ^ { 5 } S C ^ { \star } / \varepsilon ^ { 2 } )$ sample complexity asserted in Theorem 1(b) provides a first result for offline RL (and offline reduction for policy finetuning) under single-policy concentrability in finite-horizon MDPs. Second, the value estimation bound in Theorem 1(a) shows that the estimated value function $\widehat { V } _ { h } ( s )$ provided by VI-LCB is close to the optimal value ${ V } _ { h } ^ { \star } ( s )$ at every step $h \in [ H ]$ , in terms of the weighted average with $d _ { h } ^ { \pi _ { \star } } ( s )$ . Our next algorithm PEVI-ADV builds on this property so that VI-LCB can be used as a “warm-up” learning procedure that provides a high-quality value estimate.
+
+Sharp offline learning via reference-advantage decomposition We now design a new sharp algorithm PEVI-ADV which achieves an improved $\widetilde { O } ( H ^ { 3 } S C ^ { \star } / \varepsilon ^ { 2 } )$ sample complexity (for small enough $\varepsilon$ ). This improves over VI-LCB by $\widetilde O ( H ^ { 2 } )$ and is the first algorithm that matches the sample complexity lower bound. PEVI-ADV adds two new ingredients over VI-LCB in order to achieve the $\tilde { O } ( \bar { H } ^ { 2 } )$ improvement:
+
+1. We replace the Hoeffding-style bonus in VI-LCB with a Bernstein-style bonus. This shaves off one $H$ factor in the sample complexity via the total variance property (Lemma C.4).   
+2. Both VI-LCB and our PEVI-ADV use data splitting to make sure that the estimated value $\widehat { V } _ { h + 1 }$ and empirical transitions $\widehat { \mathbb { P } } _ { h }$ are estimated using different subsets of $\mathcal { D }$ , this yields conditional independence that is required in bounding concentration terms of the form $( \widehat { \mathbb { P } } _ { h } - \mathbb { P } _ { h } ) \widehat { V } _ { h + 1 }$ . However, applied naively, this data splitting induces one undesired $H$ factor in the sample complexity as we need to split $\mathcal { D }$ into $H$ folds and thus each $\mathbb { P } _ { h }$ is estimated using only $n / H$ episodes of data. As a technical crux of this algorithm, we overcome this issue by adapting the reference-advantage decomposition technique of Zhang et al. [63]. This technique proposes to learn an initial reference value function $\widehat { V } ^ { \mathrm { r e f } }$ of good quality in a certain sense, and then performing the following type of approximate value iteration (using the right-hand side as the algorithm update):
+
+$$
+\mathbb { P } _ { h } \widehat { V } _ { h + 1 } \approx \widehat { \mathbb { P } } _ { h , 0 } \widehat { V } _ { h + 1 } ^ { \mathrm { r e f } } + \widehat { \mathbb { P } } _ { h , 1 } \Big ( \widehat { V } _ { h + 1 } - \widehat { V } _ { h + 1 } ^ { \mathrm { r e f } } \Big ) .
+$$
+
+Above, $\widehat { V } _ { h + 1 } , \widehat { \mathbb { P } } _ { h , 0 }$ , and $\widehat { \mathbb { P } } _ { h , 1 }$ are estimated on three disjoint subsets of the data. The advantage of this approach is that, due to this new independence structure, $\widehat { \mathbb { P } } _ { h , 0 }$ for different $h \in [ H ]$ can be estimated on the same set of trajectories without $H$ -fold splitting, which shaves off the $H$ factor within this part. On the other hand, estimating $\widehat { \mathbb { P } } _ { h , 1 }$ still requires $H$ -fold splitting, yet this would not hurt the sample complexity if the magnitude of $( \widehat { V } _ { h + 1 } - \widehat { V } _ { h + 1 } ^ { \mathrm { r e f } } )$ is much smaller than its naive upper bound $O ( H )$ —we show this can be achieved by using VI-LCB to learn $\widehat { V } ^ { \mathrm { r e f } }$ .
+
+Require: Dataset $\mathcal { D } = \left\{ ( s _ { 1 } ^ { ( i ) } , a _ { 1 } ^ { ( i ) } , r _ { 1 } ^ { ( i ) } , \dotsc , s _ { H } ^ { ( i ) } , a _ { H } ^ { ( i ) } , r _ { H } ^ { ( i ) } ) \right\} _ { i = 1 } ^ { n }$ collected by executing $\mu$ in $M$ .
+
+1: Split the dataset $\mathcal { D }$ into ${ \mathcal { D } } _ { \mathrm { r e f } }$ , $\mathcal { D } _ { 0 }$ and $\left\{ \mathcal { D } _ { h , 1 } \right\} _ { h = 1 } ^ { H }$ uniformly at random:
+
+$$
+n _ { \mathrm { r e f } } : = \left| \mathcal { D } _ { \mathrm { r e f } } \right| = n / 3 , n _ { 0 } : = \left| \mathcal { D } _ { 0 } \right| = n / 3 , n _ { 1 , h } : = \left| \mathcal { D } _ { h , 1 } \right| : = n / ( 3 H ) ( n _ { 1 } : = n / 3 ) .
+$$
+
+2: Learn a reference value function $\widehat { V } ^ { \mathrm { r e f } } \gets \mathrm { V I - L C B } ( \mathcal { D } _ { \mathrm { r e f } } )$ via VI-LCB (Algorithm 3).
+
+3: Let $N _ { h , 0 } ( s , a )$ and $N _ { h , 0 } ( s , a , s ^ { \prime } )$ denote the visitation count of $( s , a )$ and $( s , a , s ^ { \prime } )$ at step $h$ within dataset $\mathcal { D } _ { 0 }$ . Construct empirical model estimates:
+
+$$
+\widehat { \mathbb { P } } _ { h , 0 } ( s ^ { \prime } | s , a ) \gets \frac { N _ { h , 0 } ( s , a , s ^ { \prime } ) } { N _ { h , 0 } ( s , a ) \vee 1 } , \quad \mathrm { a n d } \quad \widehat { r } _ { h , 0 } ( s , a ) \gets r _ { h } ( s , a ) \mathbb { 1 } \left\{ N _ { h , 0 } ( s , a ) \geq 1 \right\} .
+$$
+
+Similarly define $N _ { h , 1 } ( s , a ) , N _ { h , 1 } ( s , a , s ^ { \prime } ) ,$ $( \widehat { r } _ { h , 1 } , \widehat { \mathbb { P } } _ { h , 1 } )$ for all $h \in [ H ]$ based on dataset $\mathcal { D } _ { h , 1 }$
+
+4: Set $\begin{array} { r } { b _ { h , 0 } ( s , a ) \gets c \cdot \left( \sqrt { \frac { [ \widehat { \Psi } _ { h , 0 } \widehat { V } _ { h + 1 } ^ { \mathrm { r e f } } ] ( s , a ) \iota } { N _ { h , 0 } ( s , a ) \vee 1 } } + \frac { H \iota } { N _ { h , 0 } ( s , a ) \vee 1 } \right) } \end{array}$ for all $( h , s , a )$ , where $\iota : = \log ( H S A / \delta )$
+
+5: Set $\widehat { V } _ { H + 1 } ( s ) \gets 0$ for all $s \in S$ .
+
+6: for $h = H , \ldots , 1$ do
+
+8: Perform pessimistic value update for all $( s , a )$
+
+$$
+\begin{array} { r l } & { \widehat { Q } _ { h } ( s , a ) \gets \widehat { r } _ { h , 0 } ( s , a ) + \left[ \widehat { \mathbb { P } } _ { h , 0 } \widehat { V } _ { h + 1 } ^ { \mathrm { r e f } } \right] ( s , a ) - b _ { h , 0 } ( s , a ) + \left[ \widehat { \mathbb { P } } _ { h , 1 } ( \widehat { V } _ { h + 1 } - \widehat { V } _ { h + 1 } ^ { \mathrm { r e f } } ) \right] ( s , a ) - b _ { h , 1 } ( s , a ) } \\ & { \widehat { V } _ { h } ( s ) \gets \left[ \operatorname* { m a x } _ { a } \widehat { Q } _ { h } ( s , a ) \right] \vee 0 . } \\ & { \mathrm { S e t } \widehat { \pi } _ { h } ( s ) \gets \arg \operatorname* { m a x } _ { a } \widehat { Q } _ { h } ( s , a ) \mathrm { f o r } \mathrm { a l l } s \in S . } \end{array}
+$$
+
+b10: end for
+
+11: return Policy $\widehat { \pi } = \{ \widehat { \pi } _ { h } \} _ { h \in [ H ] }$
+
+We instantiate this plan by carefully using VI-LCB to learn the reference value function $\widehat { V } ^ { \mathrm { r e f } }$ , combined with tight Bernstein bonuses, to shave off another $H$ factor in the sample complexity. The full PEVI-ADV algorithm is provided in Algorithm 1. We now present its guarantee in the following theorem. The proof can be found in Appendix C.
+
+Theorem 2 (Sharp offline learning via PEVI-ADV). Suppose the reference policy $\mu$ satisfies the single-policy concentrability (Assumption $A$ ). Then with probability at least $1 - \delta$ , PEVI-ADV (Algorithm 1) outputs a policy $\widehat { \pi }$ and value estimate $\widehat { V }$ such that
+
+$$
+\begin{array} { r l } & { \operatorname* { m a x } _ { h \in [ H ] } \sum _ { s \in {  { \mathcal { S } } } } d _ { h } ^ { \pi _ { \star } } ( s ) ( V _ { h } ^ { \star } ( s ) - \widehat { V } _ { h } ( s ) ) \leq \varepsilon , } \\ & { } \\ & { V _ { 1 } ^ { \star } ( s _ { 1 } ) - V _ { 1 } ^ { \widehat { \pi } } ( s _ { 1 } ) \leq \varepsilon , } \end{array}
+$$
+
+within $n = \widetilde O \left( H ^ { 3 } S C ^ { \star } / \varepsilon ^ { 2 } + H ^ { 5 . 5 } S C ^ { \star } / \varepsilon \right)$ episodes.
+
+Near-optimal offline RL under single-policy concentrability For small enough $\varepsilon \le H ^ { - 2 . 5 }$ , Theorem 2 achieves $\widetilde { O } ( H ^ { 3 } S C ^ { \star } / \varepsilon ^ { 2 } )$ sample complexity for finding the $\varepsilon$ near-optimal policy from the offlien dataset $\mathcal { D }$ . This is the first cubic horizon dependence for offline RL under single-policy concentrability, which improves over recent works [25, 42] in this setting and resolves the open question of [42]. For $C ^ { \star } \geq 2$ , our sample complexity further matches the information-theoretical lower bound $\Omega ( H ^ { 3 } S C ^ { \star } / \varepsilon ^ { 2 } )$ up to log factors4. We remark that tight hoziron dependence has also been achieved in several recent works offline RL [58, 59, 43] which are however quite different from (and do not imply) ours in both the assumptions (on the behavior policy) and the analyses.
+
+# 4 Lower bound for policy finetuning
+
+We now switch gears to considering the policy finetuning problem with any algorithm, not necessarily restricted to the offline reduction approach.
+
+Two baselines: offline reduction $\pmb { \& }$ purely online RL A first observation is that naive offline reduction is already a strong baseline for policy finetuning, by our Theorem 2: Our PEVI-ADV algorithm only collects data with $\mu$ and does not do any online exploration, yet achieves a sharp $\bar { \tilde { O } } ( H ^ { 3 } S C ^ { \star } / \bar { \varepsilon } ^ { 2 } )$ sample complexity for finding a near-optimal policy.
+
+On the other hand, as the policy finetuning setting allows online interaction, purely online $R L$ is another baseline algorithm: Simply run any sample-efficient online RL algorithm (which typically uses optimism to encourage exploration) from scratch, and disregard the reference policy $\mu$ . Using any sharp online RL algorithm such as UCBVI [5], this approach can find an $\varepsilon$ near-optimal policy within $\widetilde { \cal O } ( H ^ { 3 } S A / \varepsilon ^ { 2 } )$ episodes of play. Note that whether this is advantageous over the offline reduction boils down to the comparison between $C ^ { \star }$ and $A$ , which makes sense intuitively. For example, $C ^ { \star } \leq o ( A )$ means that $\mu$ is perhaps close enough to $\pi _ { \star }$ so that collecting data from $\mu$ and run offline policy optimization is a stronger algorithm than exploring from scratch.
+
+Given these two baselines, it is natural to ask whether there exists an algorithm that improves over both — Can we design an algorithm that performs some amount of optimistic exploration, yet also utilizes the knowledge of $\mu$ , so as to achieve a better rate than both offline reduction and purely online RL? In this section, we provide an information-theoretic lower bound showing that, perhaps surprisingly, the answer is negative: there is an $\Omega ( H ^ { 3 } S \operatorname* { m i n } { \{ C ^ { \star } , A \} } / \varepsilon ^ { 2 } )$ sample complexity lower bound for any policy finetuning algorithm, if we still assume that $\mu$ satisfies $C ^ { \star }$ single-policy concentrability.
+
+Lower bound To formally state our lower bound, we define the class of problems
+
+$$
+\mathcal { M } _ { C ^ { \star } } : = \Bigg \{ ( M , \mu ) : ~ \mathrm { E x i s t s ~ d e t e r m i n i s t i c } ~ \pi _ { \star } ~ \mathrm { o f } ~ M \mathrm { ~ s u c h ~ t h a t ~ } \operatorname* { s u p } _ { h , s , a } \frac { d _ { h } ^ { \pi _ { \star } } ( s , a ) } { d _ { h } ^ { \mu } ( s , a ) } \leq C ^ { \star } \Bigg \} .
+$$
+
+We recall that a policy finetuning algorithm for problem $( M , \mu )$ is defined as any algorithm that can play in the MDP $M$ for $n$ episodes, has full knowledge of the reference policy $\mu$ , and outputs a policy $\widehat { \pi }$ after playing in the MDP.
+
+With these definitions ready, we now state our lower bound for policy finetuning. The proof of Theorem 3 can be found in Appendix D.
+
+Theorem 3 (Lower bound for policy finetuning). Suppose $S , H \geq 3 , A \geq 2 , C ^ { \star } \geq 2 .$ . Then, there exists an absolute constant $c _ { 0 } > 0$ such that for any $\varepsilon \le 1 / 1 2$ and any online finetuning algorithm that outputs a policy $\widehat { \pi }$ , if the number of episodes
+
+$$
+n \leq c _ { 0 } \cdot H ^ { 3 } S \operatorname* { m i n } { \{ C ^ { \star } , A \} } / \varepsilon ^ { 2 } ,
+$$
+
+then there exists a problem instance $( M , \mu ) \ \in \ { \mathcal { M } } _ { C ^ { \star } }$ on which the algorithm suffers from $\varepsilon$ - suboptimality:
+
+$$
+\mathbb { E } _ { M } \left[ V _ { 1 , M } ^ { \star } - V _ { 1 , M } ^ { \widehat { \pi } } \right] \geq \varepsilon ,
+$$
+
+where the expectation $\mathbb { E } _ { M }$ is w.r.t. the randomness during the algorithm execution within MDP M .
+
+Either offline reduction or purely online is optimal Theroem 3 shows that any policy finetuning algorithm needs to play at least $\Omega ( H ^ { 3 } S \operatorname* { m i n } \{ C ^ { \star } , A \} / \varepsilon ^ { 2 } )$ episodes in order to find an $\varepsilon$ near-optimal policy. Crucially, this implies that either a sharp offline reduction (e.g. our PEVI-ADV algorithm) or purely online RL matches the lower bound (up to log), depending on whether $C ^ { \star } \lesssim A$ . In other words, if we have the knowledge of whether $C ^ { \star } \leq A$ , choosing the right one of these two baseline algorithms will yield the optimal sample complexity. Perhaps surprisingly, this rules out the possibility of designing any algorithm “in between” that combines online exploration and knowledge of $\mu$ to improve the sample complexity, at least in the worst-case over all problems in $\mathcal { M } _ { C ^ { \star } }$ . We argue that this “no algorithm in between” phenomenon may be due to the single-policy concentrability assumption being too strong such that offline reduction already achieves a rather competitive sample
+
+Require: MDP $M$ , reference policy $\mu$
+
+6: Denote the final output of UCBVI-UPLOW as
+
+$$
+\begin{array} { r } { ( \overline { { V } } _ { h _ { \star } + 1 } , \underline { { V } } _ { h _ { \star } + 1 } , \widehat { \pi } _ { ( h _ { \star } + 1 ) : H } ^ { \mathrm { U C B } } ) \gets \mathbf { U C B V I - U P L O W } ( n _ { \mathrm { U C B } } ) . } \end{array}
+$$
+
+7: # Stage 2: Learn step $1 : h _ { \star }$ via executing $\mu +$ pessimistic offline policy optimization 8: Collect 9: Learn p $\mathcal { D }  \{ n - n _ { \mathrm { U C B } }$ episodes of data using policy he TRUNCATED-PEVI-ADV( $\mu$ up to step lgorithm 5 $h _ { \star } \}$ . $\hat { \pi } _ { 1 : h _ { \star } } ^ { \mathrm { P E V I } }$
+
+$$
+\widehat { \pi } _ { 1 : h _ { \star } } ^ { \mathrm { P E V I } }  \mathrm { T R U N C A T E D - P E V I - A D V } ( \mathcal { D } , h _ { \star } , \underline { { V } } _ { h _ { \star } + 1 } ) .
+$$
+
+$\widehat { \pi } = ( \widehat { \pi } _ { 1 : h _ { \star } } ^ { \mathrm { P E V I } } , \widehat { \pi } _ { ( h _ { \star } + 1 ) : H } ^ { \mathrm { U C B } } )$
+
+complexity $\widetilde { O } ( H ^ { 3 } S C ^ { \star } / \varepsilon ^ { 2 } )$ . We investigate policy finetuning beyond the single-policy concentrability assumption in Section 5.
+
+We also remark that Theorem 3 generalizes both the $\Omega ( H ^ { 3 } S A / \varepsilon ^ { 2 } )$ lower bound for online RL [11, 58, 13] into the policy finetuning problem, as well as the $\Omega ( \dot { H } ^ { 3 } S C ^ { \star } / \varepsilon ^ { 2 } )$ lower bound for offline RL under single-policy concentrability with $C ^ { \star } \geq 2 [ 4 2 ] ^ { 5 }$ . Further, Theorem 3 directly implies an $\Omega ( H ^ { 3 } S C ^ { \star } / \varepsilon ^ { \star } )$ lower bound for offline RL with $2 \leq C ^ { \star } \leq O ( A )$ , as any algorithm for offline policy optimization is also an algorithm for policy finetuning via the offline reduction.
+
+Proof intuition; Construction of hard instance The proof of Theorem 3 constructs a family of hard MDPs that requires solving $H S$ “independent” bandit problems with $A$ arms, similar as in existing $\Omega ( H ^ { 3 } S A / \bar { \varepsilon } ^ { 2 } )$ lower bounds for online RL [11, 58]. However, our key modification is that we let the optimal arms to be always within the first $K : = \operatorname* { m i n } \left\{ C ^ { \star } , A \right\}$ actions instead of all $A$ actions, and we define our reference policy $\mu$ to play uniformly within $[ K ]$ . This $\mu$ has the following properties:
+
+• $\mu$ satisfies $C ^ { \star }$ single-policy concentrability for any MDP in this family (Lemma D.1). • $\mu$ provides the knowledge that the optimal actions are within $[ K ]$ , but no other knowledge about the optimal actions.
+
+Therefore, with $\mu$ at hand, any policy finetuning algorithm can “gain the knowledge” that the optimal actions are within $[ K ]$ , but still needs to try all $K$ actions in order to solve each bandit problem— rigorizing this information-theoretically gives the $\Omega ( H ^ { 3 } S K / \varepsilon ^ { 2 } ) = \Omega ( H ^ { 3 } S \operatorname* { m i n } { \{ C ^ { \star } , A \} } \mathbf { \hat { / } } \varepsilon ^ { 2 } )$ lower bound.
+
+# 5 Hybrid offline/online algorithm for policy finetuning
+
+Towards circumventing the lower bound in Theorem 3, in this section, we study policy finetuning under more relaxed assumptions on the reference policy $\mu$ . A weaker $\mu$ will induce a higher sample complexity for naive offline reduction approaches, and thus yields opportunities for designing new algorithms that can potentially better utilize $\mu$ .
+
+More concretely, we consider the following relaxation: We assume $\mu$ satisfies partial concentrability only up to a certain time-step $h _ { \star } \leq H$ , and may not have any bounded concentrability at steps $h > h _ { \star }$ . We formalize this in the following
+
+Assumption B ( $h _ { \star }$ -partial concentrability). The reference policy $\mu$ satisfies the single-policy concentrability with respect to $\pi _ { \star }$ up to step $h _ { \star }$ only:
+
+$$
+\operatorname* { m a x } _ { h \leq h _ { \star } } \operatorname* { m a x } _ { s , a \in { \mathcal { S } } \times { \mathcal { A } } } { \frac { d _ { h } ^ { \pi _ { \star } } ( s , a ) } { d _ { h } ^ { \mu } ( s , a ) } } \leq C ^ { \mathrm { p a r t i a l } }
+$$
+
+(with the convention $0 / 0 = 0 ,$ ), where $\pi _ { \star }$ is some deterministic optimal policy of the MDP, and constant Cpartial .
+
+Algorithm description We design a hybrid offline/online algorithm HOOVI (presented in Algorithm 2) for policy finetuning under the partial concentrability assumption. At a high-level, the algorithm consists of two main stages:
+
+• In the first stage, it runs an online algorithm UCBVI-UPLOW which uses optimistic exploration to find a near-optimal policy ${ \widehat { \pi } } ^ { \mathrm { U C B } }$ and an accurate value estimate for steps $( h _ { \star } + 1 ) : H$ . • In the second stage, we run a TRUNCATED-PEVI-ADV algorithm, which collects data from $\mu$ and runs offline policy optimization to find a near-optimal policy $\widehat { \pi } ^ { \mathrm { { P E V I } } }$ for steps $1 : h _ { \star }$ , building on the lower value estimate $\underline { { V } } _ { h _ { \star } + 1 }$ from the first stage.
+
+This strategy makes sense intuitively as the reference policy $\mu$ does not have guarantees for steps $h _ { \star } + 1 : H$ and thus the algorithm is required to perform optimistic exploration first to get a good policy. However, additional technical cares are needed in order to make the above algorithm provably sample-efficient. The analysis of the second stage requires the online algorithm in the first stage to not only perform fast exploration (e.g. by using upper confidence bounds), but also output a lower value estimate for step $h _ { \star } + 1$ , and in addition output a final output policy that achieves at least the value of the lower value estimate at every state $s \in S$ . Such lower bounds are not directly available in standard online RL algorithms such as UCBVI [5].
+
+We resolve this by designing the UCBVI-UPLOW algorithm (detailed description in Algorithm 4), which is a modification of the Nash-VI Algorithm of Liu et al. [34] (for two-player Markov games) into the single-player case. This algorithm is particularly suitable for our purpose since it maintains both upper bounds of $V ^ { \star }$ and lower bounds for the value function of the deployed policies. Our UCBVI-UPLOW further integrates the certified policy technique of Bai et al. [7] to make sure that its output policy achieves value greater or equal than the lower bound at every state (similar guarantees can also be obtained by the policy certificate technique of Dann et al. [12]).
+
+We now state our main theoretical guarantee for the HOOVI algorithm. The proof can be found in Appendix E.
+
+Theorem 4 (Hybrid online $/$ offline learning for policy finetuning). Suppose the reference policy $\mu$ satisfies the partial concentrability (Assumption $B$ ) up to some step $h _ { \star } \leq H$ . Then for small enough $\varepsilon \le \operatorname* { m i n } \left\{ h _ { \star } ^ { - 2 . 5 } , C ^ { \mathrm { p a r t i a l } } / S \right\}$ , HOOVI (Algorithm 2) outputs a policy $\widehat { \pi }$ such that ${ \cal V } _ { 1 } ^ { \star } ( s _ { 1 } ) -$ $V _ { 1 } ^ { \widehat { \pi } } ( s _ { 1 } ) \leq \varepsilon$ with probability at least $1 - \delta$ , within
+
+$$
+n = { \tilde { O } } \left( { \frac { H ^ { 2 } h _ { \star } S C ^ { \mathrm { p a r t i a l } } + ( H - h _ { \star } ) ^ { 3 } S A ( C ^ { \mathrm { p a r t i a l } } ) ^ { 2 } } { \varepsilon ^ { 2 } } } \right)
+$$
+
+episodes of play.
+
+Comparison against offline reduction and purely online algorithms The sample complexity in Theorem 4 compares favorably against both naive offline reduction as well as purely online algorithms in certain situations. First, naive offline reduction with $\mu$ does not have any guarantee since $\mu$ is not assumed to have a finite single-policy concentrability at $h \geq h _ { \star } + 1$ . We can modify $\mu$ into $\mu ^ { \prime }$ that plays uniformly within $\mathcal { A }$ at steps $h \geq h _ { \star } + 1$ ; the single-policy concentrability coefficient of $\mu ^ { \prime }$ is guaranteed to be finite but scales exponentially as $O ( \breve { A } ^ { H - \mathbf { \breve { h } _ { \star } } } )$ in the worst case, leading to a sample complexity much worse than ours (which is polynomial in $H , S , A )$ .
+
+On the other hand, a sharp online algorithm can still achieve $\widetilde { \cal O } ( H ^ { 3 } S A / \varepsilon ^ { 2 } )$ in this setting (by optimistic exploration from scratch). Our Theorem 4 is in general incomparable with this, but can be better in cases when both $C ^ { \mathrm { p a r t i a l } }$ and $H \mathrm { ~ - ~ } h _ { \star }$ are small, e.g., if $\ { \dot { C } } ^ { \mathrm { p a r t i a l } } = o ( A )$ and $( H - h _ { \star } ) / H = o ( ( C ^ { \mathrm { p a r t i a l } } ) ^ { - 2 / 3 } )$ . This makes sense intuitively as our hybrid offline/online algorithm benefits the most if the length requiring exploration $( H - h _ { \star } )$ is small, and the partial concentrability $C ^ { \mathrm { p a r t i a l } }$ is small so that $\mu$ still has a high-quality for the first $h _ { \star }$ steps. To best of our knowledge, this is first result that characterizes when the sample complexity of such hybrid algorithms can be beneficial over purely online or offline algorithms.
+
+# 6 Conclusion & discussions
+
+This paper studies policy finetuning, a new reinforcement learning setting that allows us to compare and connect sample-efficient online and offline reinforcement learning. We establish sharp upper and lower bounds for policy finetuning under various assumptions on the reference policy. Our bounds show that the optimal policy finetuning algorithm is either offline reduction or a purely online algorithm in the specific setting where the reference policy satisfies single-policy concentrability, and we also show that a hybrid online/offline algorithm can be advantageous over both in more relaxed settings. Many directions could be of interest for future research, such as alternative assumptions on the reference policy, or policy finetuning with function approximation.
+
+Also, while our contributions are mainly theoretical, implementing or extending our policy finetuning algorithms on real-world RL tasks would be a compelling future direction. When the environment is a tabular MDP, our Algorithm 1 (offline reduction) and Algorithm 2 (hybrid offline / online RL) are readily implementable. When there is large state/action space and potentially function approximation, we believe our algorithm can be adapted, for example, by replacing all the optimistic/pessimistic value iteration steps by DQN-type algorithms [38] with positive/negative bonus functions [50]. Experimental evaluation of such algorithms would be a good direction for future work.
+
+# Acknowledgment
+
+The authors would like to thank Ming Yin, Chi Jin and David Forsyth for the many insightful discussions. NJ acknowledges funding support from the ARL Cooperative Agreement W911NF17-2-0196, NSF IIS-2112471, and Adobe Data Science Research Award. HW, CX, YB are funded through employment with Salesforce.
+
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+        "text": "Recent theoretical work studies sample-efficient reinforcement learning (RL) extensively in two settings: learning interactively in the environment (online RL), or learning from an offline dataset (offline RL). However, existing algorithms and theories for learning near-optimal policies in these two settings are rather different and disconnected. Towards bridging this gap, this paper initiates the theoretical study of policy finetuning, that is, online RL where the learner has additional access to a “reference policy” $\\mu$ close to the optimal policy $\\pi _ { \\star }$ in a certain sense. We consider the policy finetuning problem in episodic Markov Decision Processes (MDPs) with $S$ states, $A$ actions, and horizon length $H$ . We first design a sharp offline reduction algorithm—which simply executes $\\mu$ and runs offline policy optimization on the collected dataset—that finds an $\\varepsilon$ near-optimal policy within $\\bar { \\tilde { O } } ( \\bar { H ^ { 3 } } S C ^ { \\star } / \\varepsilon ^ { 2 } )$ episodes, where $C ^ { \\star }$ is the single-policy concentrability coefficient between $\\mu$ and $\\pi _ { \\star }$ . This offline result is the first that matches the sample complexity lower bound in this setting, and resolves a recent open question in offline RL. We then establish an $\\Omega ( H ^ { 3 } S \\bar { \\operatorname* { m i n } } \\{ C ^ { \\star } , A \\} / \\varepsilon ^ { 2 } )$ sample complexity lower bound for any policy finetuning algorithm, including those that can adaptively explore the environment. This implies that—perhaps surprisingly—the optimal policy finetuning algorithm is either offline reduction or a purely online RL algorithm that does not use $\\mu$ . Finally, we design a new hybrid offline/online algorithm for policy finetuning that achieves better sample complexity than both vanilla offline reduction and purely online RL algorithms, in a relaxed setting where $\\mu$ only satisfies concentrability partially up to a certain time step. Overall, our results offer a quantitative understanding on the benefit of a good reference policy, and make a step towards bridging offline and online RL. ",
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+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "text": "Reinforcement learning (RL)—where agents learn to play sequentially in an environment to maximize a cumulative reward function—has achieved great recent success in many artificial intelligence challenges such as video games playing [38, 52], large-scale strategy games (e.g. GO) [44, 45], robotic manipulation [3, 32], behavior learning in social scenarios [8], and more. In many such challenging domains, achieving human-like or superhuman performance requires training the RL agent with millions of samples (steps of acting or game playing) or more. Understanding and improving the sample efficiency of RL algorithms has been a central topic of research. ",
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+        "text": "Sample-efficient RL has been studied in a rich body of theoretical work in two main settings: online $R L$ , in which the learner has interactive access to the environment and can execute any policy; and offline $R L$ , in which the learner only has access to an “offline” dataset collected by executing some (one or many) policies within the environment, and is not allowed to further access the environment. These two settings share some common learning goals such as the sample complexity (number of episodes of playing) for finding the optimal policy. However, existing algorithms and theories in the online and offline setting seem rather different and disconnected—In online RL, state-of-the-art sample-efficient algorithms typically explore the entire environment, e.g. by using optimism to encourage visitation to unseen states and actions [9, 27, 19, 41, 21, 5, 22, 12, 23, 53]. In contrast, offline RL does not allow interactive exploration, and sample-efficient policy optimization algorithms typically focus on optimizing an unbiased (or downward biased) estimator of the value function [39, 48, 4, 40, 10, 56, 35, 58, 25, 42]. It is therefore of interest to ask whether these two types of algorithms and theories can be connected in any way. ",
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+        "type": "text",
+        "text": "Further, on the empirical end, insights and patterns from offline RL often help as well in designing online RL algorithms and improving the sample efficiency in the real world. For example, there are online RL algorithms that alternate between data collection steps using a fixed policy, and policy improvement steps by learning on the collected dataset [20]. The replay buffer in value-based algorithms can also be seen as a local form of offline (off-policy) policy optimization and are often be used in conjunction with optimistic exploration techniques [38, 18, 49]. The prevalence of these algorithms also offers practical motivations for us to look for a more unified understanding of online and offline RL in theory. These reasonings motivate us to ask the following question: ",
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+        "type": "text",
+        "text": "Can we bridge sample-efficient offline and online RL from a theoretical perspective? ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "This paper proposes policy finetuning, a new RL setting that investigates the benefit of a good initial policy in reinforcement learning, and encapsulates challenges of both online and offline RL. In the policy finetuning problem, the learner is given interactive access to the environment and asked to learn a near-optimal policy, but in addition has access to a reference policy $\\mu$ that is good in certain aspects. This setting offers great flexibility for the algorithm design: For example, the algorithm is allowed to either simply collect data from $\\mu$ and run any offline policy optimization algorithm on the collected dataset. It is also allowed to play any other policy interactively, including those that adaptively explores the environment. The policy finetuning problem offers a common playground for both offline and online types of algorithms, and has a unified performance metric (sample complexity for finding the near-optimal policy) for comparing their performance. ",
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+        "type": "text",
+        "text": "We study the policy finetuning problem theoretically in finite-horizon Markov Decision Processes (MDPs) with $H$ time steps, $S$ states, and $A$ actions. We summarize our contributions as follows. ",
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+        "text": "• We begin by considering offline reduction algorithms which simply collect data using the reference policy $\\mu$ and run an offline policy optimization algorithm on the collected dataset. This setting equivalent to offline RL with behavior policy $\\mu$ , and thus our result translates to a same result for offline RL as well. We design an algorithm PEVI-ADV that is able to find an $\\varepsilon$ -optimal policy (for small $\\varepsilon$ ) within $\\tilde { O } ( H ^ { 3 } S \\bar { C } ^ { \\star } / \\varepsilon ^ { 2 } )$ episodes of play, where $C ^ { \\star }$ is the single-policy concentrability coefficient between $\\mu$ and some optimal policy $\\pi _ { \\star }$ (Section 3). This improves over the best existing offline result by an $H ^ { 2 }$ factor in the same setting and matches the lower bound (up to log factors), thereby resolving the recent open question of [42] on tight offline RL under single-policy concentrability.   \n• Under the same assumption on $\\mu$ , we establish an $\\Omega ( H ^ { 3 } S \\operatorname* { m i n } { \\{ C ^ { \\star } , A \\} } / \\varepsilon ^ { 2 } )$ sample complexity lower bound for any policy finetuning algorithm, including those that adaptively explores the environment (Section 4). This implies that the optimal policy finetuning algorithm is either offline reduction via PEVI-ADV, or a “purely” online RL algorithm from scratch (such as UCBVI), depending on whether $C ^ { \\star } \\leq A$ . This comes rather surprising, as it rules out possibilities of combining online exploration and knowledge of $\\mu$ to further improve the sample complexity over the aforementioned two baselines.   \n• Finally, we consider policy finetuning in a more challenging setting where $\\mu$ only satisfies concentrability up to a certain time step. We design a “hybrid offline/online” algorithm HOOVI that combines online exploration and offline data collection, and show that it achieves better sample complexity than both vanilla offline reduction and purely online algorithms in certain cases (Section 5). This gives a positive example on when such hybrid algorithm designs are beneficial. ",
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+    {
+        "type": "text",
+        "text": "1.1 Related work ",
+        "text_level": 1,
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+        "text": "Sample-efficient online RL There is a long line of work on establishing provably sample-efficient online RL algorithms. A major portion of these works is concerned with the tabular setting with finitely many states and actions [9, 27, 19, 5, 11, 2, 22, 63]. For episodic MDPs with inhomogeneous transition functions with $S$ states, and $A$ actions, and horizon length $H$ , the optimal sample complexity for finding the $\\varepsilon$ near-optimal policy is $\\widetilde { \\cal O } ( H ^ { 3 } S A / \\varepsilon ^ { 2 } )$ , achieved by various algorithms such as UCBVI of Azar et al. [5] and UCB-Advantage of Zhang et al. [63]. Our paper adapts the reference-advantage decomposition technique of Zhang et al. [63] to designing sharp offline algorithms. Online RL with with large state/action spaces are also studied by using function approximation in conjunction with structural assumptions on the MDP [23, 61, 62, 1, 41, 21, 47, 53, 57, 14, 24]. ",
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+        "type": "text",
+        "text": "Offline RL Offline/batch RL studies the case where the agent only has access to an offline dataset obtained by executing a behavior policy in the environment. Sample-efficient learning results in offline RL typically work by assuming either sup-concentrability assumptions [39, 48, 4, 40, 15, 51, 10, 56]) or lower bounded exploration constants [58, 59] to ensure the sufficient coverage of offline data over all (relevant) states and actions. However, such strong coverage assumptions can often fail to hold in practice [16]. More recent works address this by using either policy constraint/regularization [16, 35, 29, 55], or the pessimism principle to optimize conservatively on the offline data [30, 60, 28, 25, 59, 42]. The policy-constraint/regularization-based approaches prevent the policy to visit states and actions that has no or low coverage from the offline data. Our proposed offline RL algorithm PEVI-ADV (Algorithm 1) is inspired by the pessimistic value iteration algorithms of [25, 42] and achieves an improved sample complexity over these work under the same single-policy concentrability assumption on the behavior policy. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Bridging online and offline RL Kalashnikov et al. [26] observed empirically that the performance of policies trained purely from offline data can be improved considerably by a small amount of additional online fine-tuning. A recent line of work studied low switching cost RL [6, 63, 17, 54]— which forbits online RL algorithms from switching its policy too often—as an interpolation between the online and offline settings. The same problem is also studied empirically as deployment-efficient RL [36, 46]. While we also attempt to bridge online and offline RL, our work differs from this line in that our policy finetuning setting allows a direct comparison between “fully offline” and “fully online” algorithms, whereas the low switching cost setting prohibits fully online algorithms. ",
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+    {
+        "type": "text",
+        "text": "2 Preliminaries ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Markov Decision Processes In this paper, we consider episodic Markov decision processes (MDPs) with time-inhomogeneous transitions, specified by $M = ( \\mathcal { S } , \\mathcal { A } , H , \\mathbb { P } , r )$ , where $s$ is the state space, $\\mathcal { A }$ is the action space, $H$ is the horizon length, $\\mathbb { P } = \\{ \\mathbb { P } _ { h } \\} _ { h = 1 } ^ { H }$ where $\\mathbb { P } _ { h } ( \\cdot | s , a ) \\in \\Delta _ { S }$ is the transition probabilities at step $h$ , and $r = \\{ r _ { h } : \\mathcal { S } \\times \\mathcal { A } \\to [ 0 , 1 ] \\} _ { h = 1 } ^ { H }$ =1  are the deterministic1 reward functions at time step $h \\in [ H ]$ . Without loss of generality, we assume that the initial state $s _ { 1 }$ is deterministic2. ",
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+        "type": "text",
+        "text": "Policies, value functions, visitation distributions A policy $\\pi = \\{ \\pi _ { h } ( \\cdot | s ) \\} _ { h \\in [ H ] , s \\in { \\mathcal S } }$ consists of distributions $\\pi _ { h } ( \\cdot | s ) \\in \\Delta _ { { \\cal A } }$ . We use $\\mathbb { E } _ { \\pi } [ \\cdot ]$ to denote the expectation with respect to the random trajectory induced by $\\pi$ in the MDP $M$ , that is, $\\left( s _ { 1 } , a _ { 1 } , r _ { 1 } , s _ { 2 } , a _ { 2 } , r _ { 2 } , \\ldots , s _ { H } , a _ { H } , r _ { H } \\right)$ , where $a _ { h } =$ $\\pi _ { h } ( s _ { h } )$ , $r _ { h } = r _ { h } ( s _ { h } , a _ { h } )$ , $s _ { h + 1 } \\sim \\mathbb { P } _ { h } ( \\cdot | s _ { h } , a _ { h } )$ . For each policy $\\pi$ , let $V _ { h } ^ { \\pi } : { \\cal S }  \\mathbb { R }$ and $Q _ { h } ^ { \\pi }$ : $S \\times \\mathcal { A }  \\mathbb { R }$ denote its value functions and Q functions at each time step $h \\in [ H ]$ , that is, ",
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+        "type": "equation",
+        "img_path": "images/7873e633643a6904bd586662797821de0a5e78a2d12e9b01ba967b991b5fab97.jpg",
+        "text": "$$\nV _ { h } ^ { \\pi } ( s ) : = \\mathbb { E } _ { \\pi } \\bigg [ \\sum _ { h ^ { \\prime } = h } ^ { H } r _ { h ^ { \\prime } } ( s _ { h ^ { \\prime } } , a _ { h ^ { \\prime } } ) \\bigg | s _ { h } = s \\bigg ] , Q _ { h } ^ { \\pi } ( s , a ) : = \\mathbb { E } _ { \\pi } \\bigg [ \\sum _ { h ^ { \\prime } = h } ^ { H } r _ { h ^ { \\prime } } ( s _ { h ^ { \\prime } } , a _ { h ^ { \\prime } } ) \\bigg | s _ { h } = s , a _ { h } = a \\bigg ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The operators $\\mathbb { P } _ { h }$ and $\\mathbb { V } _ { h }$ are defined as $[ \\mathbb { P } _ { h } V _ { h + 1 } ] ( s , a ) : = \\mathbb { E } [ V _ { h + 1 } ( s ^ { \\prime } ) | s _ { h } = s , a _ { h } = a ]$ and $[ \\mathbb { V } _ { h } V _ { h + 1 } ] ( s , a ) : = \\mathrm { V a r } [ V _ { h + 1 } ( s ^ { \\prime } ) | s _ { h } = s , a _ { h } = a ]$ for any value function $V _ { h + 1 }$ at time step $h + 1$ ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "We also use $\\widehat { \\mathbb { P } } _ { h }$ and $\\widehat { V } _ { h }$ to denote empirical versions of these operators building on estimated models (which will be clear in the context). ",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "We use $\\pi _ { \\star } : = \\arg \\operatorname* { m a x } _ { \\pi } V _ { 1 } ^ { \\pi } ( s _ { 1 } )$ to denote any optimal policy, and $V _ { h } ^ { \\star } : = V _ { h } ^ { \\pi _ { \\star } }$ and $Q _ { h } ^ { \\star } : = Q _ { h } ^ { \\pi _ { \\star } }$ to denote the value function and $\\mathrm { Q }$ function of $\\pi ^ { \\star }$ at all $h \\in [ H ]$ . Throughout this paper, our learning goal is to find an near-optimal policy $\\widehat { \\pi }$ such that ${ \\cal V } _ { 1 } ^ { \\star } ( s _ { 1 } ) \\dot { - } \\dot { \\cal V } _ { 1 } ^ { \\widehat \\pi } ( s _ { 1 } ) \\le \\varepsilon$ . ",
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+        "type": "text",
+        "text": "Finally, we let $d _ { h } ^ { \\pi }$ denote the state(-action) visitation distributions of $\\pi$ at time step $h \\in [ H ]$ : ",
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+        "type": "equation",
+        "img_path": "images/bb8236f38906f659940502f345ea6956ae293423aa6c4d209be9dc1aa8b10fe2.jpg",
+        "text": "$$\n\\begin{array} { r } { d _ { h } ^ { \\pi } ( s ) : = \\mathbb { P } ( s _ { h } = s | \\pi ) , \\mathrm { ~ a n d ~ } d _ { h } ^ { \\pi } ( s , a ) : = \\mathbb { P } ( s _ { h } = s , a _ { h } = a | \\pi ) . } \\end{array}\n$$",
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+        "type": "text",
+        "text": "Miscellaneous We use standard $O ( \\cdot )$ and $\\Omega ( \\cdot )$ notation: $A = O ( B )$ is defined as $A \\leq C B$ for some absolute constant $C > 0$ (and similarly for $\\Omega$ ). The tilded notation $A = { \\widetilde { O } } ( B )$ denotes $A \\leq C L \\cdot B$ where $L$ is a poly-logarithmic factor of problem parameters. ",
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+        "type": "text",
+        "text": "2.1 Policy Finetuning ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We now introduce the setting of policy finetuning. A policy finetuning problem consists of an MDP $M$ and a reference policy $\\mu$ . During the learning stage, the learner can perform the following two types of moves: ",
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+        "type": "text",
+        "text": "(a) Play an episode in the MDP $M$ using any policy (i.e. learner has online interactive access to $M$ ). (b) Access the values of the reference policy $\\mu _ { h } ( a | s )$ for all $( h , s , a )$ . For example, the learner can use it to sample actions $a \\sim \\mu _ { h } ( \\cdot | s )$ for any $h , s$ for arbitrarily many times during learning. ",
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+        "type": "text",
+        "text": "The goal of the learner is to output $\\varepsilon$ near-optimal policy $\\widehat { \\pi }$ within as few episodes of play (within the MDP) as possible. ",
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+        "type": "text",
+        "text": "A unique feature about the policy finetuning setting is that it allows both online interactive plays via any online RL algorithm (not necessarily using $\\mu$ ), as well as offline reduction which simply collects data by executing the reference policy $\\mu$ and do anything with the collected dataset. In particular, this means that any algorithm for offline policy optimization (based on offline datasets) also gives an algorithm for policy finetuning via this offline reduction. Therefore, policy finetuning offers a common playground for both online and offline type algorithms with a unified learning goal. ",
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+        "type": "text",
+        "text": "Assumption on reference policy Throughout most of this paper (except for Section 5), we consider the following assumption on the reference policy $\\mu$ . ",
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+        "type": "text",
+        "text": "Assumption A (Single-policy concentrability). The reference policy $\\mu$ satisfies that ",
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+    {
+        "type": "equation",
+        "img_path": "images/35c3cb98cdbff3125af0092395a4d39c6db513187b43edac27a1105173466856.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { h \\in [ H ] , ( s , a ) \\in S \\times A } \\frac { d _ { h } ^ { \\pi _ { \\star } } ( s , a ) } { d _ { h } ^ { \\mu } ( s , a ) } \\leq C ^ { \\star }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "(with the convention $0 / 0 = 0 ,$ ) for some deterministic optimal policy $\\pi _ { \\star }$ and constant $C ^ { \\star } \\geq 1$ ",
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+        "type": "text",
+        "text": "The single-policy concentrability characterizes the distance between the visitation distributions of the reference policy $\\mu$ and some optimal policy $\\pi ^ { \\star }$ . This assumption is considered in the recent work of Rashidinejad et al. [42] on offline RL and is more relaxed than previously assumed concentrability assumptions which typically requires the supremum concentrability against all possible $\\pi$ ’s to be bounded [10]. We consider this assumption as it both allows efficient offline RL algorithms [42], and is perhaps also a sensible measure of quality for the reference policy in policy finetuning. ",
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+        "type": "text",
+        "text": "3 Sharp offline learning via reference-advantage decomposition ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We begin by investigating the sharpest sample complexity for policy finetuning via the offline reduction approach. This requires us to design sharp offline RL algorithms that run on the dataset $\\mathcal { D }$ collected by executing $\\mu$ . We emphasize that this is both an interesting offline RL question on its own right, and also important for our later discussions on lower bounds and other algorithms for policy finetuning, as the sharpest sample complexity via offline reduction provides a solid baseline. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Warm-up: VI-LCB As a warm-up, we first show that a finite-horizon variant of the VILCB (Value Iteration with Lower Confidence Bounds) algorithm of Rashidinejad et al. [42] achieves sample complexity $\\widetilde { O } ( H ^ { 5 } S C ^ { \\star } / \\varepsilon ^ { 2 } )$ for finding an $\\varepsilon$ near-optimal policy. This result is similar to the $\\widetilde { O } ( S C ^ { \\star } / ( 1 - \\gamma ) ^ { 5 } \\varepsilon ^ { 2 } )$ guarantee3 for the original VI-LCB in infinite-horizon discounted MDPs [42, Theorem 6]. The main ingredients of our VI-LCB algorithm is a pessimistic value iteration procedure in which we perform value iteration on the empirical model estimated from the dataset $\\mathcal { D }$ , along with a negative Hoeffding bonus term to impose pessimism. Due to space constraints, the algorithm description (Algorithm 3) and the proof of Theorem 1 are deferred to Appendix B. ",
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+        "text": "Theorem 1 (VI-LCB for finite-horizon MDPs). Suppose the reference policy $\\mu$ satisfies the singlepolicy concentrability (Assumption $A$ ). Then with probability at least $1 - \\delta$ , VI-LCB (Algorithm 3) outputs a policy $\\widehat { \\pi }$ and value estimate $\\widehat { V }$ such that ",
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+        "text": "$$\n\\begin{array} { r l } & { ) \\operatorname* { m a x } _ { h \\in [ H ] } \\sum _ { s \\in { \\mathcal { S } } } d _ { h } ^ { \\pi _ { \\star } } ( s ) ( V _ { h } ^ { \\star } ( s ) - \\widehat { V } _ { h } ( s ) ) \\leq \\varepsilon , } \\\\ & { } \\\\ & { V _ { 1 } ^ { \\star } ( s _ { 1 } ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s _ { 1 } ) \\leq \\varepsilon , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "within $n = \\widetilde O \\bigl ( H ^ { 5 } S C ^ { \\star } / \\varepsilon ^ { 2 } \\bigr )$ episodes. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Theorem 1 serves two main purposes. First, the $\\widetilde { O } ( H ^ { 5 } S C ^ { \\star } / \\varepsilon ^ { 2 } )$ sample complexity asserted in Theorem 1(b) provides a first result for offline RL (and offline reduction for policy finetuning) under single-policy concentrability in finite-horizon MDPs. Second, the value estimation bound in Theorem 1(a) shows that the estimated value function $\\widehat { V } _ { h } ( s )$ provided by VI-LCB is close to the optimal value ${ V } _ { h } ^ { \\star } ( s )$ at every step $h \\in [ H ]$ , in terms of the weighted average with $d _ { h } ^ { \\pi _ { \\star } } ( s )$ . Our next algorithm PEVI-ADV builds on this property so that VI-LCB can be used as a “warm-up” learning procedure that provides a high-quality value estimate. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Sharp offline learning via reference-advantage decomposition We now design a new sharp algorithm PEVI-ADV which achieves an improved $\\widetilde { O } ( H ^ { 3 } S C ^ { \\star } / \\varepsilon ^ { 2 } )$ sample complexity (for small enough $\\varepsilon$ ). This improves over VI-LCB by $\\widetilde O ( H ^ { 2 } )$ and is the first algorithm that matches the sample complexity lower bound. PEVI-ADV adds two new ingredients over VI-LCB in order to achieve the $\\tilde { O } ( \\bar { H } ^ { 2 } )$ improvement: ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "1. We replace the Hoeffding-style bonus in VI-LCB with a Bernstein-style bonus. This shaves off one $H$ factor in the sample complexity via the total variance property (Lemma C.4).   \n2. Both VI-LCB and our PEVI-ADV use data splitting to make sure that the estimated value $\\widehat { V } _ { h + 1 }$ and empirical transitions $\\widehat { \\mathbb { P } } _ { h }$ are estimated using different subsets of $\\mathcal { D }$ , this yields conditional independence that is required in bounding concentration terms of the form $( \\widehat { \\mathbb { P } } _ { h } - \\mathbb { P } _ { h } ) \\widehat { V } _ { h + 1 }$ . However, applied naively, this data splitting induces one undesired $H$ factor in the sample complexity as we need to split $\\mathcal { D }$ into $H$ folds and thus each $\\mathbb { P } _ { h }$ is estimated using only $n / H$ episodes of data. As a technical crux of this algorithm, we overcome this issue by adapting the reference-advantage decomposition technique of Zhang et al. [63]. This technique proposes to learn an initial reference value function $\\widehat { V } ^ { \\mathrm { r e f } }$ of good quality in a certain sense, and then performing the following type of approximate value iteration (using the right-hand side as the algorithm update): ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6301bf8363369b3245af6f0e014891ac8da26d003c6923c0d2812a6bc3fc6748.jpg",
+        "text": "$$\n\\mathbb { P } _ { h } \\widehat { V } _ { h + 1 } \\approx \\widehat { \\mathbb { P } } _ { h , 0 } \\widehat { V } _ { h + 1 } ^ { \\mathrm { r e f } } + \\widehat { \\mathbb { P } } _ { h , 1 } \\Big ( \\widehat { V } _ { h + 1 } - \\widehat { V } _ { h + 1 } ^ { \\mathrm { r e f } } \\Big ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Above, $\\widehat { V } _ { h + 1 } , \\widehat { \\mathbb { P } } _ { h , 0 }$ , and $\\widehat { \\mathbb { P } } _ { h , 1 }$ are estimated on three disjoint subsets of the data. The advantage of this approach is that, due to this new independence structure, $\\widehat { \\mathbb { P } } _ { h , 0 }$ for different $h \\in [ H ]$ can be estimated on the same set of trajectories without $H$ -fold splitting, which shaves off the $H$ factor within this part. On the other hand, estimating $\\widehat { \\mathbb { P } } _ { h , 1 }$ still requires $H$ -fold splitting, yet this would not hurt the sample complexity if the magnitude of $( \\widehat { V } _ { h + 1 } - \\widehat { V } _ { h + 1 } ^ { \\mathrm { r e f } } )$ is much smaller than its naive upper bound $O ( H )$ —we show this can be achieved by using VI-LCB to learn $\\widehat { V } ^ { \\mathrm { r e f } }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Require: Dataset $\\mathcal { D } = \\left\\{ ( s _ { 1 } ^ { ( i ) } , a _ { 1 } ^ { ( i ) } , r _ { 1 } ^ { ( i ) } , \\dotsc , s _ { H } ^ { ( i ) } , a _ { H } ^ { ( i ) } , r _ { H } ^ { ( i ) } ) \\right\\} _ { i = 1 } ^ { n }$ collected by executing $\\mu$ in $M$ . ",
+        "bbox": [
+            176,
+            108,
+            803,
+            135
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "1: Split the dataset $\\mathcal { D }$ into ${ \\mathcal { D } } _ { \\mathrm { r e f } }$ , $\\mathcal { D } _ { 0 }$ and $\\left\\{ \\mathcal { D } _ { h , 1 } \\right\\} _ { h = 1 } ^ { H }$ uniformly at random: ",
+        "bbox": [
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+            136,
+            665,
+            154
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e5097df3ca06abf8dbc946d659fc8a788c2286d2d2a80e141d6ae92a2678e9e9.jpg",
+        "text": "$$\nn _ { \\mathrm { r e f } } : = \\left| \\mathcal { D } _ { \\mathrm { r e f } } \\right| = n / 3 , n _ { 0 } : = \\left| \\mathcal { D } _ { 0 } \\right| = n / 3 , n _ { 1 , h } : = \\left| \\mathcal { D } _ { h , 1 } \\right| : = n / ( 3 H ) ( n _ { 1 } : = n / 3 ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            157,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "2: Learn a reference value function $\\widehat { V } ^ { \\mathrm { r e f } } \\gets \\mathrm { V I - L C B } ( \\mathcal { D } _ { \\mathrm { r e f } } )$ via VI-LCB (Algorithm 3). ",
+        "bbox": [
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+            181,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3: Let $N _ { h , 0 } ( s , a )$ and $N _ { h , 0 } ( s , a , s ^ { \\prime } )$ denote the visitation count of $( s , a )$ and $( s , a , s ^ { \\prime } )$ at step $h$ within dataset $\\mathcal { D } _ { 0 }$ . Construct empirical model estimates: ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f6bbfea71cb04ec9125ac2ee517d180cb78a7ce89b26e41ae4b0ee6a7cf350e9.jpg",
+        "text": "$$\n\\widehat { \\mathbb { P } } _ { h , 0 } ( s ^ { \\prime } | s , a ) \\gets \\frac { N _ { h , 0 } ( s , a , s ^ { \\prime } ) } { N _ { h , 0 } ( s , a ) \\vee 1 } , \\quad \\mathrm { a n d } \\quad \\widehat { r } _ { h , 0 } ( s , a ) \\gets r _ { h } ( s , a ) \\mathbb { 1 } \\left\\{ N _ { h , 0 } ( s , a ) \\geq 1 \\right\\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            243,
+            231,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Similarly define $N _ { h , 1 } ( s , a ) , N _ { h , 1 } ( s , a , s ^ { \\prime } ) ,$ $( \\widehat { r } _ { h , 1 } , \\widehat { \\mathbb { P } } _ { h , 1 } )$ for all $h \\in [ H ]$ based on dataset $\\mathcal { D } _ { h , 1 }$ ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4: Set $\\begin{array} { r } { b _ { h , 0 } ( s , a ) \\gets c \\cdot \\left( \\sqrt { \\frac { [ \\widehat { \\Psi } _ { h , 0 } \\widehat { V } _ { h + 1 } ^ { \\mathrm { r e f } } ] ( s , a ) \\iota } { N _ { h , 0 } ( s , a ) \\vee 1 } } + \\frac { H \\iota } { N _ { h , 0 } ( s , a ) \\vee 1 } \\right) } \\end{array}$ for all $( h , s , a )$ , where $\\iota : = \\log ( H S A / \\delta )$ ",
+        "bbox": [
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+            294,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5: Set $\\widehat { V } _ { H + 1 } ( s ) \\gets 0$ for all $s \\in S$ . ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "6: for $h = H , \\ldots , 1$ do ",
+        "bbox": [
+            181,
+            344,
+            341,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "8: Perform pessimistic value update for all $( s , a )$ ",
+        "bbox": [
+            181,
+            391,
+            522,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/03804901a64d6ec413dda5beaa8c33e95f6a754ecb3acc678e41a1f27ec71595.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\widehat { Q } _ { h } ( s , a ) \\gets \\widehat { r } _ { h , 0 } ( s , a ) + \\left[ \\widehat { \\mathbb { P } } _ { h , 0 } \\widehat { V } _ { h + 1 } ^ { \\mathrm { r e f } } \\right] ( s , a ) - b _ { h , 0 } ( s , a ) + \\left[ \\widehat { \\mathbb { P } } _ { h , 1 } ( \\widehat { V } _ { h + 1 } - \\widehat { V } _ { h + 1 } ^ { \\mathrm { r e f } } ) \\right] ( s , a ) - b _ { h , 1 } ( s , a ) } \\\\ & { \\widehat { V } _ { h } ( s ) \\gets \\left[ \\operatorname* { m a x } _ { a } \\widehat { Q } _ { h } ( s , a ) \\right] \\vee 0 . } \\\\ & { \\mathrm { S e t } \\widehat { \\pi } _ { h } ( s ) \\gets \\arg \\operatorname* { m a x } _ { a } \\widehat { Q } _ { h } ( s , a ) \\mathrm { f o r } \\mathrm { a l l } s \\in S . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            411,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "b10: end for ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "11: return Policy $\\widehat { \\pi } = \\{ \\widehat { \\pi } _ { h } \\} _ { h \\in [ H ] }$ ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We instantiate this plan by carefully using VI-LCB to learn the reference value function $\\widehat { V } ^ { \\mathrm { r e f } }$ , combined with tight Bernstein bonuses, to shave off another $H$ factor in the sample complexity. The full PEVI-ADV algorithm is provided in Algorithm 1. We now present its guarantee in the following theorem. The proof can be found in Appendix C. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2 (Sharp offline learning via PEVI-ADV). Suppose the reference policy $\\mu$ satisfies the single-policy concentrability (Assumption $A$ ). Then with probability at least $1 - \\delta$ , PEVI-ADV (Algorithm 1) outputs a policy $\\widehat { \\pi }$ and value estimate $\\widehat { V }$ such that ",
+        "bbox": [
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+            609,
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e5f8f7d074a7daa9d44224e48408764c5d8c3405543a64f4319f94bddf53150c.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\operatorname* { m a x } _ { h \\in [ H ] } \\sum _ { s \\in {  { \\mathcal { S } } } } d _ { h } ^ { \\pi _ { \\star } } ( s ) ( V _ { h } ^ { \\star } ( s ) - \\widehat { V } _ { h } ( s ) ) \\leq \\varepsilon , } \\\\ & { } \\\\ & { V _ { 1 } ^ { \\star } ( s _ { 1 } ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s _ { 1 } ) \\leq \\varepsilon , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "within $n = \\widetilde O \\left( H ^ { 3 } S C ^ { \\star } / \\varepsilon ^ { 2 } + H ^ { 5 . 5 } S C ^ { \\star } / \\varepsilon \\right)$ episodes. ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Near-optimal offline RL under single-policy concentrability For small enough $\\varepsilon \\le H ^ { - 2 . 5 }$ , Theorem 2 achieves $\\widetilde { O } ( H ^ { 3 } S C ^ { \\star } / \\varepsilon ^ { 2 } )$ sample complexity for finding the $\\varepsilon$ near-optimal policy from the offlien dataset $\\mathcal { D }$ . This is the first cubic horizon dependence for offline RL under single-policy concentrability, which improves over recent works [25, 42] in this setting and resolves the open question of [42]. For $C ^ { \\star } \\geq 2$ , our sample complexity further matches the information-theoretical lower bound $\\Omega ( H ^ { 3 } S C ^ { \\star } / \\varepsilon ^ { 2 } )$ up to log factors4. We remark that tight hoziron dependence has also been achieved in several recent works offline RL [58, 59, 43] which are however quite different from (and do not imply) ours in both the assumptions (on the behavior policy) and the analyses. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 Lower bound for policy finetuning ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We now switch gears to considering the policy finetuning problem with any algorithm, not necessarily restricted to the offline reduction approach. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Two baselines: offline reduction $\\pmb { \\& }$ purely online RL A first observation is that naive offline reduction is already a strong baseline for policy finetuning, by our Theorem 2: Our PEVI-ADV algorithm only collects data with $\\mu$ and does not do any online exploration, yet achieves a sharp $\\bar { \\tilde { O } } ( H ^ { 3 } S C ^ { \\star } / \\bar { \\varepsilon } ^ { 2 } )$ sample complexity for finding a near-optimal policy. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "On the other hand, as the policy finetuning setting allows online interaction, purely online $R L$ is another baseline algorithm: Simply run any sample-efficient online RL algorithm (which typically uses optimism to encourage exploration) from scratch, and disregard the reference policy $\\mu$ . Using any sharp online RL algorithm such as UCBVI [5], this approach can find an $\\varepsilon$ near-optimal policy within $\\widetilde { \\cal O } ( H ^ { 3 } S A / \\varepsilon ^ { 2 } )$ episodes of play. Note that whether this is advantageous over the offline reduction boils down to the comparison between $C ^ { \\star }$ and $A$ , which makes sense intuitively. For example, $C ^ { \\star } \\leq o ( A )$ means that $\\mu$ is perhaps close enough to $\\pi _ { \\star }$ so that collecting data from $\\mu$ and run offline policy optimization is a stronger algorithm than exploring from scratch. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Given these two baselines, it is natural to ask whether there exists an algorithm that improves over both — Can we design an algorithm that performs some amount of optimistic exploration, yet also utilizes the knowledge of $\\mu$ , so as to achieve a better rate than both offline reduction and purely online RL? In this section, we provide an information-theoretic lower bound showing that, perhaps surprisingly, the answer is negative: there is an $\\Omega ( H ^ { 3 } S \\operatorname* { m i n } { \\{ C ^ { \\star } , A \\} } / \\varepsilon ^ { 2 } )$ sample complexity lower bound for any policy finetuning algorithm, if we still assume that $\\mu$ satisfies $C ^ { \\star }$ single-policy concentrability. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Lower bound To formally state our lower bound, we define the class of problems ",
+        "bbox": [
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+            446,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/aae97b6bd8b27064e6832f5ed965a267d4775d9eb74dd94cd8d78d72ec133645.jpg",
+        "text": "$$\n\\mathcal { M } _ { C ^ { \\star } } : = \\Bigg \\{ ( M , \\mu ) : ~ \\mathrm { E x i s t s ~ d e t e r m i n i s t i c } ~ \\pi _ { \\star } ~ \\mathrm { o f } ~ M \\mathrm { ~ s u c h ~ t h a t ~ } \\operatorname* { s u p } _ { h , s , a } \\frac { d _ { h } ^ { \\pi _ { \\star } } ( s , a ) } { d _ { h } ^ { \\mu } ( s , a ) } \\leq C ^ { \\star } \\Bigg \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We recall that a policy finetuning algorithm for problem $( M , \\mu )$ is defined as any algorithm that can play in the MDP $M$ for $n$ episodes, has full knowledge of the reference policy $\\mu$ , and outputs a policy $\\widehat { \\pi }$ after playing in the MDP. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "With these definitions ready, we now state our lower bound for policy finetuning. The proof of Theorem 3 can be found in Appendix D. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3 (Lower bound for policy finetuning). Suppose $S , H \\geq 3 , A \\geq 2 , C ^ { \\star } \\geq 2 .$ . Then, there exists an absolute constant $c _ { 0 } > 0$ such that for any $\\varepsilon \\le 1 / 1 2$ and any online finetuning algorithm that outputs a policy $\\widehat { \\pi }$ , if the number of episodes ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8289bb00773e69057493f9c137673f555e072fba6a64ae8993e48e3b9cd9db28.jpg",
+        "text": "$$\nn \\leq c _ { 0 } \\cdot H ^ { 3 } S \\operatorname* { m i n } { \\{ C ^ { \\star } , A \\} } / \\varepsilon ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "then there exists a problem instance $( M , \\mu ) \\ \\in \\ { \\mathcal { M } } _ { C ^ { \\star } }$ on which the algorithm suffers from $\\varepsilon$ - suboptimality: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ecd5b38c546f5c899fd67c32c941af297d097102dc7e9bf08346f159d934c020.jpg",
+        "text": "$$\n\\mathbb { E } _ { M } \\left[ V _ { 1 , M } ^ { \\star } - V _ { 1 , M } ^ { \\widehat { \\pi } } \\right] \\geq \\varepsilon ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where the expectation $\\mathbb { E } _ { M }$ is w.r.t. the randomness during the algorithm execution within MDP M . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Either offline reduction or purely online is optimal Theroem 3 shows that any policy finetuning algorithm needs to play at least $\\Omega ( H ^ { 3 } S \\operatorname* { m i n } \\{ C ^ { \\star } , A \\} / \\varepsilon ^ { 2 } )$ episodes in order to find an $\\varepsilon$ near-optimal policy. Crucially, this implies that either a sharp offline reduction (e.g. our PEVI-ADV algorithm) or purely online RL matches the lower bound (up to log), depending on whether $C ^ { \\star } \\lesssim A$ . In other words, if we have the knowledge of whether $C ^ { \\star } \\leq A$ , choosing the right one of these two baseline algorithms will yield the optimal sample complexity. Perhaps surprisingly, this rules out the possibility of designing any algorithm “in between” that combines online exploration and knowledge of $\\mu$ to improve the sample complexity, at least in the worst-case over all problems in $\\mathcal { M } _ { C ^ { \\star } }$ . We argue that this “no algorithm in between” phenomenon may be due to the single-policy concentrability assumption being too strong such that offline reduction already achieves a rather competitive sample ",
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+        "type": "text",
+        "text": "Require: MDP $M$ , reference policy $\\mu$ ",
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+        "type": "text",
+        "text": "6: Denote the final output of UCBVI-UPLOW as ",
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+        "img_path": "images/38aecbd747a8359b025d92074bc42138789afaf8c87435c50b24c72fe719285d.jpg",
+        "text": "$$\n\\begin{array} { r } { ( \\overline { { V } } _ { h _ { \\star } + 1 } , \\underline { { V } } _ { h _ { \\star } + 1 } , \\widehat { \\pi } _ { ( h _ { \\star } + 1 ) : H } ^ { \\mathrm { U C B } } ) \\gets \\mathbf { U C B V I - U P L O W } ( n _ { \\mathrm { U C B } } ) . } \\end{array}\n$$",
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+        "text": "7: # Stage 2: Learn step $1 : h _ { \\star }$ via executing $\\mu +$ pessimistic offline policy optimization 8: Collect 9: Learn p $\\mathcal { D }  \\{ n - n _ { \\mathrm { U C B } }$ episodes of data using policy he TRUNCATED-PEVI-ADV( $\\mu$ up to step lgorithm 5 $h _ { \\star } \\}$ . $\\hat { \\pi } _ { 1 : h _ { \\star } } ^ { \\mathrm { P E V I } }$ ",
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+        "img_path": "images/b6ba15af63ffccca718f90fa9b56c4223b5136a7c54450b442011be714f609a7.jpg",
+        "text": "$$\n\\widehat { \\pi } _ { 1 : h _ { \\star } } ^ { \\mathrm { P E V I } }  \\mathrm { T R U N C A T E D - P E V I - A D V } ( \\mathcal { D } , h _ { \\star } , \\underline { { V } } _ { h _ { \\star } + 1 } ) .\n$$",
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+        "type": "text",
+        "text": "$\\widehat { \\pi } = ( \\widehat { \\pi } _ { 1 : h _ { \\star } } ^ { \\mathrm { P E V I } } , \\widehat { \\pi } _ { ( h _ { \\star } + 1 ) : H } ^ { \\mathrm { U C B } } )$ ",
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+        "type": "text",
+        "text": "complexity $\\widetilde { O } ( H ^ { 3 } S C ^ { \\star } / \\varepsilon ^ { 2 } )$ . We investigate policy finetuning beyond the single-policy concentrability assumption in Section 5. ",
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+        "type": "text",
+        "text": "We also remark that Theorem 3 generalizes both the $\\Omega ( H ^ { 3 } S A / \\varepsilon ^ { 2 } )$ lower bound for online RL [11, 58, 13] into the policy finetuning problem, as well as the $\\Omega ( \\dot { H } ^ { 3 } S C ^ { \\star } / \\varepsilon ^ { 2 } )$ lower bound for offline RL under single-policy concentrability with $C ^ { \\star } \\geq 2 [ 4 2 ] ^ { 5 }$ . Further, Theorem 3 directly implies an $\\Omega ( H ^ { 3 } S C ^ { \\star } / \\varepsilon ^ { \\star } )$ lower bound for offline RL with $2 \\leq C ^ { \\star } \\leq O ( A )$ , as any algorithm for offline policy optimization is also an algorithm for policy finetuning via the offline reduction. ",
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+        "type": "text",
+        "text": "Proof intuition; Construction of hard instance The proof of Theorem 3 constructs a family of hard MDPs that requires solving $H S$ “independent” bandit problems with $A$ arms, similar as in existing $\\Omega ( H ^ { 3 } S A / \\bar { \\varepsilon } ^ { 2 } )$ lower bounds for online RL [11, 58]. However, our key modification is that we let the optimal arms to be always within the first $K : = \\operatorname* { m i n } \\left\\{ C ^ { \\star } , A \\right\\}$ actions instead of all $A$ actions, and we define our reference policy $\\mu$ to play uniformly within $[ K ]$ . This $\\mu$ has the following properties: ",
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+        "text": "• $\\mu$ satisfies $C ^ { \\star }$ single-policy concentrability for any MDP in this family (Lemma D.1). • $\\mu$ provides the knowledge that the optimal actions are within $[ K ]$ , but no other knowledge about the optimal actions. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Therefore, with $\\mu$ at hand, any policy finetuning algorithm can “gain the knowledge” that the optimal actions are within $[ K ]$ , but still needs to try all $K$ actions in order to solve each bandit problem— rigorizing this information-theoretically gives the $\\Omega ( H ^ { 3 } S K / \\varepsilon ^ { 2 } ) = \\Omega ( H ^ { 3 } S \\operatorname* { m i n } { \\{ C ^ { \\star } , A \\} } \\mathbf { \\hat { / } } \\varepsilon ^ { 2 } )$ lower bound. ",
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+        "type": "text",
+        "text": "5 Hybrid offline/online algorithm for policy finetuning ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Towards circumventing the lower bound in Theorem 3, in this section, we study policy finetuning under more relaxed assumptions on the reference policy $\\mu$ . A weaker $\\mu$ will induce a higher sample complexity for naive offline reduction approaches, and thus yields opportunities for designing new algorithms that can potentially better utilize $\\mu$ . ",
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+        "type": "text",
+        "text": "More concretely, we consider the following relaxation: We assume $\\mu$ satisfies partial concentrability only up to a certain time-step $h _ { \\star } \\leq H$ , and may not have any bounded concentrability at steps $h > h _ { \\star }$ . We formalize this in the following ",
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+        "text": "Assumption B ( $h _ { \\star }$ -partial concentrability). The reference policy $\\mu$ satisfies the single-policy concentrability with respect to $\\pi _ { \\star }$ up to step $h _ { \\star }$ only: ",
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+        "type": "equation",
+        "img_path": "images/00aa51480e3252a0217ca5cc8641a95a1f0a15c01cb3fd0ca29b24de3cdf5ef0.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { h \\leq h _ { \\star } } \\operatorname* { m a x } _ { s , a \\in { \\mathcal { S } } \\times { \\mathcal { A } } } { \\frac { d _ { h } ^ { \\pi _ { \\star } } ( s , a ) } { d _ { h } ^ { \\mu } ( s , a ) } } \\leq C ^ { \\mathrm { p a r t i a l } }\n$$",
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+        "type": "text",
+        "text": "(with the convention $0 / 0 = 0 ,$ ), where $\\pi _ { \\star }$ is some deterministic optimal policy of the MDP, and constant Cpartial . ",
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+        "type": "text",
+        "text": "Algorithm description We design a hybrid offline/online algorithm HOOVI (presented in Algorithm 2) for policy finetuning under the partial concentrability assumption. At a high-level, the algorithm consists of two main stages: ",
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+        "type": "text",
+        "text": "• In the first stage, it runs an online algorithm UCBVI-UPLOW which uses optimistic exploration to find a near-optimal policy ${ \\widehat { \\pi } } ^ { \\mathrm { U C B } }$ and an accurate value estimate for steps $( h _ { \\star } + 1 ) : H$ . • In the second stage, we run a TRUNCATED-PEVI-ADV algorithm, which collects data from $\\mu$ and runs offline policy optimization to find a near-optimal policy $\\widehat { \\pi } ^ { \\mathrm { { P E V I } } }$ for steps $1 : h _ { \\star }$ , building on the lower value estimate $\\underline { { V } } _ { h _ { \\star } + 1 }$ from the first stage. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This strategy makes sense intuitively as the reference policy $\\mu$ does not have guarantees for steps $h _ { \\star } + 1 : H$ and thus the algorithm is required to perform optimistic exploration first to get a good policy. However, additional technical cares are needed in order to make the above algorithm provably sample-efficient. The analysis of the second stage requires the online algorithm in the first stage to not only perform fast exploration (e.g. by using upper confidence bounds), but also output a lower value estimate for step $h _ { \\star } + 1$ , and in addition output a final output policy that achieves at least the value of the lower value estimate at every state $s \\in S$ . Such lower bounds are not directly available in standard online RL algorithms such as UCBVI [5]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We resolve this by designing the UCBVI-UPLOW algorithm (detailed description in Algorithm 4), which is a modification of the Nash-VI Algorithm of Liu et al. [34] (for two-player Markov games) into the single-player case. This algorithm is particularly suitable for our purpose since it maintains both upper bounds of $V ^ { \\star }$ and lower bounds for the value function of the deployed policies. Our UCBVI-UPLOW further integrates the certified policy technique of Bai et al. [7] to make sure that its output policy achieves value greater or equal than the lower bound at every state (similar guarantees can also be obtained by the policy certificate technique of Dann et al. [12]). ",
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+    {
+        "type": "text",
+        "text": "We now state our main theoretical guarantee for the HOOVI algorithm. The proof can be found in Appendix E. ",
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+        "type": "text",
+        "text": "Theorem 4 (Hybrid online $/$ offline learning for policy finetuning). Suppose the reference policy $\\mu$ satisfies the partial concentrability (Assumption $B$ ) up to some step $h _ { \\star } \\leq H$ . Then for small enough $\\varepsilon \\le \\operatorname* { m i n } \\left\\{ h _ { \\star } ^ { - 2 . 5 } , C ^ { \\mathrm { p a r t i a l } } / S \\right\\}$ , HOOVI (Algorithm 2) outputs a policy $\\widehat { \\pi }$ such that ${ \\cal V } _ { 1 } ^ { \\star } ( s _ { 1 } ) -$ $V _ { 1 } ^ { \\widehat { \\pi } } ( s _ { 1 } ) \\leq \\varepsilon$ with probability at least $1 - \\delta$ , within ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/015566f49ca90ad433d4834392f4309449b8b55819547302198c2e6e6e3ef509.jpg",
+        "text": "$$\nn = { \\tilde { O } } \\left( { \\frac { H ^ { 2 } h _ { \\star } S C ^ { \\mathrm { p a r t i a l } } + ( H - h _ { \\star } ) ^ { 3 } S A ( C ^ { \\mathrm { p a r t i a l } } ) ^ { 2 } } { \\varepsilon ^ { 2 } } } \\right)\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "episodes of play. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Comparison against offline reduction and purely online algorithms The sample complexity in Theorem 4 compares favorably against both naive offline reduction as well as purely online algorithms in certain situations. First, naive offline reduction with $\\mu$ does not have any guarantee since $\\mu$ is not assumed to have a finite single-policy concentrability at $h \\geq h _ { \\star } + 1$ . We can modify $\\mu$ into $\\mu ^ { \\prime }$ that plays uniformly within $\\mathcal { A }$ at steps $h \\geq h _ { \\star } + 1$ ; the single-policy concentrability coefficient of $\\mu ^ { \\prime }$ is guaranteed to be finite but scales exponentially as $O ( \\breve { A } ^ { H - \\mathbf { \\breve { h } _ { \\star } } } )$ in the worst case, leading to a sample complexity much worse than ours (which is polynomial in $H , S , A )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "On the other hand, a sharp online algorithm can still achieve $\\widetilde { \\cal O } ( H ^ { 3 } S A / \\varepsilon ^ { 2 } )$ in this setting (by optimistic exploration from scratch). Our Theorem 4 is in general incomparable with this, but can be better in cases when both $C ^ { \\mathrm { p a r t i a l } }$ and $H \\mathrm { ~ - ~ } h _ { \\star }$ are small, e.g., if $\\ { \\dot { C } } ^ { \\mathrm { p a r t i a l } } = o ( A )$ and $( H - h _ { \\star } ) / H = o ( ( C ^ { \\mathrm { p a r t i a l } } ) ^ { - 2 / 3 } )$ . This makes sense intuitively as our hybrid offline/online algorithm benefits the most if the length requiring exploration $( H - h _ { \\star } )$ is small, and the partial concentrability $C ^ { \\mathrm { p a r t i a l } }$ is small so that $\\mu$ still has a high-quality for the first $h _ { \\star }$ steps. To best of our knowledge, this is first result that characterizes when the sample complexity of such hybrid algorithms can be beneficial over purely online or offline algorithms. ",
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+    {
+        "type": "text",
+        "text": "6 Conclusion & discussions ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "This paper studies policy finetuning, a new reinforcement learning setting that allows us to compare and connect sample-efficient online and offline reinforcement learning. We establish sharp upper and lower bounds for policy finetuning under various assumptions on the reference policy. Our bounds show that the optimal policy finetuning algorithm is either offline reduction or a purely online algorithm in the specific setting where the reference policy satisfies single-policy concentrability, and we also show that a hybrid online/offline algorithm can be advantageous over both in more relaxed settings. Many directions could be of interest for future research, such as alternative assumptions on the reference policy, or policy finetuning with function approximation. ",
+        "bbox": [
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+        "page_idx": 9
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+    {
+        "type": "text",
+        "text": "Also, while our contributions are mainly theoretical, implementing or extending our policy finetuning algorithms on real-world RL tasks would be a compelling future direction. When the environment is a tabular MDP, our Algorithm 1 (offline reduction) and Algorithm 2 (hybrid offline / online RL) are readily implementable. When there is large state/action space and potentially function approximation, we believe our algorithm can be adapted, for example, by replacing all the optimistic/pessimistic value iteration steps by DQN-type algorithms [38] with positive/negative bonus functions [50]. Experimental evaluation of such algorithms would be a good direction for future work. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgment ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The authors would like to thank Ming Yin, Chi Jin and David Forsyth for the many insightful discussions. NJ acknowledges funding support from the ARL Cooperative Agreement W911NF17-2-0196, NSF IIS-2112471, and Adobe Data Science Research Award. HW, CX, YB are funded through employment with Salesforce. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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+        "type": "text",
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parse/train/SkFAWax0-/SkFAWax0-.md

@@ -0,0 +1,300 @@
+# VOICELOOP: VOICE FITTING AND SYNTHESIS VIA A PHONOLOGICAL LOOP
+
+Yaniv Taigman, Lior Wolf, Adam Polyak and Eliya Nachmani
+
+Facebook AI Research {yaniv, wolf, adampolyak, eliyan} $@$ fb.com
+
+# ABSTRACT
+
+We present a new neural text to speech (TTS) method that is able to transform text to speech in voices that are sampled in the wild. Unlike other systems, our solution is able to deal with unconstrained voice samples and without requiring aligned phonemes or linguistic features. The network architecture is simpler than those in the existing literature and is based on a novel shifting buffer working memory. The same buffer is used for estimating the attention, computing the output audio, and for updating the buffer itself. The input sentence is encoded using a context-free lookup table that contains one entry per character or phoneme. The speakers are similarly represented by a short vector that can also be fitted to new identities, even with only a few samples. Variability in the generated speech is achieved by priming the buffer prior to generating the audio. Experimental results on several datasets demonstrate convincing capabilities, making TTS accessible to a wider range of applications. In order to promote reproducibility, we release our source code and models1.
+
+# 1 INTRODUCTION
+
+We study the task of mimicking a person’s voice based on samples that are captured in-the-wild. As far as we know, no other solution exists for this highly applicable learning problem. While the current systems are mostly based on carefully collected or curated audio samples, our method is able to employ the audio of public speeches (from youtube), despite a large amount of background noise and clapping and even with an inaccurate automatic transcript. Moreover, almost all in-the-wild videos contain multiple other speakers that become challenging voice sample outliers and, in some cases, the videos are shot with home equipment and are of reduced quality.
+
+Our method, called VoiceLoop, is inspired by a working-memory model known as the phonological loop (Baddeley, 1986). The loop holds verbal information for short periods of time. It comprises both a phonological store, where information is constantly being replaced, and a rehearsal process, which maintains longer-term representations in the phonological store.
+
+In our method, we construct a phonological store by employing a shifting buffer that is best seen as a matrix $S ~ \in \mathbb { R } ^ { d \times k }$ with columns $\bar { S } [ 1 ] \ldots S [ k ]$ . At every time point, all columns shift to the right $( S [ i + 1 ] = S [ i ]$ for $1 \leq i < k \}$ ), column $k$ is discarded, and a new representation vector $u$ is placed in the first position $( S [ 1 ] = u )$ ). $u$ is a function of four parameters, among which are the latest “spoken” output and the buffer $S$ itself. The buffer is, therefore, constantly refreshed with new information, similar to the phonological store, and the mechanism that creates the representations reuses the existing information in the buffer, thus creating long term dependencies.
+
+The two other input parameters of the network that computes the new representation $u$ are the identity of the speaker and the current attention-mediated context. The identity is captured by a learned embedding and is stored in a lookup table (for the individuals in the training set) or fitted (for new individuals). The usage of this embedding for the phonological store means that it influences the dynamic behavior of the store, the attention mechanism and the output process. Since the last process requires heavy personalization, it also receives the identity embedding directly.
+
+![](images/a0456ecc35aa3e52127fdd48e495db69d8c185454a419ec899a6e62f7b9bc2e2.jpg)  
+Figure 1: An overview of the VoiceLoop architecture. The reader combines the encoding of the sentence’s phonemes using the attention weights to create the current context. A new representation is created by a shallow network that receives the context, the speaker ID, the previous output, and the buffer. The new representation is inserted into the buffer and the earliest vector in the buffer is discarded. The output is obtained by another shallow network that receives the buffer and the speaker as inputs. Once trained, fitting a new voice is done by freezing the network, except for the speaker embedding.
+
+The input sentences in our system are represented as a list of phonemes. Each phoneme out of the 42 in the dictionary being employed, is encoded as a short vector. The encoding of an input sentence is the list of vectors which corresponds to its list of phonemes. The context, either through a Recurrent Neural Network (RNN) or triphones, is not used.
+
+At each time point, the encodings of the phonemes are weighted and then summed, using a vector of attention weights, to form the current context vector. As attention mechanism, we employ the Graves attention model (Graves, 2013), which ensures a monotonic increase in the position along the sequence of input phonemes.
+
+A few properties of our methods stand out in the landscape of neural text to speech work: (i) Instead of conventional RNNs, we propose to employ a memory buffer. (ii) The same memory is shared between all processes and is repeatedly used to make all inferences. (iii) We employ shallow fully-connected networks for all computations. (iv) The input encoding part of the “reader” mechanism is extremely simple.
+
+We hypothesize that these properties make our architecture more robust than existing methods and allow us to mimic speakers based on noisy and limited training data. Moreover, since the output is more directly linked to the inputs, we are able to fit new speakers using relatively short audio sequences coupled with automatically generated text.
+
+Finally, the output of our system is deterministic, given its input. However, multiple intonations are readily generated by employing priming, which involves initializing the buffer $S$ prior to the synthesis process.
+
+Experimentally, we evaluate our method in two ways. For TTS quality, we follow the standard Mean Opinion Score (MOS) experiment done by Arik et al. (2017a). For speaker identification, we train a multi-class network which achieves near-perfect performance on a real validation set, and test it against generated ones.
+
+# 2 PREVIOUS WORK
+
+Text to speech (TTS) methods can be mostly classified into four families: rule-based, concatenative, statistical-parametric (mostly HMM based), and neural. HMM-based methods (Zen et al., 2009) require careful collection of the samples, or as recently attempted by Baljekar & Black, filtering of noisy samples for in-the-wild application. Concatenative methods are somewhat less restrictive but still require tens of minutes of clean and well transcribed samples from the target voice. Emerging neural methods may hold the (currently unrealized) promise of allowing the imitation of new speakers, based on limited and unconstrained samples captured in the wild.
+
+Very recent neural TTS systems include the Deep Voice systems DV1 (Arik et al., 2017b) & DV2 (Arik et al., 2017a), WaveNet (Oord et al., 2016), Char2Wav (Sotelo et al., 2017), and Tacotron (Wang et al., 2017). The DV2 system is a well-engineered system, which includes specialized subsystems for segmenting phonemes, predicting phoneme duration, and predicting the fundamental frequency. Each subsystem includes stacked bidirectional recurrent networks, multilayer fully connected networks and many residual connections. This stands in stark contrast to our system, which employs a single shared memory, one output process, and shallow fully connected networks.
+
+DV2 is the only other current method that models multiple speakers in a single network. However, in contrast to our results, there are three critical differences: (a) There are no in-the-wild experiments; (b) no fitting to a new speaker that did not appear in the training set is shown possible; and (c) the authors employ a large private set and delegate the attention problem to sub-systems, including strong ground-truth alignment between phonemes, waveforms and linguistic features. The linguistic features, which comprise of phone duration, syllable stress, number of syllables in a word and position of the current syllable in a phrase, are also used during inference for generating the samples (used in the subjective Mean Opinion Score tasks as well). In contrast, our method learns “where to read” from the input. Note that (a) and (b) are crucial capabilities in making TTS accessible to a wide range of applications, in particular when casually and efficiently modeling non-professional speakers. The need for professionally collected datasets and the lack of post-training fitting could be inherent to the DV2 architecture, since it has a large number of speaker-dependent modules, whereas we fit a new speaker in a single place.
+
+The Tacotron system employs a multi-stage encoder-decoder architecture with multiple RNNs and a block called CBHG (Lee et al., 2016) components, with each CBHG containing multiple convolutional layers, a highway network (Srivastava et al., 2015), and a bidirectional GRU (Cho et al., 2014). The output is a synthesized spectrogram, from which the audio is reconstructed by the GriffinLim (Griffin & Lim, 1984) method. Trained on a large private training set recorded by a professional single speaker, the Tacotron system is able to read raw text (characters and not phonemes). While Tacotron was not trained for multiple speakers, Arik et al. (2017a) have done so and report a high level of sensitivity to the choice of parameters and a need to incorporate the input embedding in many network sites. The Char2Wav architecture employs RNNs for both the reader and the generator. As an attention mechanism, the Graves positional attention mechanism (Graves, 2013) is used. The same attention mechanism is used in our work. However, in our case, the parameters of the attention model are based on the shared memory store (the buffer). Similarly to our method, the network was also trained to predict vocoder features. In addition, for added quality, the vocoder was replaced by a SampleRNN network (Mehri et al., 2016). In contrast to the above mentioned systems, which employ RNNs, the WaveNet architecture is based on stacks of dilated convolutions, which are termed “causal” for not looking into the future. The output audio is generated sample by sample, which, at typical sampling rates of thousands of hertz, is too slow for current TTS applications. Wavenet has shown single-speaker TTS capabilities, but not multi-speaker.
+
+Waveforms Synthesis There is currently no TTS method which can synthesize waveforms from scratch. WaveNet, DV1, DV2, Char2wav and Tacotron were all conditioned on top of lower level generators. Wavenet was conditioned on F0 vocoder features, as well as linguistic features extracted from separately trained RNN-based text representations. SampleRNNs were employed on top of vocoders. Tacotron synthesized spectograms from mel-spectograms, approximating waveforms using Griffin-Lim. As observed by DV2, small errors in the spectrogram generation result in unnatural (metallic) noise in the reconstruction. Further audio processing can be used to alleviate them, but to a limited extent. Better results were achieved (Arik et al., 2017a) by replacing Griffin-Lim with a Wavenet-like net conditioned on the generated spectogram and speaker.
+
+Table 1: The components of the VoiceLoop model   
+
+<table><tr><td>Symbol</td><td>Description</td><td>Computed as:</td></tr><tr><td>St ∈Rdxk 2aiaier ut∈Rd EeRdpxl zERds Kt,βt,Yt ∈RC Ht,O²,Y∈RC at∈Rl Ct ∈Rdp</td><td>buffer at time t new representation for the buffer embedding of the input sequence embedding of the current speaker attention model parameters attention GMM parameters attention vector at time t context vector at time t</td><td>St[1]= ut; St[i+1]= St-1[i] Nu([St-1,[ct +tanh(Fuz),Ot-1]l) E[i]=LUTp[si] LUTs[id] orSec. 3.2 Na(St-1) μt= μt-1+et,σ²=eβt,γt=sm(Yt） See Eq.3,4 Ct =Eαt</td></tr><tr><td>Ot ERd Nu :kd+dp+do→d Nrureees Na :kd →3c N:kd →do LUTp ∈Rdp×42 LUT∈RdsXN Fu:ds→dp F:ds→do</td><td>output vector at time t buffer update network attention network output network embedding of each phoneme embedding of the speakers projection of the speaker for update projection of the speaker for output</td><td>N(St)+Foz</td></tr><tr><td>d sraaeieetr k d ds C S1...St,1≤s≤42 1 N</td><td>dimensionality of the buffer capacity of the buffer dim.of the input embedding LUT dim.of the vocoder feature vector dim. of the speaker embedding # GMM component (attention model) input sequence length of the input sequence number of speakers in the training set</td><td>dp+do 20 256 63 dp 10</td></tr></table>
+
+Our system was designed with simplicity in mind in order to promote robustness and reproducibility. We focus on modeling the underlying generation process and do not integrate or condition explicitly for waveforms synthesis. Instead, we employ the WORLD (Morise et al., 2016) vocoder (D4C edition) for feature extraction and waveform synthesis. While this bounds the achievable quality, we also experimented with adding WaveNet and SampleRNN. However, the added performance did not seem to justify the extra effort, especially for in-the-wild voice training data, where we observed no improvement.
+
+Differentiable Memory The differentiable buffer architecture, in which a new representation is added at every step, and the last vector added is discarded in a FIFO manner, is novel as far as we know. There are multiple other network models in the literature that are augmented by an external memory structure, e.g., (Joulin & Mikolov, 2015; Sukhbaatar et al., 2015; Graves et al., 2014). However, to our knowledge, our work is one of very few applications of such memory networks outside in practice.
+
+Perhaps the closest model to our work is Stack RNN by Joulin & Mikolov (2015), in which the network is augmented with an infinite stack to which a state vector can be added (PUSH) or removed (POP) at every time step. Unlike our model, only the top of the stack is read each time.
+
+# 3 THE ARCHITECTURE
+
+The architecture of the VoiceLoop model is depicted in Fig. 1 and the components of the architecture are listed in Tab. 1. The forward pass of the network has four steps, which are run sequentially. Following a context-free encoding of the input sequence and an encoding of the speaker, the buffer at time t, $\mathbf { \bar { \mathbf { } } } { S _ { t } } \in \mathbb { R } ^ { d \times k }$ , plays a major role in all of the remaining steps and links between the other components of each step. It also carries the error signal from the output to the earlier steps.
+
+Step I: Encoding the speaker and the input sentence Every speaker is represented by a vector $z$ During training, the vectors of the training speakers are stored in a lookup table $L U T _ { s }$ which maps a running id number to a representation of dimensionality $d _ { s }$ . For new speakers, which are being fitted after the network was trained, the vector $z$ is computed by the straightforward optimization process described in Sec. 3.2.
+
+The input sentence is converted to a sequence of phonemes $s _ { 1 } , s _ { 2 } , \ldots , s _ { l }$ by employing the CMU pronouncing dictionary (Weide, 1998). The number of phonemes in this dictionary is 40, to which two items are added to indicate pauses of different lengths. Each $s _ { i }$ is then mapped separately to an encoding that is based on a trained lookup table $L U T _ { p }$ . This results in an encoding matrix $E$ of size $d _ { p } \times l$ , where $d _ { p }$ is the size of the encoding, and $l$ is the sequence length.
+
+Step II: Computing the context Similar to (Sotelo et al., 2017; Chorowski et al., 2015), we employ the Graves Gaussian Mixture Model (GMM)-based monotonic attention mechanism. At each output time point $t = 1 , 2 , \dots$ , the attention network $N _ { a }$ receives the buffer from the previous time step $S _ { t - 1 }$ as input and outputs the GMM priors $\gamma _ { t }$ , shifts $\kappa _ { t }$ , and log-variances $\beta _ { t }$ . For a GMM with $c$ components, each of these is a vector in $\mathbb { R } ^ { c }$ . $N _ { a }$ has one hidden layer, of dimensionality $\textstyle { \frac { d k } { 1 0 } }$ and a ReLU activation function for the hidden layer.
+
+The attention is then computed as follows:
+
+$$
+\gamma _ { t } ^ { \prime } [ i ] = \frac { e x p ( \gamma _ { t } [ i ] ) } { \sum _ { j } e x p ( \gamma _ { t } [ j ] ) } , i = 1 , 2 , \ldots , c
+$$
+
+i.e., the softmax function is applied to the priors. The means of the GMMs are increased:
+
+$$
+\mu _ { t } = \mu _ { t - 1 } + e x p ( \kappa _ { t } ) ,
+$$
+
+and the variances are computed as $\sigma _ { t } ^ { 2 } = e x p ( \beta _ { t } )$ . For each GMM component $1 \leq i \leq c$ and each point along the input sequence $1 \le j \le l$ , we then compute:
+
+$$
+\phi [ i , j ] = \frac { \gamma _ { t } ^ { \prime } [ i ] } { \sqrt { 2 \pi \sigma _ { t } ^ { 2 } [ i ] } } e x p ( - \frac { ( j - \mu _ { t } [ i ] ) ^ { 2 } } { 2 \sigma _ { t } ^ { 2 } [ i ] } )
+$$
+
+The attention weights $\alpha _ { t }$ are computed for each location in the sequence by summing along all $c$ components:
+
+$$
+\alpha _ { t } [ j ] = \sum _ { i = 1 } ^ { c } \phi [ i , j ]
+$$
+
+The context vector $c _ { t }$ is then computed as weighted sum of the columns of the input sequence embedding matrix $E$ as $c _ { t } = E \alpha _ { t }$ . The loss function of the entire model depends on the attention vector through this context vector. The GMM is differentiable with respect to mean, std and weight, and these are updated, during training, through backpropagation.
+
+Step III: Updating the buffer At each time step, a new representation vector $u$ of dimensionality $d$ is added to the buffer at the first location $S _ { t } [ 1 ]$ , the last column of the buffer from the previous time step $S _ { t - 1 } [ k ]$ is discarded, and the rest are copied $S _ { t } [ i + 1 ] = S _ { t - 1 } [ i ]$ for $i = 1 , \ldots , k - 1$ .
+
+In our implementation, the number of features in the buffer $d$ is the sum of the dimensionality of the embedding of the phonemes $d _ { p }$ and the output’s dimensionality $d _ { o }$ . This choice was made so that a direct comparison to a buffer that does not employ an update network can be performed. In this case, $u$ is simply the concatenation of the current context vector $c _ { t }$ and the output from the previous time step $o _ { t - 1 }$ . It soon became very clear that this loop-less buffer update leads to poor results, emphasizing the role of using information of the buffer $S$ itself in the update process.
+
+The vector $u$ is, therefore, computed using a shallow fully connected network $N _ { u }$ , with one hidden layer of a size that is the tenth of the input dimensionality and a ReLU activation function.
+
+The network receives as input the buffer $S _ { t - 1 }$ , the context vector $c _ { t }$ , and the previous output $o _ { t - 1 }$ The new vector $u$ is also made speaker dependent by adding a projection of the speaker embedding $z$ to the context vector. This projection is followed by a hyperbolic tangent activation function, in order to maintain scale. Therefore,
+
+$$
+C _ { t } = [ c _ { t } + t a n h ( F _ { u } z ) , o _ { t - 1 } ]
+$$
+
+![](images/f6ad362c76523e8b3b0cd6dd526216f23c413d9f98cb5391787d3aaaf06b2497.jpg)  
+Figure 2: Memory Location Significance. For each of the three networks $N _ { u }$ , $N _ { a }$ and $N _ { o }$ , we average the absolute values of the weights to the hidden layer across all hidden neurons and across the $d$ rows of the buffer. The result is a measure of the relative importance of each column of the buffer. Best viewed in color.
+
+$$
+u = N _ { u } ( [ S _ { t - 1 } , C _ { t } ] ) ,
+$$
+
+where $[ a , b ]$ is the concatenation of the two column vectors $a$ and $b$ to one column vector, or the concatenation of two matrices $a$ and $b$ side by side.
+
+Another way in which we allow the speaker to influence the generated output is by initializing the buffer based on the speaker’s embedding. Specifically, in our implementation, the speaker embedding size $d _ { s }$ is the same as the phoneme embedding size $d _ { p }$ and we set the top part of the buffer $S _ { 0 }$ to be $z$ repeated $k$ times. The lower part of size $d _ { o } \times k$ is set to zero.
+
+Step IV: Generating the output The output is generated using a network $N _ { o }$ that is of the same architecture as $N _ { a }$ and $N _ { u }$ and a projection of the user by a learned matrix $F _ { o }$ :
+
+$$
+o _ { t } = N _ { o } ( S _ { t } + F _ { o } z )
+$$
+
+Memory Location Significance In order to better understand the behavior of the buffer, we consider the relative role of each buffer location $1 , 2 , \ldots , k$ on the activations of $N _ { u } , N _ { a }$ , and $N _ { o }$ Specifically, we average the absolute values of the weights from the input (buffer elements) to the hidden layer. The averaging is performed across all $d$ features and ${ \frac { d k } { \frac { 1 0 } { \alpha } } }$ hidden units, and provides one value per each location. As can be seen in Fig 2, the weights of the latest elements are more prominent, especially, as expected, for the output network $N _ { o }$ . However, even the rightmost column has a relative contribution that is at least one third of the leftmost column. This supports the utility of our buffer architecture, in which all memory locations are equal inputs to the downstream fully connected networks.
+
+# 3.1 TRAINING
+
+In our current implementation, the output is a vector of vocoder features of dimensionality $d _ { o } = 6 3$ . Similar to (Sotelo et al., 2017), these features were computed using the Merlin toolkit (Wu et al., 2016). During training, the output at each time frame $t$ is compared to the vocoder features of the ground truth data $Y _ { t }$ using the MSE loss: $\frac { 1 } { d _ { o } } \Vert Y _ { t } - o _ { t } \Vert ^ { 2 }$ . This loss requires an exact temporal alignment of the input and the output sequence. However, human speech is not deterministic and one cannot expect a deterministic method to predict the ground truth. For example, even the same speaker cannot replicate her voice to completely remove the MSE loss since there is variability when repeating the same sentence. Teacher forcing solves this since it eliminates most of the drift and enforces a specific way of uttering the sentence.
+
+In conventional teacher forcing, during training, the input to the network $N _ { u }$ is $Y _ { t - 1 }$ and not $o _ { t - 1 }$ This holds the danger of teaching the network to predict only one time frame ahead, which would create a drift in the output when run on test data. We, therefore, employ a variant of the teacher-forcing technique, which uses the following input to $N _ { u }$ as the previous output
+
+$$
+\frac { o _ { t - 1 } + Y _ { t - 1 } } { 2 } + \eta ,
+$$
+
+where $\eta$ is a random noise vector. When training starts, the predicted output $o _ { t - 1 }$ is by itself a source of noise. As training progresses, it becomes more similar to $Y _ { t - 1 }$ . However, the systematic difference between the two allows the network to better fit the situation that occurs at test time.
+
+During training, a forward pass on all of the output sequences is performed (without truncation), followed by a backward pass.
+
+Efficiency The full model contains 9.3 million parameters and runs near real-time on an Intel Xeon E5 single-core CPU and 5 times faster when on M40 NVIDIA GPU, including vocoder CPU decoding. This was benchmarked with our publicly available python PyTorch implementation. Therefore, even without special optimizations, engineering VoiceLoop to run on a mobile client is possible, similar to existing non-neural TTS client solutions (e.g. Android’s text-to-speech APK).
+
+# 3.2 FITTING A NEW PERSON
+
+Different people exhibit different patterns and present various mannerisms in their speech. Therefore, learning to fit these factors from a limited amount of speech is a challenging task. The goal of speaker mimicking TTS is to be able to mimic a new person based on a relatively short voice sample. Ideally, the new voice would be captured by the parameters of the speaker embedding $z$ , without the need to retrain the network. Naturally, enough variability in the population of the training speakers is needed in order to support this. To fit a new speaker, we are given voice samples and transcribed text. We then employ the training procedure, where the weights of all networks and projections $( N _ { a } , N _ { u } , N _ { o } , L U T _ { p } , F _ { u } , F _ { o } )$ are kept fixed and only vector $z$ is learned (using SGD) to form the embedding of the new speaker.
+
+The same training procedure as detailed in Sec. 3.1 is employed for fitting a new person, including the application of teacher-forcing. We find that the fitting process is very stable with regards to voice characteristics such as pitch. We also noticed that the accent in the new sample needs to be relatively close to the accents presented in the training samples. See Sec. 3.2 for fitting experiments.
+
+# 3.3 GENERATING VARIABILITY
+
+As mentioned, natural speech is not deterministic and each time a sentence is said, it is said in a different way. For simplicity, our method does not employ a random component, such as a variational autoencoder. However, we can generate different outputs by employing priming (Graves, 2013). In this technique, the initial buffer $S _ { 0 }$ is initialized based on an initial process in which another word or sentence is run through the system. One can expect that a sentence from the training set that is said in excitement, would paint the buffer differently than one that is flatter. Experimenting with this technique, demonstrates that we are indeed able to achieve the desired level of variability. However, the direct link between the nature of the priming sequence and the generated output is only anecdotal at this point.
+
+# 4 EXPERIMENTS
+
+We make use of multiple datasets. First, for comparing with existing single speaker techniques, we employ single speaker literature datasets. Second, we employ various subsets of the VCTK dataset (Veaux et al., 2017) for various multi-speaker training and/or fitting experiments. Third, we create a dataset that is composed from four to five public speeches of four public figures. The data was downloaded from youtube, where these speeches are publicly available and were automatically transcribed. Samples generated by our method are available on the project’s website https: //github.com/facebookresearch/loop.
+
+The MOS measure for the proposed method was computed using the crowdMOS toolkit by P. Ribeiro et al. (2011) and Amazon Mechanical Turk. All samples were presented at 16kHz and the raters were told that they are presented with the results of the different algorithms. At least 20 raters participated in each such experiment, with $9 5 \%$ confidence intervals. We restricted all experiments to North American raters.
+
+# 4.1 SINGLE SPEAKER EXPERIMENTS
+
+The single speaker experiments took place on the LJ (Ito, 2017a), the Nancy corpus from the 2011 Blizzard Challenge (King & Karaiskos, 2011), and the English audiobook data for the 2013 Blizzard Challenge (King & Karaiskos, 2013). Our method was compared to the ground truth as well as to Char2Wav and to Tacotron. The Char2Wav system was trained by us using the authors’ implementation available at https://github.com/sotelo/parrot. The training of the Char2Wav model, in each experiment, was optimized by measuring the loss on the validation set, over the following hyperparameters: initial learning rate of $\left[ 1 e - 2 , 1 e - 3 , 1 e - 4 \right]$ , source noise standard deviation $( [ 1 , 2 , 4 ] )$ , batch-size ([16, 32, 64]) and the length of each training sample $( [ 1 0 e 2 , 1 0 e 4 ] )$ .
+
+The Tacotron models were pretrained models available from the best public implementation we could find, which is by Ito (2017b). This re-implementation has models only for the LJ and the Nancy datasets. Note that Tacotron has raised a lot of attention and considerable effort was put by the community to replicate the paper’s results. However, there would very likely be a different choice of hyper-parameters between such re-implementations and the one of the authors.
+
+The MOS scores are shown in Tab. 2. These were computed using the “same_sentence” option of crowdMOS, following DV2 (personal communication). As can be seen, our single speaker results are better than those of the other two algorithms across datasets, but still somewhat lower than the ground truth results.
+
+It is interesting to note that on Blizzard 2011, our results are better than Tacotron (reimplementation) but not significantly better than Char2Wav, while on Blizzard 2013 it is significantly better than both. This can be attributed to the clean nature of Blizzard 2011, for which Char2Wav is robust enough, and demonstrates our method’s robustness to noise.
+
+Tab. 3 presents Mel Cepstral Distortion (MCD) scores. This is an automatic, albeit limited, method of testing compatibility between two audio sequences. Since the sequences are not aligned, we employ MCD DTW, which uses dynamic time warping (DTW) to align the sequences. As can be seen, in this metric too, our method outperforms the baseline methods. The single except is Tacotron’s lower distortion on the LJ dataset. However, as shown in Tab. 2, Tacotron is not competitive on this dataset.
+
+Table 2: Single Speaker MOS Scores (Mean ± SD)   
+
+<table><tr><td>Method</td><td>LJ</td><td>Blizzard 2011</td><td>Blizzard 2013</td></tr><tr><td>Tacotron (re-impl)</td><td>2.06 ± 1.02</td><td>2.15 ± 1.10</td><td>N/A</td></tr><tr><td>Char2wav</td><td>3.42 ± 1.14</td><td>3.33 ± 1.06</td><td>2.03 ± 1.16</td></tr><tr><td>VoiceLoop</td><td>3.69 ± 1.04</td><td>3.38 ± 1.00</td><td>3.40 ± 1.03</td></tr><tr><td>Ground truth</td><td>4.60 ± 0.71</td><td>4.56 ± 0.67</td><td>4.80 ± 0.50</td></tr></table>
+
+Table 3: Single Speaker MCD Scores (Mean ± SD; lower is better)   
+
+<table><tr><td>Method</td><td>LJ</td><td>Blizzard 2011</td><td>Blizzard 2013</td></tr><tr><td>Tacotron (re-impl)</td><td>12.82 ± 1.41</td><td>14.60 ± 7.02</td><td>N/A</td></tr><tr><td>Char2wav</td><td>19.41 ± 5.15</td><td>13.97 ± 4.93</td><td>18.72 ± 6.41</td></tr><tr><td>VoiceLoop</td><td>14.42 ± 1.39</td><td>8.86 ±1.22</td><td>8.67 ± 1.26</td></tr></table>
+
+# 4.2 MULTI-SPEAKER EXPERIMENTS
+
+Multi-speaker experiments were performed on the VCTK dataset (Veaux et al., 2017). The 109 speakers were divided into four different nested subsets: 22 North American speakers, both male and females; and 65, 85 and 101 random selection of speakers, where the remaining eight speakers were left out for validation. Each subset was shuffled into train and test sets. Different models were trained to each of the subsets. Qualitatively, the models provide distinguished voices, and as can be seen in Fig. 3, the generated voice samples display a different dynamic behavior for different speakers.
+
+![](images/9ec1c8ec5ba0ce9ff107e31130d9098f2d9274b357e12f53f06e3803b35e9cd8.jpg)  
+Figure 3: Top: The attention probabilities obtained when mimicking three different North American speakers from VCTK using the same sentence: “but there is no eye contact”. The $\mathbf { X } ^ { } -$ -axis is the time along the generated audio. The y-axis depicts the sequence of phonemes. Dots indicate the maximal response along time for each phoneme, illustrating learned phoneme duration differences between identities (not given during training). Bottom: The 4-th Mel-cepstrum for the three generated sentences (dashed) as well as the ground-truth (solid) of the leftmost speaker. Best viewed in zoom.
+
+Table 4: Multi-speaker MOS scores (Mean ± SE)   
+
+<table><tr><td>Method</td><td>VCTK22</td><td>VCTK65</td><td>VCTK85</td><td>VCTK101</td></tr><tr><td>Char2wav</td><td>2.84 ± 1.20</td><td>2.85 ± 1.19</td><td>2.76 ± 1.19</td><td>2.66 ± 1.16</td></tr><tr><td>VoiceLoop</td><td>3.57 ± 1.08</td><td>3.40 ± 1.00</td><td>3.13 ± 1.17</td><td>3.33 ± 1.10</td></tr><tr><td>GT</td><td>4.61 ± 0.75</td><td>4.59 ± 0.72</td><td>4.64 ± 0.64</td><td>4.63 ± 0.66</td></tr></table>
+
+In our experiments, we employ the author’s implementation of Char2Wav mentioned above as baseline. Note that while the Char2Wav paper did not present multi-speaker results, the open implementation is more general and includes this option.
+
+Sentences from the test set of VCTK are employed for testing. Following DV2 (private communication), the MOS results were computed using the “diff_sentences” option of the crowdMOS toolkit, and are depicted in Tab. 4. As can be seen, our multi-speaker method shows a considerable advantage over the Char2Wav system across all VCTK subsets, but is not as good as the ground truth. These results are consistent with the MCD scores as reported in Tab. 5.
+
+Speaker Identification The capability of the system to generate distinguished voices that match the original voices was tested, as was done in DV2, using a speaker classifier. We train a multi-class convolutional network on the ground-truth training set of multiple speakers, and test on the generated ones. The network gets as input an arbitrary size of vocoder samples, performs five convolutional layers of 3x3 filters over 32 batch-normalized channels, followed by max-pooling, average pooling over time, two fully-connected layers, and ending with a softmax of the number of classes tested. All intermediate layers were linearly rectified.
+
+The identification results are shown in Tab. 6. The VoiceLoop results are more accurate than the results on the VCTK test split, despite using the same text. This might indicate that the voices generated are more similar to the training voices than the natural variability that is present in the dataset. The Char2Wav results are considerably lower.
+
+Table 5: Multi-speaker MCD scores (Mean ± SE; lower is better)   
+
+<table><tr><td>Method</td><td>VCTK22</td><td>VCTK65</td><td>VCTK85</td><td>VCTK101</td></tr><tr><td>Char2wav</td><td>15.71 ± 1.82</td><td>15.1 ± 1.45</td><td>15.23 ± 1.49</td><td>15.06 ± 1.32</td></tr><tr><td>VoiceLoop</td><td>13.74 ± 0.98</td><td>14.1 ± 0.94</td><td>14.16 ± 0.87</td><td>14.22 ± 0.88</td></tr></table>
+
+Table 6: Multi-Speaker Identification Top-1 Accuracy (%)   
+
+<table><tr><td>Method</td><td>VCTK85</td><td>VCTK101</td></tr><tr><td>VCTK test split</td><td>98.25</td><td>97.16</td></tr><tr><td>Char2Wav on test split sentences</td><td>75.70</td><td>81.63</td></tr><tr><td>VoiceLoop on test split sentences</td><td>100</td><td>99.76</td></tr></table>
+
+# 4.3 NEW SPEAKER FITTING EXPERIMENTS
+
+Our system is the only published system that is capable of post-training fitting of new speakers. In order to experiment with this capability, we employ the VoiceLoop model trained on VCTK85 and experiment on the remaining 16 speakers one by one, where only the speaker embedding $z$ gets updated. While TTS systems typically require several hours of data to model a single speaker (Zen et al., 2009), our fitting set contains only 23.65 minutes per speaker on average.
+
+![](images/91ceb44e61dcff3043cbbf2aa4dd6e1394d679764444c8de47326ccc710dba71.jpg)  
+Figure 4: Fitting new speaker embeddings to an existing VoiceLoop model. The graph plots top1 identification accuracy with respect to a sample set length (in minutes) per speaker. Scores were averaged over 5 splits each. The “Full training” horizontal line is the top-1 accuracy for the corresponding speakers, when trained together with the model from scratch. The leftmost datapoint is for two sentences (about 10sec) per speaker.
+
+As described in 3.2, we randomly initialize a new embedding for every new speaker and update only its weights during back-propagation on the fitting data. The newly fitted speakers achieve ${ \bf 3 . 0 8 \pm }$ 0.95 MOS, suggesting that the generation mechanism has not deteriorated below a “fair” level by the new entries.
+
+Similar to the multi-speaker case, we train classifiers for the corresponding identities on ground-truth data, but test on the fitted ones, achieving $8 7 . 6 \%$ top-1 identification accuracy. Despite lower rates than those in Tab. 6, generations of fitted identities are still reasonably discriminative. We conjecture that training VoiceLoop on a larger set of speakers (e.g. LibriSpeech Panayotov et al. (2015)) will be able to represent unseen identities better.
+
+Fitting Data Size The performance of fitting a new identity clearly relies on the length of the sample that is available for that speaker. In order to understand the influence of the sample size, we repeated the above fitting process for the 16 speakers, but capped the available fitting data per speaker. Specifically, we experimented with a maximal amount of training data of 1, 5, 10, 15 and 20 minutes of voice for each speaker. Instead of cutting the last sentence in the middle, it was removed in case that the threshold was crossed. We repeated this fitting process 5 times, each time fitting a different set of samples at a particular limit.
+
+In Fig. 4 we report identification accuracies for each limit. Surprisingly, even with two sentences per speaker, totaling about 10 seconds in average, we can fit a new speaker into VoiceLoop such that the speaker is identifiable at $6 4 . 4 \%$ top-1 identification rate.
+
+# 4.4 IN THE WILD EXPERIMENTS
+
+To demonstrate the flexibility of our method, we downloaded several publicly available videos from youtube. We picked four different known speakers (see samples page), and for each we retrieved the top four to five results, provided that they are longer than 20 minutes. We extracted the audio and its associated (youtube’s) automatically transcribed text. The total amount of data is 6.2 hours, which we then segmented into 8000 segments. Each segment length is around three seconds, similar to the datasets used in the experiments above. Both the data and its corresponding text are noisy: some of the samples include panel discussions and others with questions from various reporters. Sometimes, microphone echo was observed, or relatively low quality audio originated from mobile video conference sessions. We then trained on this data a VoiceLoop model from scratch, using exactly the same training procedure used by the other experiments. This achieved MOS is ${ \bf 2 . 9 7 \bar { \pm } }$ 1.03, and top-1 accuracy of $9 5 . 8 1 \%$ .
+
+We also demonstrate priming (Sec. 3.3) on this dataset. Even for the same speaker, multiple intonations can be generated by initializing $S _ { 0 }$ in different ways. This capability is depicted in Fig. 5 and in the samples page.
+
+# 5 DISCUSSION
+
+Employing web-based in-the-wild training data means that the network is trained on mixed data that contains both speech and other sources. For example, our samples contain a considerable amount of clapping and laughs. Moreover, public speeches contain a larger than usual amount of dramatic prosody and methodological pauses (the same is also true with audiobooks). As our experiments show, our method is mostly robust to these, since it is able to model the voices despite of these difficulties and without replicating the background noises in the synthesized output. The baseline model of Char2Wav was not able to properly model the voices of the youtube dataset and presented clapping sounds in its output.
+
+The architectural simplicity of our system is likely to be the reason for its robustness. Another advantage that stems directly from it, is its computational efficiency. Based on a few shallow networks and on an iterative process that does not consider future samples, our method can generate voice on mobile devices in speeds far exceeding real-time. For comparison, deep voice (Arik et al., 2017b) is posed as a real time neural TTS system, and it achieves a rate of up to 2.7 times real-time on a Intel Xeon E5-2660 v3 Haswell CPU, running 6 concurrent threads (GPU does not provide speedup for the inference of the deep voice system).
+
+![](images/5d1173b5d35905582c3853492c246eded508e9ae4fac068b49997462f06f90db.jpg)  
+Figure 5: Same input, different intonations. A single in the wild speaker saying the sentence “priming is done like that ”, where each time $S _ { 0 }$ is initialized differently. (a) Without priming. (b) Priming with the word “I". (c) Priming with the word “had”. (d) Priming with the word “must”. (e) Priming with the word “bye”. The figure shows the raw waveform, spectrogram, and F0 estimation (include voicedness) in the first, second and third rows respectively. From the spectrogram plots we can observe different duration for some phonemes. The F0 estimation of (c) and (d) shows that the speaker talks in higher tone while in (b) and (e) we can observe lower tone of the speaker. This demonstrates how priming changes the intonations of the model outputs.
+
+The link we form to the model of Baddeley (1986) is by way of analogy and, to be clear, does not imply that we implement this model as is. Specifically, by phonological features, we mean a joint (mixed) representation, in memory, of sound based information and language based information, which is a unique characteristic of our model in comparison to previous work. The short term memory in Baddleley’s model is analog to our buffer and the analog to the rehearsal mechanism is the recursive way in which our buffer is updated. Namely, the new element in the buffer $( u )$ is calculated based on the entire buffer. As noted in Sec. 3, without this dependency on the buffer, our model becomes completely ineffective.
+
+While we employ the loop-updated buffer for the task of speech synthesis, the model is quite general. For example, we have employed the buffer for machine translation from English to French using a dot product based attention model (Bahdanau et al., 2014). The discrete nature of the output means that an output embedding had to be added, but the overall structure remained the same. The performance seemed at least similar to the baseline RNN attention model. However, no attempt has yet been made to achieve state of the art results on existing benchmarks. Surprisingly, relatively large buffer sizes (9) seem to produce better results, despite the input and the output being relatively short. Staying in the realm of voice, the buffer model can be readily used to form a transformation in the other direction (from speech to text), and applied to audio denoising.
+
+# 6 CONCLUSION
+
+We present a new memory architecture that serves as an effective working memory module. Building on this, we are able to present a neural TTS solution of an architecture that is less complex than those found in the recent literature. It also does not require any alignment between phonemes and acoustics or linguistic features as inputs. Using the new architecture, we are able to present, for the first time as far as we know, multi-speaker TTS that is based on unconstrained samples collected from public speeches. Our work also presents a unique ability to fit new speakers (post-training), which is demonstrated even for very limited sample size.
+
+# REFERENCES
+
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+
+Sercan O Arik, Mike Chrzanowski, Adam Coates, Gregory Diamos, Andrew Gibiansky, Yongguo Kang, Xian Li, John Miller, Jonathan Raiman, Shubho Sengupta, et al. Deep voice: Real-time neural text-to-speech. In Proc. of the 34th International Conference on Machine Learning (ICML), 2017b.
+
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+
+Keith Ito. Tacotron speech synthesis implemented in tensorflow, with samples and a pre-trained model, 2017b. URL https://github.com/keithito/tacotron.
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parse/train/a7APmM4B9d/a7APmM4B9d.md

@@ -0,0 +1,287 @@
+# Decision Transformer: Reinforcement Learning via Sequence Modeling
+
+Lili Chen∗,1, Kevin $\mathbf { L u } ^ { * , 1 }$ , Aravind Rajeswaran2, Kimin Lee1, Aditya Grover2,3, Michael Laskin1, Pieter Abbeel1, Aravind Srinivas†,4, Igor Mordatch†,5
+
+∗equal contribution †equal advising
+
+1UC Berkeley 2Facebook AI Research 3UCLA 4OpenAI 5Google Brain
+
+{lilichen, kzl}@berkeley.edu
+
+# Abstract
+
+We introduce a framework that abstracts Reinforcement Learning (RL) as a sequence modeling problem. This allows us to draw upon the simplicity and scalability of the Transformer architecture, and associated advances in language modeling such as GPT- $\mathbf { X }$ and BERT. In particular, we present Decision Transformer, an architecture that casts the problem of RL as conditional sequence modeling. Unlike prior approaches to RL that fit value functions or compute policy gradients, Decision Transformer simply outputs the optimal actions by leveraging a causally masked Transformer. By conditioning an autoregressive model on the desired return (reward), past states, and actions, our Decision Transformer model can generate future actions that achieve the desired return. Despite its simplicity, Decision Transformer matches or exceeds the performance of state-of-the-art model-free offline RL baselines on Atari, OpenAI Gym, and Key-to-Door tasks.
+
+![](images/1a24cd0765714aa601fab185d61614e5c237741df36ea350b02c6e428b196d29.jpg)  
+Figure 1: Decision Transformer architecture1. States, actions, and returns are fed into modalityspecific linear embeddings and a positional episodic timestep encoding is added. Tokens are fed into a GPT architecture which predicts actions autoregressively using a causal self-attention mask.
+
+![](images/0797a951d243f2314131b11d4b2498080d4ff7e99afe42059cf1db90aad8dccc.jpg)  
+Figure 2: Illustrative example of finding shortest path for a fixed graph (left) posed as reinforcement learning. Training dataset consists of random walk trajectories and their per-node returns-to-go (middle). Conditioned on a starting state and generating largest possible return at each node, Decision Transformer sequences optimal paths.
+
+# 1 Introduction
+
+Recent work has shown transformers [1] can model large-scale distributions of semantic concepts, including capable zero-shot generalization in language [2] and impressive out-of-distribution image generation [3]. This stands in sharp contrast to much work in reinforcement learning (RL), which learns a single policy to model a particular narrow behavior distribution. Given the diversity of applications and impact of transformer models, we seek to examine their application to sequential decision making problems. In particular, instead of using transformers as an architectural choice for traditional RL algorithms [4, 5], we seek to study if trajectory modeling (analogous to language modeling) can serve as a replacement for conventional RL algorithms.
+
+We consider the following shift in paradigm: instead of training a policy through conventional RL algorithms like temporal difference (TD) learning [6], the dominant paradigm in RL, we will train transformer models on collected experience using a sequence modeling objective. This will allow us to bypass the need for bootstrapping to propagate returns – thereby avoiding one of the “deadly triad” [6] known to destabilize RL. It also avoids the need for discounting future rewards, as typically done in TD-learning, which can induce undesirable short-sighted behaviors. Additionally, we can make use of existing transformer frameworks widely used in language and vision that are easy to scale, utilizing a large body of work studying stable training of transformer models; this approach removes the need for specialized RL frameworks by appealing only to commonplace supervised learning systems. Given their demonstrated ability to model long sequences and wide data distributions, transformers also have other advantages. Transformers can perform credit assignment directly via self-attention, in contrast to Bellman backups which slowly propagate rewards and are prone to “distractor” signals [7]. This can enable transformers to still work effectively in the presence of sparse or distracting rewards. Furthermore, a transformer modeling approach can model a wide distribution of behaviors, enabling better generalization and transfer. While “upside-down” reinforcement learning (UDRL) [8, 9, 10] also uses a supervised loss conditioned on a target return, our work is motivated by sequence modeling rather than supervised learning and seeks to benefit from modeling long sequences of behaviors. See Section 6 for more discussions about related works.
+
+We explore our hypothesis by considering offline RL, where we will task agents with learning policies from suboptimal data – producing maximally effective behavior from fixed, limited experience. This task is traditionally challenging due to error propagation and value overestimation [11]. However, it is a natural task when training with a sequence modeling objective. By training an autoregressive model on sequences of states, actions, and returns, we reduce policy sampling to autoregressive generative modeling. We can specify the expertise of the policy – which “skill” to query – by manually setting the return tokens, acting as a prompt for generation.
+
+Illustrative example. To get an intuition for our proposal, consider the task of finding a shortest path on a directed graph posed as an RL problem. The reward is 0 when at the goal node and $- 1$ otherwise. We train a GPT [12] model to predict next token in a sequence of returns-to-go (sum of future rewards), states, and actions. Training only on random walk data – with no expert demonstrations – we can at test time generate optimal trajectories by adding a prior to generate highest possible returns (see more details and empirical results in the Appendix) and subsequently generate actions conditioned on that. Thus, by combining the tools of sequence modeling with hindsight return information, we achieve policy improvement without the need for dynamic programming.
+
+Motivated by this observation, we propose Decision Transformer, where we use the GPT architecture to autoregressively model trajectories (shown in Figure 1). We study whether sequence modeling can perform policy optimization by evaluating Decision Transformer on offline RL benchmarks in Atari [13], OpenAI Gym [14], and Key-to-Door [15] environments. We show that – without using dynamic programming – Decision Transformer performs comparably on these benchmarks to state-of-the-art model-free offline RL algorithms [16, 17]. Furthermore, in tasks where long-term credit assignment is required, Decision Transformer capably outperforms RL algorithms. With this work, we hope to bridge vast recent progress in transformer models with RL problems.
+
+# 2 Preliminaries
+
+# 2.1 Offline reinforcement learning
+
+We consider learning in a Markov decision process (MDP) described by the tuple $( \boldsymbol { S } , \mathcal { A } , \boldsymbol { P } , \mathcal { R } )$ . The MDP tuple consists of states $s \in S$ , actions $a \in { \mathcal { A } }$ , transition dynamics $P ( s ^ { \prime } | s , a )$ , and a reward function $r = \mathcal { R } ( s , a )$ . We use $s _ { t }$ , $a _ { t }$ , and $r _ { t } = \mathcal { R } ( s _ { t } , a _ { t } )$ to denote the state, action, and reward at timestep $t$ , respectively. The goal in reinforcement learning is to learn a policy which maximizes the expected return $\mathbb { E } \left[ \sum _ { t = 1 } ^ { T } r _ { t } \right]$ in an MDP. In offline reinforcement learning, instead of obtaining data via environment interactions, we only have access to some fixed limited dataset consisting of trajectories from the environment. This setting is harder as it removes the ability for agents to explore the environment and collect additional feedback.
+
+# 2.2 Transformers
+
+Transformers were proposed by Vaswani et al. [1] as an architecture to efficiently model sequences. They consist of stacked self-attention layers with residual connections. Each self-attention layer receives $n$ embeddings $\{ x _ { i } \} _ { i = 1 } ^ { n }$ corresponding to unique input tokens, and outputs $n$ embeddings $\{ z _ { i } \} _ { i = 1 } ^ { n }$ , preserving the input dimensions. The $i$ -th token is mapped via linear transformations to a key $k _ { i }$ , query $q _ { i }$ , and value $v _ { i }$ . The $i$ -th output of the self-attention layer is given by weighting the values $v _ { j }$ by the normalized dot product between the query $q _ { i }$ and other keys $k _ { j }$ :
+
+$$
+z _ { i } = \sum _ { j = 1 } ^ { n } \operatorname { s o f t m a x } ( \{ \langle q _ { i } , k _ { j ^ { \prime } } \rangle \} _ { j ^ { \prime } = 1 } ^ { n } ) _ { j } \cdot v _ { j } .
+$$
+
+This allows the layer to assign “credit” by implicitly forming state-return associations via similarity of the query and key vectors (maximizing the dot product). In this work, we use the GPT architecture [12], which modifies the transformer architecture with a causal self-attention mask to enable autoregressive generation, replacing the summation/softmax over the $n$ tokens with only the previous tokens in the sequence $( j \in \bar { [ 1 , i ] } )$ . We defer the other architecture details to the original papers.
+
+# 3 Method
+
+In this section, we present Decision Transformer, which models trajectories autoregressively with minimal modification to the transformer architecture, as summarized in Figure 1 and Algorithm 1.
+
+Trajectory representation. The key desiderata in our choice of trajectory representation are (a) it should enable transformers to learn meaningful patterns and (b) we should be able to conditionally generate actions at test time. It is nontrivial to model rewards since we would like the model to generate actions based on future desired returns, rather than past rewards. As a result, instead of modeling the rewards directly, we model the returns-to-go $\begin{array} { r } { \overline { { \boldsymbol { R } } } _ { t } = \sum _ { t ^ { \prime } = t } ^ { T } \boldsymbol { r } _ { t ^ { \prime } } } \end{array}$ . This leads to the following trajectory representation which is amenable to autoregressive training and generation:
+
+$$
+\tau = \left( \widehat { R } _ { 1 } , s _ { 1 } , a _ { 1 } , \widehat { R } _ { 2 } , s _ { 2 } , a _ { 2 } , \ldots , \widehat { R } _ { T } , s _ { T } , a _ { T } \right) .
+$$
+
+Architecture. We feed the last $K$ timesteps into Decision Transformer, for a total of $3 K$ tokens (one for each modality: return-to-go, state, or action). To obtain token embeddings, we learn a linear layer for each modality, which projects raw inputs to the embedding dimension, followed by layer normalization [18]. For environments with visual inputs, the state is fed into a convolutional encoder instead of a linear layer. Additionally, an embedding for each timestep is learned and added to each token – note this is different than the standard positional embedding used by transformers, as one timestep corresponds to three tokens. The tokens are then processed by a GPT [12] model, which predicts future action tokens via autoregressive modeling.
+
+Training. We sample minibatches of sequence length $K$ from the dataset. The prediction head corresponding to the input token $s _ { t }$ is trained to predict $a _ { t }$ – either with cross-entropy loss for discrete actions or mean-squared error for continuous actions – and the losses for each timestep are averaged. We did not find predicting the states or returns-to-go to be necessary for good performance, although it is possible (as shown in Section 5.3) and would be an interesting study for future work.
+
+Evaluation. During evaluation rollouts, we specify a target return based on our desired performance (e.g., specify maximum possible return to generate expert behavior) as well as the environment starting state, to initialize generation. After executing the generated action, we decrement the target return by the achieved reward and obtain the next state. We repeat this process of generating actions and applying them to obtain the next return-to-go and state until episode termination.
+
+# Algorithm 1 Decision Transformer Pseudocode (for continuous actions)
+
+# R, s, a, t: returns -to -go , states , actions , or timesteps   
+# K: context length ( length of each input to DecisionTransformer )   
+# transformer : transformer with causal masking (GPT)   
+# embed_s , embed_a , embed_R : linear embedding layers   
+# embed_t : learned episode positional embedding   
+# pred_a : linear action prediction layer
+
+# # main model
+
+def DecisionTransformer (R , s , a , t ):
+
+# compute embeddings for tokens   
+pos_embedding $=$ embed_t ( t ) # per - timestep ( note : not per - token )   
+s_embedding $=$ embed_s ( s ) $^ +$ pos_embedding   
+a_embedding $=$ embed_a ( a ) $^ +$ pos_embedding   
+R_embedding $=$ embed_R ( R ) $^ +$ pos_embedding
+
+# interleave tokens as (R_1 , s_1 , a_1 R_K , s_K ) input_embeds $=$ stack ( R_embedding , s_embedding , a_embedding )
+
+# use transformer to get hidden states hidden_states $=$ transformer ( input_embeds $=$ input_embeds )
+
+# select hidden states for action prediction tokens a_hidden $=$ unstack ( hidden_states ). actions
+
+# predict action return pred_a ( a_hidden )
+
+# training loop
+
+for (R , s , a , t ) in dataloader : # dims : ( batch_size , K, dim ) a_preds $=$ DecisionTransformer (R , s , a , t ) loss $=$ mean (( a_preds - a ) $* * 2$ ) # L2 loss for continuous actions optimizer . zero_grad (); loss . backward (); optimizer . step ()
+
+# evaluation loop
+
+target_return $\mathbf { \lambda } = \mathbf { \bar { \lambda } } 1$ # for instance , expert - level return R , s , a , t , done $=$ [ target_return ] , [ env . reset ()] , [] , [1] , False while not done : # autoregressive generation / sampling
+
+action $=$ DecisionTransformer (R , s , a , t )[ -1] # for cts actions new_s , r , done , _ $=$ env . step ( action )
+
+# append new tokens to sequence R = R + [ R [ -1] - r] # decrement returns -to -go with reward s , a , t = s + [ new_s ] , a $^ +$ [ action ] , t + [ len ( R )] R , s , a , t $=$ R [ - K :] , ... # only keep context length of K
+
+![](images/6be7ce5512d46b7431e828fe71c83f457f669167743b6fc373ae4c4399521a50.jpg)  
+Figure 3: Results comparing Decision Transformer (ours) to TD learning (CQL) and behavior cloning across Atari, OpenAI Gym, and Minigrid. On a diverse set of tasks, Decision Transformer performs comparably or better than traditional approaches.
+
+# 4 Evaluations on offline RL benchmarks
+
+In this section, we investigate if Decision Transformer can perform well compared to standard TD and imitation learning approaches for offline RL. TD learning algorithms represent the conventional stateof-the-art, while imitation learning algorithms have similar formulations to Decision Transformer. The exact algorithms depend on the environment but our motivations are as follows:
+
+• TD learning: most of these methods use an action-space constraint or value pessimism, and will be the most faithful comparison to Decision Transformer, representing standard RL methods. A state-of-the-art model-free method is Conservative Q-Learning (CQL) [17] which serves as our primary comparison. In addition, we also compare against other prior model-free RL algorithms like BEAR [19] and BRAC [20]. • Imitation learning: this regime similarly uses supervised losses for training, rather than Bellman backups. We use behavior cloning here, and include a more detailed discussion in Section 5.1.
+
+We evaluate on both discrete (Atari [13]) and continuous (OpenAI Gym [14]) control tasks. The former requires long-term credit assignment, while the latter requires fine-grained continuous control, representing a diverse set of tasks. Our main results are summarized in Figure 3, where we show averaged expert normalized performance for each domain.
+
+# 4.1 Atari
+
+The Atari benchmark is challenging due to its high-dimensional visual inputs and difficulty of credit assignment arising from the delay between actions and resulting rewards. We evaluate our method on $1 \%$ of all samples in the DQN-replay dataset as per Agarwal et al. [16], representing 500 thousand of the 50 million transitions observed by an online DQN agent [21] during training; we report the mean and standard deviation of 3 seeds. We normalize scores based on a professional gamer, following the protocol of Hafner et al. [22], where 100 represents the professional gamer score and 0 represents a random policy.
+
+We compare to CQL [17], REM [16], and QR-DQN [23] on four Atari tasks (Breakout, Qbert, Pong, and Seaquest) that are evaluated in Agarwal et al. [16]. We use context lengths of $K = 3 0$ for Decision Transformer (except $K = 5 0$ for Pong); for results with different values of $K$ see the supplementary material. We also report the performance of behavior cloning (BC), which utilizes
+
+<table><tr><td>Game</td><td>DT (Ours)</td><td>CQL</td><td>QR-DQN</td><td>REM</td><td>BC</td></tr><tr><td>Breakout</td><td>267.5 ± 97.5</td><td>211.1</td><td>21.1</td><td>32.1</td><td>138.9 ± 61.7</td></tr><tr><td>Qbert</td><td>15.1 ± 11.4</td><td>104.2</td><td>1.7</td><td>1.4</td><td>17.3 ± 14.7</td></tr><tr><td>Pong</td><td>106.1 ±8.1</td><td>111.9</td><td>20.0</td><td>39.1</td><td>85.2 ± 20.0</td></tr><tr><td>Seaquest</td><td>2.4± 0.7</td><td>1.7</td><td>1.4</td><td>1.0</td><td>2.1± 0.3</td></tr></table>
+
+Table 1: Gamer-normalized scores for the $1 \%$ DQN-replay Atari dataset. We report the mean and variance across 3 seeds. Best mean scores are highlighted in bold. Decision Transformer (DT) performs comparably to CQL on 3 out of 4 games, and outperforms other baselines in most games.
+
+the same network architecture and hyperparameters as Decision Transformer but does not have return-to-go conditioning2. For CQL, REM, and QR-DQN baselines, we report numbers directly from the CQL paper. We show results in Table 1. Our method is competitive with CQL in 3 out of 4 games and outperforms or matches REM, QR-DQN, and BC on all 4 games.
+
+# 4.2 OpenAI Gym
+
+In this section, we consider the continuous control tasks from the D4RL benchmark [24]. We also consider a 2D reacher environment that is not part of the benchmark, and generate the datasets using a similar methodology to the D4RL benchmark. Reacher is a goal-conditioned task and has sparse rewards, so it represents a different setting than the standard locomotion environments (HalfCheetah, Hopper, and Walker). The different dataset settings are described below.
+
+1. Medium: 1 million timesteps generated by a “medium” policy that achieves approximately one-third the score of an expert policy.   
+2. Medium-Replay: the replay buffer of an agent trained to the performance of a medium policy (approximately $2 5 \mathrm { k } { - } 4 0 0 \mathrm { k }$ timesteps in our environments).   
+3. Medium-Expert: 1 million timesteps generated by the medium policy concatenated with 1 million timesteps generated by an expert policy.
+
+We compare to CQL [17], BEAR [19], BRAC [20], and AWR [25]. CQL represents the state-ofthe-art in model-free offline RL, an instantiation of TD learning with value pessimism. Score are normalized so that 100 represents an expert policy, as per Fu et al. [24]. CQL numbers are reported from the original paper; BC numbers are run by us; and the other methods are reported from the D4RL paper. Our results are shown in Table 2. Decision Transformer achieves the highest scores in a majority of the tasks and is competitive with the state of the art in the remaining tasks.
+
+<table><tr><td>Dataset</td><td>Environment</td><td>DT (Ours)</td><td>CQL</td><td>BEAR</td><td>BRAC-v</td><td>AWR</td><td>BC</td></tr><tr><td>Medium-Expert</td><td>HalfCheetah</td><td>86.8 ± 1.3</td><td>62.4</td><td>53.4</td><td>41.9</td><td>52.7</td><td>59.9</td></tr><tr><td>Medium-Expert</td><td>Hopper</td><td>107.6 ± 1.8</td><td>111.0</td><td>96.3</td><td>0.8</td><td>27.1</td><td>79.6</td></tr><tr><td>Medium-Expert</td><td>Walker</td><td>108.1 ±0.2</td><td>98.7</td><td>40.1</td><td>81.6</td><td>53.8</td><td>36.6</td></tr><tr><td>Medium-Expert</td><td>Reacher</td><td>89.1 ±1.3</td><td>30.6</td><td>-</td><td>-</td><td>1</td><td>73.3</td></tr><tr><td>Medium</td><td>HalfCheetah</td><td>42.6 ± 0.1</td><td>44.4</td><td>41.7</td><td>46.3</td><td>37.4</td><td>43.1</td></tr><tr><td>Medium</td><td>Hopper</td><td>67.6 ± 1.0</td><td>58.0</td><td>52.1</td><td>31.1</td><td>35.9</td><td>63.9</td></tr><tr><td>Medium</td><td>Walker</td><td>74.0 ± 1.4</td><td>79.2</td><td>59.1</td><td>81.1</td><td>17.4</td><td>77.3</td></tr><tr><td>Medium</td><td>Reacher</td><td>51.2 ± 3.4</td><td>26.0</td><td>1</td><td>1</td><td>1</td><td>48.9</td></tr><tr><td>Medium-Replay</td><td>HalfCheetah</td><td>36.6 ± 0.8</td><td>46.2</td><td>38.6</td><td>47.7</td><td>40.3</td><td>4.3</td></tr><tr><td>Medium-Replay</td><td>Hopper</td><td>82.7 ± 7.0</td><td>48.6</td><td>33.7</td><td>0.6</td><td>28.4</td><td>27.6</td></tr><tr><td>Medium-Replay</td><td>Walker</td><td>66.6 ± 3.0</td><td>26.7</td><td>19.2</td><td>0.9</td><td>15.5</td><td>36.9</td></tr><tr><td>Medium-Replay</td><td>Reacher</td><td>18.0 ± 2.4</td><td>19.0</td><td>1</td><td>1</td><td>1</td><td>5.4</td></tr><tr><td colspan="2">Average (Without Reacher)</td><td>74.7</td><td>63.9</td><td>48.2</td><td>36.9</td><td>34.3</td><td>46.4</td></tr><tr><td colspan="2">Average (All Settings)</td><td>69.2</td><td>54.2</td><td>1</td><td>-</td><td>1</td><td>47.7</td></tr></table>
+
+Table 2: Results for D4RL datasets4. We report the mean and variance for three seeds. Decision Transformer (DT) outperforms conventional RL algorithms on almost all tasks.
+
+# 5 Discussion
+
+# 5.1 Does Decision Transformer perform behavior cloning on a subset of the data?
+
+In this section, we seek to gain insight into whether Decision Transformer can be thought of as performing imitation learning on a subset of the data with a certain return. To investigate this, we propose a new method, Percentile Behavior Cloning $( \% \mathrm { B C } )$ , where we run behavior cloning on only the top $X \%$ of timesteps in the dataset, ordered by episode returns. The percentile $X \%$ interpolates between standard BC $X = 1 0 0 \%$ ) that trains on the entire dataset and only cloning the best observed trajectory $( X  0 \%$ ), trading off between better generalization by training on more data with training a specialized model that focuses on a desirable subset of the data.
+
+<table><tr><td>Dataset</td><td>Environment</td><td>DT (Ours)</td><td>10%BC</td><td>25%BC</td><td>40%BC</td><td>100%BC</td><td>CQL</td></tr><tr><td>Medium</td><td>HalfCheetah</td><td>42.6 ± 0.1</td><td>42.9</td><td>43.0</td><td>43.1</td><td>43.1</td><td>44.4</td></tr><tr><td>Medium</td><td>Hopper</td><td>67.6 ± 1.0</td><td>65.9</td><td>65.2</td><td>65.3</td><td>63.9</td><td>58.0</td></tr><tr><td>Medium</td><td>Walker</td><td>74.0 ± 1.4</td><td>78.8</td><td>80.9</td><td>78.8</td><td>77.3</td><td>79.2</td></tr><tr><td>Medium</td><td>Reacher</td><td>51.2 ± 3.4</td><td>51.0</td><td>48.9</td><td>58.2</td><td>58.4</td><td>26.0</td></tr><tr><td>Medium-Replay</td><td>HalfCheetah</td><td>36.6 ± 0.8</td><td>40.8</td><td>40.9</td><td>41.1</td><td>4.3</td><td>46.2</td></tr><tr><td>Medium-Replay</td><td>Hopper</td><td>82.7 ± 7.0</td><td>70.6</td><td>58.6</td><td>31.0</td><td>27.6</td><td>48.6</td></tr><tr><td>Medium-Replay</td><td>Walker</td><td>66.6 ± 3.0</td><td>70.4</td><td>67.8</td><td>67.2</td><td>36.9</td><td>26.7</td></tr><tr><td>Medium-Replay</td><td>Reacher</td><td>18.0 ± 2.4</td><td>33.1</td><td>16.2</td><td>10.7</td><td>5.4</td><td>19.0</td></tr><tr><td colspan="2">Average</td><td>56.1</td><td>56.7</td><td>52.7</td><td>49.4</td><td>39.5</td><td>43.5</td></tr></table>
+
+Table 3: Comparison between Decision Transformer (DT) and Percentile Behavior Cloning $( \% \mathrm { B C } )$
+
+We show full results comparing $\% \mathrm { B C }$ to Decision Transformer and CQL in Table 3, sweeping over $X \in [ 1 0 \% , 2 5 \% , 4 0 \% , 1 \dot { 0 } \dot { 0 } \% ]$ . Note that while both $\% \mathrm { B C }$ and DT introduce hyperparameters, returns are human interpretable and it is relatively natural for humans to specify a desired return compared to choosing an optimal subset for cloning. When data is plentiful – as in the D4RL regime – we find $\% \mathrm { B C }$ can match or beat other offline RL methods. On most environments, Decision Transformer is competitive with the performance of the best $\% \mathrm { B C }$ , indicating it can hone in on a particular subset after training on the entire dataset distribution.
+
+In contrast, when we study low data regimes – such as Atari, where we use $1 \%$ of a replay buffer as the dataset – $\mathbf { \nabla } \cdot \% \mathbf { B } \mathbf { C }$ is weak (shown in Table 4). This suggests that in scenarios with relatively low amounts of data, Decision Transformer can outperform $\% \mathrm { B C }$ by using all trajectories in the dataset to improve generalization, even if those trajectories are dissimilar from the return conditioning target. Our results indicate that Decision Transformer can be more effective than simply performing imitation learning on a subset of the dataset. On the tasks we considered, Decision Transformer either outperforms or is competitive to $\% \mathrm { B C }$ , without the confound of having to select the optimal subset.
+
+<table><tr><td>Game</td><td>DT (Ours)</td><td>10%BC</td><td>25%BC</td><td>40%BC</td><td>100%BC</td></tr><tr><td>Breakout</td><td>267.5 ± 97.5</td><td>28.5±8.2</td><td>73.5 ± 6.4</td><td>108.2 ± 67.5</td><td>138.9 ± 61.7</td></tr><tr><td>Qbert</td><td>15.1 ± 11.4</td><td>6.6 ± 1.7</td><td>16.0 ± 13.8</td><td>11.8± 5.8</td><td>17.3 ± 14.7</td></tr><tr><td>Pong</td><td>106.1 ±8.1</td><td>2.5± 0.2</td><td>13.3 ± 2.7</td><td>72.7 ± 13.3</td><td>85.2 ± 20.0</td></tr><tr><td>Seaquest</td><td>2.4 ± 0.7</td><td>1.1 ± 0.2</td><td>1.1 ± 0.2</td><td>1.6 ± 0.4</td><td>2.1 ± 0.3</td></tr></table>
+
+Table 4: $\% \mathrm { B C }$ scores for Atari. We report the mean and variance across 3 seeds. Decision Transformer (DT) outperforms all versions of $\% \mathrm { B C }$ in most games.
+
+# 5.2 How well does Decision Transformer model the distribution of returns?
+
+We evaluate the ability of Decision Transformer to understand return-to-go tokens by varying the desired target return over a wide range – evaluating the multi-task distribution modeling capability of transformers. Figure 4 shows the average sampled return accumulated by the agent over the course of the evaluation episode for varying values of target return. On every task, the desired target returns and the true observed returns are highly correlated. On some tasks like Pong, HalfCheetah and Walker, Decision Transformer generates trajectories that almost perfectly match the desired returns (as indicated by the overlap with the oracle line). Furthermore, on some Atari tasks like Seaquest, we can prompt the Decision Transformer with higher returns than the maximum episode return available in the dataset, demonstrating that Decision Transformer is sometimes capable of extrapolation.
+
+# 5.3 Does Decision Transformer perform effective long-term credit assignment?
+
+To evaluate long-term credit assignment capabilities of our model, we consider a variant of the Key-to-Door environment proposed in Mesnard et al. [15]. This is a grid-based environment with a sequence of three phases: (1) in the first phase, the agent is placed in a room with a key; (2) then, the agent is placed in an empty room; (3) and finally, the agent is placed in a room with a door. The agent receives a binary reward when reaching the door in the third phase, but only if it picked up the key in the first phase. This problem is difficult for credit assignment because credit must be propagated from the beginning to the end of the episode, skipping over actions taken in the middle.
+
+![](images/c1156eaeb2971806e2c5eb8d09680aa47f4f5cf207c581cf8c22a3d777eebeb5.jpg)  
+Figure 4: Sampled (evaluation) returns accumulated by Decision Transformer when conditioned on the specified target (desired) returns. Top: Atari. Bottom: D4RL medium-replay datasets.
+
+We train on datasets of trajectories generated by applying random actions and report success rates in Table 5. Furthermore, for the Key-to-Door environment we use the entire episode length as the context, rather than having a fixed content window as in the other environments. Methods that use highsight return information: our Decision Transformer model and $\% \mathrm { B C }$ (trained only on successful episodes) are able to learn effective policies – producing near-optimal paths, despite only training on random walks. TD learning (CQL) cannot effectively propagate $\mathrm { Q }$ -values over the long horizons involved and gets poor performance.
+
+Table 5: Success rate for Key-to-Door environment. Methods using hindsight (Decision Transformer, $\% \mathbf { B } \mathbf { C } _ { \epsilon }$ ) can learn successful policies, while TD learning struggles to perform credit assignment.   
+
+<table><tr><td>Dataset</td><td>DT (Ours)</td><td>CQL</td><td>BC</td><td>%BC</td><td>Random</td></tr><tr><td>1K Random Trajectories</td><td>71.8%</td><td>13.1%</td><td>1.4%</td><td>69.9%</td><td>3.1%</td></tr><tr><td>10K Random Trajectories</td><td>94.6%</td><td>13.3%</td><td>1.6%</td><td>95.1%</td><td>3.1%</td></tr></table>
+
+# 5.4 Can transformers be accurate critics in sparse reward settings?
+
+In previous sections, we established that decision transformer can produce effective policies (actors). We now evaluate whether transformer models can also be effective critics. We modify Decision Transformer to output return tokens in addition to action tokens on the Key-to-Door environment. We find that the transformer continuously updates reward probability based on events during the episode, shown in Figure 5 (Left). Furthermore, we find the transformer attends to critical events in the episode (picking up the key or reaching the door), shown in Figure 5 (Right), indicating formation of state-reward associations as discussed in Raposo et al. [26] and enabling accurate value prediction.
+
+# 5.5 Does Decision Transformer perform well in sparse reward settings?
+
+A known weakness of TD learning algorithms is that they require densely populated rewards in order to perform well, which can be unrealistic and/or expensive. In contrast, Decision Transformer can improve robustness in these settings since it makes minimal assumptions on the density of the reward. To evaluate this, we consider a delayed return version of the D4RL benchmarks where the agent does not receive any rewards along the trajectory, and instead receives the cumulative reward of the trajectory in the final timestep. Our results for delayed returns are shown in Table 6. Delayed returns minimally affect Decision Transformer; and due to the nature of the training process, while imitation learning methods are reward agnostic. While TD learning collapses, Decision Transformer and $\% \mathrm { B C }$ still perform well, indicating that Decision Transformer can be more robust to delayed rewards.
+
+![](images/2cf047d6f6123e6feee315d97b0f7169c03581ba80dfd8637dd3453482f91ce9.jpg)  
+Figure 5: Left: Averages of running return probabilities predicted by the transformer model for three types of episode outcomes. Right: Transformer attention weights from all timesteps superimposed for a particular successful episode. The model attends to steps near pivotal events in the episode, such as picking up the key and reaching the door.
+
+<table><tr><td></td><td>Environment</td><td colspan="2">Delayed (Sparse)</td><td colspan="2">Agnostic</td><td colspan="2">Original (Dense)</td></tr><tr><td>Dataset</td><td></td><td>DT (Ours)</td><td>CQL</td><td>BC</td><td>%BC</td><td>DT (Ours)</td><td>CQL</td></tr><tr><td>Medium-Expert</td><td>Hopper</td><td>107.3 ± 3.5</td><td>9.0</td><td>59.9</td><td>102.6</td><td>107.6</td><td>111.0</td></tr><tr><td>Medium</td><td>Hopper</td><td>60.7± 4.5</td><td>5.2</td><td>63.9</td><td>65.9</td><td>67.6</td><td>58.0</td></tr><tr><td>Medium-Replay</td><td>Hopper</td><td>78.5 ± 3.7</td><td>2.0</td><td>27.6</td><td>70.6</td><td>82.7</td><td>48.6</td></tr></table>
+
+Table 6: Results for D4RL datasets with delayed (sparse) reward. Decision Transformer (DT) and imitation learning are minimally affected by the removal of dense rewards, while CQL fails.
+
+# 5.6 Additional Discussions
+
+For more discussions see the supplementary material.
+
+# 6 Related work
+
+Offline reinforcement learning. To mitigate the impact of distribution shift in offline RL, prior algorithms either (a) constrain the policy action space [27, 28, 29] or (b) incorporate value pessimism [27, 17], or (c) incorporate pessimism into learned dynamics models [30, 31]. Since we do not use Decision Transformers to explicitly learn the dynamics model, we primarily compare against model-free algorithms; adding a dynamics model tends to improve the performance of model-free algorithms. Another line of work explores learning wide behavior distribution from an offline dataset by learning a task-agnostic set of skills, either with likelihood-based approaches [32, 33, 34, 35] or by maximizing mutual information [36, 37, 38]. Our work is similar to the likelihood-based approaches, which do not use iterative Bellman updates – although we use a simpler sequence modeling objective instead of a variational method, and use rewards for conditional generation of behaviors.
+
+Supervised learning in reinforcement learning settings. Some prior methods for reinforcement learning bear more resemblance to static supervised learning, such as Q-learning [39, 40], which still uses iterative backups, or likelihood-based methods such as behavior cloning, which do not (discussed in previous section). Recent work [8, 9, 10] studies “upside-down” reinforcement learning (UDRL), which are similar to our method in seeking to model behaviors with a supervised loss conditioned on the target return. A key difference in our work is the shift of motivation to sequence modeling rather than supervised learning: while the practical methods differ primarily in the context length and architecture, sequence modeling enables behavior modeling even without access to the reward, in a similar style to language [12] or images [41], and is known to scale well [2]. The method proposed by Kumar et al. [9] is most similar to our method with $K = 1$ , which we find sequence modeling/long contexts to outperform (see supplementary material). Ghosh et al. [42] extends prior UDRL methods to use state goal conditioning, rather than rewards, and Paster et al. [43] further use an LSTM with state goal conditioning for goal-conditoned online RL settings. Concurrent to our work, Janner et al. [44] propose Trajectory Transformer, which is similar to Decision Transformer but additionally uses state and return prediction, as well as discretization, which incorporates model-based components.
+
+We believe that their experiments, in addition to our results, highlight the potential for sequence modeling to be a generally applicable idea for reinforcement learning.
+
+Credit assignment. Many works have studied better credit assignment via state-association, learning an architecture which decomposes the reward function such that certain “important” states comprise most of the credit [45, 46, 15]. They use the learned reward function to change the reward of an actorcritic algorithm to help propagate signal over long horizons. In particular, similar to our long-term setting, some works have specifically shown such state-associative architectures can perform better in delayed reward settings [47, 7, 48, 26]. In contrast, we allow these properties to naturally emerge in a transformer architecture, without having to explicitly learn a reward function or a critic.
+
+Conditional language generation. Various works have studied guided generation for images [49] and language [50, 51]. Several works [52, 53, 54, 55, 56, 57] have explored training or fine-tuning of models for controllable text generation. Class-conditional language models can also be used to learn disciminators to guide generation [58, 50, 59, 60]. However, these approaches mostly assume constant “classes”, while in reinforcement learning the reward signal is time-varying. Furthermore, it is more natural to prompt the model desired target return and continuously decrease it by the observed rewards over time, since the transformer model and environment jointly generate the trajectory.
+
+Attention and transformer models. Transformers [1] have been applied successfully to many tasks in natural language processing [61, 12] and computer vision [62, 63]. However, transformers are relatively unstudied in RL, mostly due to differing nature of the problem, such as higher variance in training. Zambaldi et al. [5] showed that augmenting transformers with relational reasoning improve performance in combinatorial environments and Ritter et al. [64] showed iterative selfattention allowed for RL agents to better utilize episodic memories. Parisotto et al. [4] discussed design decisions for more stable training of transformers in the high-variance RL setting. Unlike our work, these still use actor-critic algorithms for optimization, focusing on novelty in architecture. Additionally, in imitation learning, some works have studied transformers as a replacement for LSTMs: Dasari and Gupta [65] study one-shot imitation learning, and Abramson et al. [66] combine language and image modalities for text-conditioned behavior generation.
+
+# 7 Conclusion
+
+We proposed Decision Transformer, seeking to unify ideas in language modeling and RL. On standard offline RL benchmarks, we showed DT can match or outperform strong algorithms designed explicitly for offline RL with minimal modifications from standard language modeling architectures.
+
+Societal impact. For real-world applications, it is important to understand the types of errors transformers make in MDP settings and possible negative consequences. It will also be important to consider the datasets we train on, which can potentially add destructive biases, particularly as we consider studying augmenting RL agents with more data which may come from questionable sources.
+
+Limitations. We introduced our paradigm shift and showed its potential in our experiments, but there is significant room for more research in this direction. The current architecture requires considerations of context length and return-to-go hyperparameters, and we show results on standard RL benchmarks; future work could improve the architecture and demonstrate results in more complex environments and tasks. We used a simple supervised loss that was effective in our experiments, but applications to large-scale datasets could benefit from self-supervised pretraining tasks. In addition, one could consider more sophisticated embeddings for returns, states, and actions. While we do not directly evaluate scaling and generalization, we utilize a method known to scale generalize well in domains such as language and vision, and we are excited about larger RL systems built upon our framework.
+
+# 8 Acknowledgements
+
+This research was supported by Berkeley Deep Drive, Open Philanthropy, and the National Science Foundation under NSF:NRI #2024675. Part of this work was completed when Aravind Rajeswaran was a PhD student at the University of Washington, where he was supported by the J.P. Morgan PhD Fellowship in AI (2020-21). We also thank Luke Metz, Daniel Freeman, and anonymous reviewers for valuable feedback and discussions, as well as Justin Fu for assistance in setting up D4RL benchmarks, and Aviral Kumar for assistance with the CQL baselines and hyperparameters.
+
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+        "text": "Lili Chen∗,1, Kevin $\\mathbf { L u } ^ { * , 1 }$ , Aravind Rajeswaran2, Kimin Lee1, Aditya Grover2,3, Michael Laskin1, Pieter Abbeel1, Aravind Srinivas†,4, Igor Mordatch†,5 ",
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+        "text": "1UC Berkeley 2Facebook AI Research 3UCLA 4OpenAI 5Google Brain ",
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+        "text": "We introduce a framework that abstracts Reinforcement Learning (RL) as a sequence modeling problem. This allows us to draw upon the simplicity and scalability of the Transformer architecture, and associated advances in language modeling such as GPT- $\\mathbf { X }$ and BERT. In particular, we present Decision Transformer, an architecture that casts the problem of RL as conditional sequence modeling. Unlike prior approaches to RL that fit value functions or compute policy gradients, Decision Transformer simply outputs the optimal actions by leveraging a causally masked Transformer. By conditioning an autoregressive model on the desired return (reward), past states, and actions, our Decision Transformer model can generate future actions that achieve the desired return. Despite its simplicity, Decision Transformer matches or exceeds the performance of state-of-the-art model-free offline RL baselines on Atari, OpenAI Gym, and Key-to-Door tasks. ",
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+            "Figure 1: Decision Transformer architecture1. States, actions, and returns are fed into modalityspecific linear embeddings and a positional episodic timestep encoding is added. Tokens are fed into a GPT architecture which predicts actions autoregressively using a causal self-attention mask. "
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+        "text": "1 Introduction ",
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+        "text": "Recent work has shown transformers [1] can model large-scale distributions of semantic concepts, including capable zero-shot generalization in language [2] and impressive out-of-distribution image generation [3]. This stands in sharp contrast to much work in reinforcement learning (RL), which learns a single policy to model a particular narrow behavior distribution. Given the diversity of applications and impact of transformer models, we seek to examine their application to sequential decision making problems. In particular, instead of using transformers as an architectural choice for traditional RL algorithms [4, 5], we seek to study if trajectory modeling (analogous to language modeling) can serve as a replacement for conventional RL algorithms. ",
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+        "text": "We consider the following shift in paradigm: instead of training a policy through conventional RL algorithms like temporal difference (TD) learning [6], the dominant paradigm in RL, we will train transformer models on collected experience using a sequence modeling objective. This will allow us to bypass the need for bootstrapping to propagate returns – thereby avoiding one of the “deadly triad” [6] known to destabilize RL. It also avoids the need for discounting future rewards, as typically done in TD-learning, which can induce undesirable short-sighted behaviors. Additionally, we can make use of existing transformer frameworks widely used in language and vision that are easy to scale, utilizing a large body of work studying stable training of transformer models; this approach removes the need for specialized RL frameworks by appealing only to commonplace supervised learning systems. Given their demonstrated ability to model long sequences and wide data distributions, transformers also have other advantages. Transformers can perform credit assignment directly via self-attention, in contrast to Bellman backups which slowly propagate rewards and are prone to “distractor” signals [7]. This can enable transformers to still work effectively in the presence of sparse or distracting rewards. Furthermore, a transformer modeling approach can model a wide distribution of behaviors, enabling better generalization and transfer. While “upside-down” reinforcement learning (UDRL) [8, 9, 10] also uses a supervised loss conditioned on a target return, our work is motivated by sequence modeling rather than supervised learning and seeks to benefit from modeling long sequences of behaviors. See Section 6 for more discussions about related works. ",
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+        "type": "text",
+        "text": "We explore our hypothesis by considering offline RL, where we will task agents with learning policies from suboptimal data – producing maximally effective behavior from fixed, limited experience. This task is traditionally challenging due to error propagation and value overestimation [11]. However, it is a natural task when training with a sequence modeling objective. By training an autoregressive model on sequences of states, actions, and returns, we reduce policy sampling to autoregressive generative modeling. We can specify the expertise of the policy – which “skill” to query – by manually setting the return tokens, acting as a prompt for generation. ",
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+        "text": "Illustrative example. To get an intuition for our proposal, consider the task of finding a shortest path on a directed graph posed as an RL problem. The reward is 0 when at the goal node and $- 1$ otherwise. We train a GPT [12] model to predict next token in a sequence of returns-to-go (sum of future rewards), states, and actions. Training only on random walk data – with no expert demonstrations – we can at test time generate optimal trajectories by adding a prior to generate highest possible returns (see more details and empirical results in the Appendix) and subsequently generate actions conditioned on that. Thus, by combining the tools of sequence modeling with hindsight return information, we achieve policy improvement without the need for dynamic programming. ",
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+        "text": "Motivated by this observation, we propose Decision Transformer, where we use the GPT architecture to autoregressively model trajectories (shown in Figure 1). We study whether sequence modeling can perform policy optimization by evaluating Decision Transformer on offline RL benchmarks in Atari [13], OpenAI Gym [14], and Key-to-Door [15] environments. We show that – without using dynamic programming – Decision Transformer performs comparably on these benchmarks to state-of-the-art model-free offline RL algorithms [16, 17]. Furthermore, in tasks where long-term credit assignment is required, Decision Transformer capably outperforms RL algorithms. With this work, we hope to bridge vast recent progress in transformer models with RL problems. ",
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+        "type": "text",
+        "text": "2 Preliminaries ",
+        "text_level": 1,
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+        "text": "2.1 Offline reinforcement learning ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We consider learning in a Markov decision process (MDP) described by the tuple $( \\boldsymbol { S } , \\mathcal { A } , \\boldsymbol { P } , \\mathcal { R } )$ . The MDP tuple consists of states $s \\in S$ , actions $a \\in { \\mathcal { A } }$ , transition dynamics $P ( s ^ { \\prime } | s , a )$ , and a reward function $r = \\mathcal { R } ( s , a )$ . We use $s _ { t }$ , $a _ { t }$ , and $r _ { t } = \\mathcal { R } ( s _ { t } , a _ { t } )$ to denote the state, action, and reward at timestep $t$ , respectively. The goal in reinforcement learning is to learn a policy which maximizes the expected return $\\mathbb { E } \\left[ \\sum _ { t = 1 } ^ { T } r _ { t } \\right]$ in an MDP. In offline reinforcement learning, instead of obtaining data via environment interactions, we only have access to some fixed limited dataset consisting of trajectories from the environment. This setting is harder as it removes the ability for agents to explore the environment and collect additional feedback. ",
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+        "type": "text",
+        "text": "2.2 Transformers ",
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+        "type": "text",
+        "text": "Transformers were proposed by Vaswani et al. [1] as an architecture to efficiently model sequences. They consist of stacked self-attention layers with residual connections. Each self-attention layer receives $n$ embeddings $\\{ x _ { i } \\} _ { i = 1 } ^ { n }$ corresponding to unique input tokens, and outputs $n$ embeddings $\\{ z _ { i } \\} _ { i = 1 } ^ { n }$ , preserving the input dimensions. The $i$ -th token is mapped via linear transformations to a key $k _ { i }$ , query $q _ { i }$ , and value $v _ { i }$ . The $i$ -th output of the self-attention layer is given by weighting the values $v _ { j }$ by the normalized dot product between the query $q _ { i }$ and other keys $k _ { j }$ : ",
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+    {
+        "type": "equation",
+        "img_path": "images/45ad9c3100be001f8f49eccde368e07a614ebaeb981f5d7b9157b67ee92e9f6f.jpg",
+        "text": "$$\nz _ { i } = \\sum _ { j = 1 } ^ { n } \\operatorname { s o f t m a x } ( \\{ \\langle q _ { i } , k _ { j ^ { \\prime } } \\rangle \\} _ { j ^ { \\prime } = 1 } ^ { n } ) _ { j } \\cdot v _ { j } .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "This allows the layer to assign “credit” by implicitly forming state-return associations via similarity of the query and key vectors (maximizing the dot product). In this work, we use the GPT architecture [12], which modifies the transformer architecture with a causal self-attention mask to enable autoregressive generation, replacing the summation/softmax over the $n$ tokens with only the previous tokens in the sequence $( j \\in \\bar { [ 1 , i ] } )$ . We defer the other architecture details to the original papers. ",
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+        "type": "text",
+        "text": "3 Method ",
+        "text_level": 1,
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+        "text": "In this section, we present Decision Transformer, which models trajectories autoregressively with minimal modification to the transformer architecture, as summarized in Figure 1 and Algorithm 1. ",
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+        "type": "text",
+        "text": "Trajectory representation. The key desiderata in our choice of trajectory representation are (a) it should enable transformers to learn meaningful patterns and (b) we should be able to conditionally generate actions at test time. It is nontrivial to model rewards since we would like the model to generate actions based on future desired returns, rather than past rewards. As a result, instead of modeling the rewards directly, we model the returns-to-go $\\begin{array} { r } { \\overline { { \\boldsymbol { R } } } _ { t } = \\sum _ { t ^ { \\prime } = t } ^ { T } \\boldsymbol { r } _ { t ^ { \\prime } } } \\end{array}$ . This leads to the following trajectory representation which is amenable to autoregressive training and generation: ",
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+        "type": "equation",
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+        "text": "$$\n\\tau = \\left( \\widehat { R } _ { 1 } , s _ { 1 } , a _ { 1 } , \\widehat { R } _ { 2 } , s _ { 2 } , a _ { 2 } , \\ldots , \\widehat { R } _ { T } , s _ { T } , a _ { T } \\right) .\n$$",
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+        "type": "text",
+        "text": "Architecture. We feed the last $K$ timesteps into Decision Transformer, for a total of $3 K$ tokens (one for each modality: return-to-go, state, or action). To obtain token embeddings, we learn a linear layer for each modality, which projects raw inputs to the embedding dimension, followed by layer normalization [18]. For environments with visual inputs, the state is fed into a convolutional encoder instead of a linear layer. Additionally, an embedding for each timestep is learned and added to each token – note this is different than the standard positional embedding used by transformers, as one timestep corresponds to three tokens. The tokens are then processed by a GPT [12] model, which predicts future action tokens via autoregressive modeling. ",
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+        "type": "text",
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+        "text": "Training. We sample minibatches of sequence length $K$ from the dataset. The prediction head corresponding to the input token $s _ { t }$ is trained to predict $a _ { t }$ – either with cross-entropy loss for discrete actions or mean-squared error for continuous actions – and the losses for each timestep are averaged. We did not find predicting the states or returns-to-go to be necessary for good performance, although it is possible (as shown in Section 5.3) and would be an interesting study for future work. ",
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+    {
+        "type": "text",
+        "text": "Evaluation. During evaluation rollouts, we specify a target return based on our desired performance (e.g., specify maximum possible return to generate expert behavior) as well as the environment starting state, to initialize generation. After executing the generated action, we decrement the target return by the achieved reward and obtain the next state. We repeat this process of generating actions and applying them to obtain the next return-to-go and state until episode termination. ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 Decision Transformer Pseudocode (for continuous actions) ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "# R, s, a, t: returns -to -go , states , actions , or timesteps   \n# K: context length ( length of each input to DecisionTransformer )   \n# transformer : transformer with causal masking (GPT)   \n# embed_s , embed_a , embed_R : linear embedding layers   \n# embed_t : learned episode positional embedding   \n# pred_a : linear action prediction layer ",
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+    {
+        "type": "text",
+        "text": "# main model ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "def DecisionTransformer (R , s , a , t ): ",
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+    {
+        "type": "text",
+        "text": "# compute embeddings for tokens   \npos_embedding $=$ embed_t ( t ) # per - timestep ( note : not per - token )   \ns_embedding $=$ embed_s ( s ) $^ +$ pos_embedding   \na_embedding $=$ embed_a ( a ) $^ +$ pos_embedding   \nR_embedding $=$ embed_R ( R ) $^ +$ pos_embedding ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "# interleave tokens as (R_1 , s_1 , a_1 R_K , s_K ) input_embeds $=$ stack ( R_embedding , s_embedding , a_embedding ) ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "# use transformer to get hidden states hidden_states $=$ transformer ( input_embeds $=$ input_embeds ) ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "# select hidden states for action prediction tokens a_hidden $=$ unstack ( hidden_states ). actions ",
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+    {
+        "type": "text",
+        "text": "# predict action return pred_a ( a_hidden ) ",
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+    {
+        "type": "text",
+        "text": "# training loop ",
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+    {
+        "type": "text",
+        "text": "for (R , s , a , t ) in dataloader : # dims : ( batch_size , K, dim ) a_preds $=$ DecisionTransformer (R , s , a , t ) loss $=$ mean (( a_preds - a ) $* * 2$ ) # L2 loss for continuous actions optimizer . zero_grad (); loss . backward (); optimizer . step () ",
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+    {
+        "type": "text",
+        "text": "# evaluation loop ",
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+    {
+        "type": "text",
+        "text": "target_return $\\mathbf { \\lambda } = \\mathbf { \\bar { \\lambda } } 1$ # for instance , expert - level return R , s , a , t , done $=$ [ target_return ] , [ env . reset ()] , [] , [1] , False while not done : # autoregressive generation / sampling ",
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+        "type": "text",
+        "text": "action $=$ DecisionTransformer (R , s , a , t )[ -1] # for cts actions new_s , r , done , _ $=$ env . step ( action ) ",
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+    {
+        "type": "text",
+        "text": "# append new tokens to sequence R = R + [ R [ -1] - r] # decrement returns -to -go with reward s , a , t = s + [ new_s ] , a $^ +$ [ action ] , t + [ len ( R )] R , s , a , t $=$ R [ - K :] , ... # only keep context length of K ",
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+    {
+        "type": "image",
+        "img_path": "images/6be7ce5512d46b7431e828fe71c83f457f669167743b6fc373ae4c4399521a50.jpg",
+        "image_caption": [
+            "Figure 3: Results comparing Decision Transformer (ours) to TD learning (CQL) and behavior cloning across Atari, OpenAI Gym, and Minigrid. On a diverse set of tasks, Decision Transformer performs comparably or better than traditional approaches. "
+        ],
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+    {
+        "type": "text",
+        "text": "4 Evaluations on offline RL benchmarks ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we investigate if Decision Transformer can perform well compared to standard TD and imitation learning approaches for offline RL. TD learning algorithms represent the conventional stateof-the-art, while imitation learning algorithms have similar formulations to Decision Transformer. The exact algorithms depend on the environment but our motivations are as follows: ",
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+        "type": "text",
+        "text": "• TD learning: most of these methods use an action-space constraint or value pessimism, and will be the most faithful comparison to Decision Transformer, representing standard RL methods. A state-of-the-art model-free method is Conservative Q-Learning (CQL) [17] which serves as our primary comparison. In addition, we also compare against other prior model-free RL algorithms like BEAR [19] and BRAC [20]. • Imitation learning: this regime similarly uses supervised losses for training, rather than Bellman backups. We use behavior cloning here, and include a more detailed discussion in Section 5.1. ",
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+        "text": "We evaluate on both discrete (Atari [13]) and continuous (OpenAI Gym [14]) control tasks. The former requires long-term credit assignment, while the latter requires fine-grained continuous control, representing a diverse set of tasks. Our main results are summarized in Figure 3, where we show averaged expert normalized performance for each domain. ",
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+    {
+        "type": "text",
+        "text": "4.1 Atari ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The Atari benchmark is challenging due to its high-dimensional visual inputs and difficulty of credit assignment arising from the delay between actions and resulting rewards. We evaluate our method on $1 \\%$ of all samples in the DQN-replay dataset as per Agarwal et al. [16], representing 500 thousand of the 50 million transitions observed by an online DQN agent [21] during training; we report the mean and standard deviation of 3 seeds. We normalize scores based on a professional gamer, following the protocol of Hafner et al. [22], where 100 represents the professional gamer score and 0 represents a random policy. ",
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+    {
+        "type": "text",
+        "text": "We compare to CQL [17], REM [16], and QR-DQN [23] on four Atari tasks (Breakout, Qbert, Pong, and Seaquest) that are evaluated in Agarwal et al. [16]. We use context lengths of $K = 3 0$ for Decision Transformer (except $K = 5 0$ for Pong); for results with different values of $K$ see the supplementary material. We also report the performance of behavior cloning (BC), which utilizes ",
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+    {
+        "type": "table",
+        "img_path": "images/c09a063a1b24fe2e8f3d693a626260be800c58ccba5fba82ee97e1f3080480bf.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Game</td><td>DT (Ours)</td><td>CQL</td><td>QR-DQN</td><td>REM</td><td>BC</td></tr><tr><td>Breakout</td><td>267.5 ± 97.5</td><td>211.1</td><td>21.1</td><td>32.1</td><td>138.9 ± 61.7</td></tr><tr><td>Qbert</td><td>15.1 ± 11.4</td><td>104.2</td><td>1.7</td><td>1.4</td><td>17.3 ± 14.7</td></tr><tr><td>Pong</td><td>106.1 ±8.1</td><td>111.9</td><td>20.0</td><td>39.1</td><td>85.2 ± 20.0</td></tr><tr><td>Seaquest</td><td>2.4± 0.7</td><td>1.7</td><td>1.4</td><td>1.0</td><td>2.1± 0.3</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 1: Gamer-normalized scores for the $1 \\%$ DQN-replay Atari dataset. We report the mean and variance across 3 seeds. Best mean scores are highlighted in bold. Decision Transformer (DT) performs comparably to CQL on 3 out of 4 games, and outperforms other baselines in most games. ",
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+    {
+        "type": "text",
+        "text": "the same network architecture and hyperparameters as Decision Transformer but does not have return-to-go conditioning2. For CQL, REM, and QR-DQN baselines, we report numbers directly from the CQL paper. We show results in Table 1. Our method is competitive with CQL in 3 out of 4 games and outperforms or matches REM, QR-DQN, and BC on all 4 games. ",
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+    {
+        "type": "text",
+        "text": "4.2 OpenAI Gym ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we consider the continuous control tasks from the D4RL benchmark [24]. We also consider a 2D reacher environment that is not part of the benchmark, and generate the datasets using a similar methodology to the D4RL benchmark. Reacher is a goal-conditioned task and has sparse rewards, so it represents a different setting than the standard locomotion environments (HalfCheetah, Hopper, and Walker). The different dataset settings are described below. ",
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+    {
+        "type": "text",
+        "text": "1. Medium: 1 million timesteps generated by a “medium” policy that achieves approximately one-third the score of an expert policy.   \n2. Medium-Replay: the replay buffer of an agent trained to the performance of a medium policy (approximately $2 5 \\mathrm { k } { - } 4 0 0 \\mathrm { k }$ timesteps in our environments).   \n3. Medium-Expert: 1 million timesteps generated by the medium policy concatenated with 1 million timesteps generated by an expert policy. ",
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+    {
+        "type": "text",
+        "text": "We compare to CQL [17], BEAR [19], BRAC [20], and AWR [25]. CQL represents the state-ofthe-art in model-free offline RL, an instantiation of TD learning with value pessimism. Score are normalized so that 100 represents an expert policy, as per Fu et al. [24]. CQL numbers are reported from the original paper; BC numbers are run by us; and the other methods are reported from the D4RL paper. Our results are shown in Table 2. Decision Transformer achieves the highest scores in a majority of the tasks and is competitive with the state of the art in the remaining tasks. ",
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+    {
+        "type": "table",
+        "img_path": "images/4058c54f4182eee535eaf3daa1c872ea7d3a5f74f8a6e9d88b74b1ad6b603bf9.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 2: Results for D4RL datasets4. We report the mean and variance for three seeds. Decision Transformer (DT) outperforms conventional RL algorithms on almost all tasks. "
+        ],
+        "table_body": "<table><tr><td>Dataset</td><td>Environment</td><td>DT (Ours)</td><td>CQL</td><td>BEAR</td><td>BRAC-v</td><td>AWR</td><td>BC</td></tr><tr><td>Medium-Expert</td><td>HalfCheetah</td><td>86.8 ± 1.3</td><td>62.4</td><td>53.4</td><td>41.9</td><td>52.7</td><td>59.9</td></tr><tr><td>Medium-Expert</td><td>Hopper</td><td>107.6 ± 1.8</td><td>111.0</td><td>96.3</td><td>0.8</td><td>27.1</td><td>79.6</td></tr><tr><td>Medium-Expert</td><td>Walker</td><td>108.1 ±0.2</td><td>98.7</td><td>40.1</td><td>81.6</td><td>53.8</td><td>36.6</td></tr><tr><td>Medium-Expert</td><td>Reacher</td><td>89.1 ±1.3</td><td>30.6</td><td>-</td><td>-</td><td>1</td><td>73.3</td></tr><tr><td>Medium</td><td>HalfCheetah</td><td>42.6 ± 0.1</td><td>44.4</td><td>41.7</td><td>46.3</td><td>37.4</td><td>43.1</td></tr><tr><td>Medium</td><td>Hopper</td><td>67.6 ± 1.0</td><td>58.0</td><td>52.1</td><td>31.1</td><td>35.9</td><td>63.9</td></tr><tr><td>Medium</td><td>Walker</td><td>74.0 ± 1.4</td><td>79.2</td><td>59.1</td><td>81.1</td><td>17.4</td><td>77.3</td></tr><tr><td>Medium</td><td>Reacher</td><td>51.2 ± 3.4</td><td>26.0</td><td>1</td><td>1</td><td>1</td><td>48.9</td></tr><tr><td>Medium-Replay</td><td>HalfCheetah</td><td>36.6 ± 0.8</td><td>46.2</td><td>38.6</td><td>47.7</td><td>40.3</td><td>4.3</td></tr><tr><td>Medium-Replay</td><td>Hopper</td><td>82.7 ± 7.0</td><td>48.6</td><td>33.7</td><td>0.6</td><td>28.4</td><td>27.6</td></tr><tr><td>Medium-Replay</td><td>Walker</td><td>66.6 ± 3.0</td><td>26.7</td><td>19.2</td><td>0.9</td><td>15.5</td><td>36.9</td></tr><tr><td>Medium-Replay</td><td>Reacher</td><td>18.0 ± 2.4</td><td>19.0</td><td>1</td><td>1</td><td>1</td><td>5.4</td></tr><tr><td colspan=\"2\">Average (Without Reacher)</td><td>74.7</td><td>63.9</td><td>48.2</td><td>36.9</td><td>34.3</td><td>46.4</td></tr><tr><td colspan=\"2\">Average (All Settings)</td><td>69.2</td><td>54.2</td><td>1</td><td>-</td><td>1</td><td>47.7</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "5 Discussion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "5.1 Does Decision Transformer perform behavior cloning on a subset of the data? ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we seek to gain insight into whether Decision Transformer can be thought of as performing imitation learning on a subset of the data with a certain return. To investigate this, we propose a new method, Percentile Behavior Cloning $( \\% \\mathrm { B C } )$ , where we run behavior cloning on only the top $X \\%$ of timesteps in the dataset, ordered by episode returns. The percentile $X \\%$ interpolates between standard BC $X = 1 0 0 \\%$ ) that trains on the entire dataset and only cloning the best observed trajectory $( X  0 \\%$ ), trading off between better generalization by training on more data with training a specialized model that focuses on a desirable subset of the data. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/30e324cb7f286a944c6ae8968e8be730623cd41a282d871352fae6a88e317a56.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 3: Comparison between Decision Transformer (DT) and Percentile Behavior Cloning $( \\% \\mathrm { B C } )$ "
+        ],
+        "table_body": "<table><tr><td>Dataset</td><td>Environment</td><td>DT (Ours)</td><td>10%BC</td><td>25%BC</td><td>40%BC</td><td>100%BC</td><td>CQL</td></tr><tr><td>Medium</td><td>HalfCheetah</td><td>42.6 ± 0.1</td><td>42.9</td><td>43.0</td><td>43.1</td><td>43.1</td><td>44.4</td></tr><tr><td>Medium</td><td>Hopper</td><td>67.6 ± 1.0</td><td>65.9</td><td>65.2</td><td>65.3</td><td>63.9</td><td>58.0</td></tr><tr><td>Medium</td><td>Walker</td><td>74.0 ± 1.4</td><td>78.8</td><td>80.9</td><td>78.8</td><td>77.3</td><td>79.2</td></tr><tr><td>Medium</td><td>Reacher</td><td>51.2 ± 3.4</td><td>51.0</td><td>48.9</td><td>58.2</td><td>58.4</td><td>26.0</td></tr><tr><td>Medium-Replay</td><td>HalfCheetah</td><td>36.6 ± 0.8</td><td>40.8</td><td>40.9</td><td>41.1</td><td>4.3</td><td>46.2</td></tr><tr><td>Medium-Replay</td><td>Hopper</td><td>82.7 ± 7.0</td><td>70.6</td><td>58.6</td><td>31.0</td><td>27.6</td><td>48.6</td></tr><tr><td>Medium-Replay</td><td>Walker</td><td>66.6 ± 3.0</td><td>70.4</td><td>67.8</td><td>67.2</td><td>36.9</td><td>26.7</td></tr><tr><td>Medium-Replay</td><td>Reacher</td><td>18.0 ± 2.4</td><td>33.1</td><td>16.2</td><td>10.7</td><td>5.4</td><td>19.0</td></tr><tr><td colspan=\"2\">Average</td><td>56.1</td><td>56.7</td><td>52.7</td><td>49.4</td><td>39.5</td><td>43.5</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We show full results comparing $\\% \\mathrm { B C }$ to Decision Transformer and CQL in Table 3, sweeping over $X \\in [ 1 0 \\% , 2 5 \\% , 4 0 \\% , 1 \\dot { 0 } \\dot { 0 } \\% ]$ . Note that while both $\\% \\mathrm { B C }$ and DT introduce hyperparameters, returns are human interpretable and it is relatively natural for humans to specify a desired return compared to choosing an optimal subset for cloning. When data is plentiful – as in the D4RL regime – we find $\\% \\mathrm { B C }$ can match or beat other offline RL methods. On most environments, Decision Transformer is competitive with the performance of the best $\\% \\mathrm { B C }$ , indicating it can hone in on a particular subset after training on the entire dataset distribution. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In contrast, when we study low data regimes – such as Atari, where we use $1 \\%$ of a replay buffer as the dataset – $\\mathbf { \\nabla } \\cdot \\% \\mathbf { B } \\mathbf { C }$ is weak (shown in Table 4). This suggests that in scenarios with relatively low amounts of data, Decision Transformer can outperform $\\% \\mathrm { B C }$ by using all trajectories in the dataset to improve generalization, even if those trajectories are dissimilar from the return conditioning target. Our results indicate that Decision Transformer can be more effective than simply performing imitation learning on a subset of the dataset. On the tasks we considered, Decision Transformer either outperforms or is competitive to $\\% \\mathrm { B C }$ , without the confound of having to select the optimal subset. ",
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+    {
+        "type": "table",
+        "img_path": "images/f7b3e0dda54d8e04dec597f56f451d1866f4fd896d6e53dc669b8062bb2381af.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 4: $\\% \\mathrm { B C }$ scores for Atari. We report the mean and variance across 3 seeds. Decision Transformer (DT) outperforms all versions of $\\% \\mathrm { B C }$ in most games. "
+        ],
+        "table_body": "<table><tr><td>Game</td><td>DT (Ours)</td><td>10%BC</td><td>25%BC</td><td>40%BC</td><td>100%BC</td></tr><tr><td>Breakout</td><td>267.5 ± 97.5</td><td>28.5±8.2</td><td>73.5 ± 6.4</td><td>108.2 ± 67.5</td><td>138.9 ± 61.7</td></tr><tr><td>Qbert</td><td>15.1 ± 11.4</td><td>6.6 ± 1.7</td><td>16.0 ± 13.8</td><td>11.8± 5.8</td><td>17.3 ± 14.7</td></tr><tr><td>Pong</td><td>106.1 ±8.1</td><td>2.5± 0.2</td><td>13.3 ± 2.7</td><td>72.7 ± 13.3</td><td>85.2 ± 20.0</td></tr><tr><td>Seaquest</td><td>2.4 ± 0.7</td><td>1.1 ± 0.2</td><td>1.1 ± 0.2</td><td>1.6 ± 0.4</td><td>2.1 ± 0.3</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5.2 How well does Decision Transformer model the distribution of returns? ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We evaluate the ability of Decision Transformer to understand return-to-go tokens by varying the desired target return over a wide range – evaluating the multi-task distribution modeling capability of transformers. Figure 4 shows the average sampled return accumulated by the agent over the course of the evaluation episode for varying values of target return. On every task, the desired target returns and the true observed returns are highly correlated. On some tasks like Pong, HalfCheetah and Walker, Decision Transformer generates trajectories that almost perfectly match the desired returns (as indicated by the overlap with the oracle line). Furthermore, on some Atari tasks like Seaquest, we can prompt the Decision Transformer with higher returns than the maximum episode return available in the dataset, demonstrating that Decision Transformer is sometimes capable of extrapolation. ",
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+    {
+        "type": "text",
+        "text": "5.3 Does Decision Transformer perform effective long-term credit assignment? ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To evaluate long-term credit assignment capabilities of our model, we consider a variant of the Key-to-Door environment proposed in Mesnard et al. [15]. This is a grid-based environment with a sequence of three phases: (1) in the first phase, the agent is placed in a room with a key; (2) then, the agent is placed in an empty room; (3) and finally, the agent is placed in a room with a door. The agent receives a binary reward when reaching the door in the third phase, but only if it picked up the key in the first phase. This problem is difficult for credit assignment because credit must be propagated from the beginning to the end of the episode, skipping over actions taken in the middle. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/c1156eaeb2971806e2c5eb8d09680aa47f4f5cf207c581cf8c22a3d777eebeb5.jpg",
+        "image_caption": [
+            "Figure 4: Sampled (evaluation) returns accumulated by Decision Transformer when conditioned on the specified target (desired) returns. Top: Atari. Bottom: D4RL medium-replay datasets. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We train on datasets of trajectories generated by applying random actions and report success rates in Table 5. Furthermore, for the Key-to-Door environment we use the entire episode length as the context, rather than having a fixed content window as in the other environments. Methods that use highsight return information: our Decision Transformer model and $\\% \\mathrm { B C }$ (trained only on successful episodes) are able to learn effective policies – producing near-optimal paths, despite only training on random walks. TD learning (CQL) cannot effectively propagate $\\mathrm { Q }$ -values over the long horizons involved and gets poor performance. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/b098074bab96a0d61716bb08dc399340315f12820eb4c04ac2b973cf59169be0.jpg",
+        "table_caption": [
+            "Table 5: Success rate for Key-to-Door environment. Methods using hindsight (Decision Transformer, $\\% \\mathbf { B } \\mathbf { C } _ { \\epsilon }$ ) can learn successful policies, while TD learning struggles to perform credit assignment. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>DT (Ours)</td><td>CQL</td><td>BC</td><td>%BC</td><td>Random</td></tr><tr><td>1K Random Trajectories</td><td>71.8%</td><td>13.1%</td><td>1.4%</td><td>69.9%</td><td>3.1%</td></tr><tr><td>10K Random Trajectories</td><td>94.6%</td><td>13.3%</td><td>1.6%</td><td>95.1%</td><td>3.1%</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5.4 Can transformers be accurate critics in sparse reward settings? ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In previous sections, we established that decision transformer can produce effective policies (actors). We now evaluate whether transformer models can also be effective critics. We modify Decision Transformer to output return tokens in addition to action tokens on the Key-to-Door environment. We find that the transformer continuously updates reward probability based on events during the episode, shown in Figure 5 (Left). Furthermore, we find the transformer attends to critical events in the episode (picking up the key or reaching the door), shown in Figure 5 (Right), indicating formation of state-reward associations as discussed in Raposo et al. [26] and enabling accurate value prediction. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5.5 Does Decision Transformer perform well in sparse reward settings? ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A known weakness of TD learning algorithms is that they require densely populated rewards in order to perform well, which can be unrealistic and/or expensive. In contrast, Decision Transformer can improve robustness in these settings since it makes minimal assumptions on the density of the reward. To evaluate this, we consider a delayed return version of the D4RL benchmarks where the agent does not receive any rewards along the trajectory, and instead receives the cumulative reward of the trajectory in the final timestep. Our results for delayed returns are shown in Table 6. Delayed returns minimally affect Decision Transformer; and due to the nature of the training process, while imitation learning methods are reward agnostic. While TD learning collapses, Decision Transformer and $\\% \\mathrm { B C }$ still perform well, indicating that Decision Transformer can be more robust to delayed rewards. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/2cf047d6f6123e6feee315d97b0f7169c03581ba80dfd8637dd3453482f91ce9.jpg",
+        "image_caption": [
+            "Figure 5: Left: Averages of running return probabilities predicted by the transformer model for three types of episode outcomes. Right: Transformer attention weights from all timesteps superimposed for a particular successful episode. The model attends to steps near pivotal events in the episode, such as picking up the key and reaching the door. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/30abdf3c2bb5d25d47c7adf3a2f6138477792c0148cf52e9f901c6b39aaf9f6e.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 6: Results for D4RL datasets with delayed (sparse) reward. Decision Transformer (DT) and imitation learning are minimally affected by the removal of dense rewards, while CQL fails. "
+        ],
+        "table_body": "<table><tr><td></td><td>Environment</td><td colspan=\"2\">Delayed (Sparse)</td><td colspan=\"2\">Agnostic</td><td colspan=\"2\">Original (Dense)</td></tr><tr><td>Dataset</td><td></td><td>DT (Ours)</td><td>CQL</td><td>BC</td><td>%BC</td><td>DT (Ours)</td><td>CQL</td></tr><tr><td>Medium-Expert</td><td>Hopper</td><td>107.3 ± 3.5</td><td>9.0</td><td>59.9</td><td>102.6</td><td>107.6</td><td>111.0</td></tr><tr><td>Medium</td><td>Hopper</td><td>60.7± 4.5</td><td>5.2</td><td>63.9</td><td>65.9</td><td>67.6</td><td>58.0</td></tr><tr><td>Medium-Replay</td><td>Hopper</td><td>78.5 ± 3.7</td><td>2.0</td><td>27.6</td><td>70.6</td><td>82.7</td><td>48.6</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "5.6 Additional Discussions ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For more discussions see the supplementary material. ",
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+    {
+        "type": "text",
+        "text": "6 Related work ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Offline reinforcement learning. To mitigate the impact of distribution shift in offline RL, prior algorithms either (a) constrain the policy action space [27, 28, 29] or (b) incorporate value pessimism [27, 17], or (c) incorporate pessimism into learned dynamics models [30, 31]. Since we do not use Decision Transformers to explicitly learn the dynamics model, we primarily compare against model-free algorithms; adding a dynamics model tends to improve the performance of model-free algorithms. Another line of work explores learning wide behavior distribution from an offline dataset by learning a task-agnostic set of skills, either with likelihood-based approaches [32, 33, 34, 35] or by maximizing mutual information [36, 37, 38]. Our work is similar to the likelihood-based approaches, which do not use iterative Bellman updates – although we use a simpler sequence modeling objective instead of a variational method, and use rewards for conditional generation of behaviors. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Supervised learning in reinforcement learning settings. Some prior methods for reinforcement learning bear more resemblance to static supervised learning, such as Q-learning [39, 40], which still uses iterative backups, or likelihood-based methods such as behavior cloning, which do not (discussed in previous section). Recent work [8, 9, 10] studies “upside-down” reinforcement learning (UDRL), which are similar to our method in seeking to model behaviors with a supervised loss conditioned on the target return. A key difference in our work is the shift of motivation to sequence modeling rather than supervised learning: while the practical methods differ primarily in the context length and architecture, sequence modeling enables behavior modeling even without access to the reward, in a similar style to language [12] or images [41], and is known to scale well [2]. The method proposed by Kumar et al. [9] is most similar to our method with $K = 1$ , which we find sequence modeling/long contexts to outperform (see supplementary material). Ghosh et al. [42] extends prior UDRL methods to use state goal conditioning, rather than rewards, and Paster et al. [43] further use an LSTM with state goal conditioning for goal-conditoned online RL settings. Concurrent to our work, Janner et al. [44] propose Trajectory Transformer, which is similar to Decision Transformer but additionally uses state and return prediction, as well as discretization, which incorporates model-based components. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We believe that their experiments, in addition to our results, highlight the potential for sequence modeling to be a generally applicable idea for reinforcement learning. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Credit assignment. Many works have studied better credit assignment via state-association, learning an architecture which decomposes the reward function such that certain “important” states comprise most of the credit [45, 46, 15]. They use the learned reward function to change the reward of an actorcritic algorithm to help propagate signal over long horizons. In particular, similar to our long-term setting, some works have specifically shown such state-associative architectures can perform better in delayed reward settings [47, 7, 48, 26]. In contrast, we allow these properties to naturally emerge in a transformer architecture, without having to explicitly learn a reward function or a critic. ",
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+        "page_idx": 9
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+    {
+        "type": "text",
+        "text": "Conditional language generation. Various works have studied guided generation for images [49] and language [50, 51]. Several works [52, 53, 54, 55, 56, 57] have explored training or fine-tuning of models for controllable text generation. Class-conditional language models can also be used to learn disciminators to guide generation [58, 50, 59, 60]. However, these approaches mostly assume constant “classes”, while in reinforcement learning the reward signal is time-varying. Furthermore, it is more natural to prompt the model desired target return and continuously decrease it by the observed rewards over time, since the transformer model and environment jointly generate the trajectory. ",
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+    {
+        "type": "text",
+        "text": "Attention and transformer models. Transformers [1] have been applied successfully to many tasks in natural language processing [61, 12] and computer vision [62, 63]. However, transformers are relatively unstudied in RL, mostly due to differing nature of the problem, such as higher variance in training. Zambaldi et al. [5] showed that augmenting transformers with relational reasoning improve performance in combinatorial environments and Ritter et al. [64] showed iterative selfattention allowed for RL agents to better utilize episodic memories. Parisotto et al. [4] discussed design decisions for more stable training of transformers in the high-variance RL setting. Unlike our work, these still use actor-critic algorithms for optimization, focusing on novelty in architecture. Additionally, in imitation learning, some works have studied transformers as a replacement for LSTMs: Dasari and Gupta [65] study one-shot imitation learning, and Abramson et al. [66] combine language and image modalities for text-conditioned behavior generation. ",
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+    {
+        "type": "text",
+        "text": "7 Conclusion ",
+        "text_level": 1,
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+    {
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+        "text": "We proposed Decision Transformer, seeking to unify ideas in language modeling and RL. On standard offline RL benchmarks, we showed DT can match or outperform strong algorithms designed explicitly for offline RL with minimal modifications from standard language modeling architectures. ",
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+    {
+        "type": "text",
+        "text": "Societal impact. For real-world applications, it is important to understand the types of errors transformers make in MDP settings and possible negative consequences. It will also be important to consider the datasets we train on, which can potentially add destructive biases, particularly as we consider studying augmenting RL agents with more data which may come from questionable sources. ",
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+    {
+        "type": "text",
+        "text": "Limitations. We introduced our paradigm shift and showed its potential in our experiments, but there is significant room for more research in this direction. The current architecture requires considerations of context length and return-to-go hyperparameters, and we show results on standard RL benchmarks; future work could improve the architecture and demonstrate results in more complex environments and tasks. We used a simple supervised loss that was effective in our experiments, but applications to large-scale datasets could benefit from self-supervised pretraining tasks. In addition, one could consider more sophisticated embeddings for returns, states, and actions. While we do not directly evaluate scaling and generalization, we utilize a method known to scale generalize well in domains such as language and vision, and we are excited about larger RL systems built upon our framework. ",
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+    {
+        "type": "text",
+        "text": "8 Acknowledgements ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "This research was supported by Berkeley Deep Drive, Open Philanthropy, and the National Science Foundation under NSF:NRI #2024675. Part of this work was completed when Aravind Rajeswaran was a PhD student at the University of Washington, where he was supported by the J.P. Morgan PhD Fellowship in AI (2020-21). We also thank Luke Metz, Daniel Freeman, and anonymous reviewers for valuable feedback and discussions, as well as Justin Fu for assistance in setting up D4RL benchmarks, and Aviral Kumar for assistance with the CQL baselines and hyperparameters. ",
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+        "type": "text",
+        "text": "References   \n[1] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in Neural Information Processing Systems, 2017.   \n[2] Tom B Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Language models are few-shot learners. arXiv preprint arXiv:2005.14165, 2020.   \n[3] Aditya Ramesh, Mikhail Pavlov, Gabriel Goh, Scott Gray, Chelsea Voss, Alec Radford, Mark Chen, and Ilya Sutskever. Zero-shot text-to-image generation. arXiv preprint arXiv:2102.12092, 2021.   \n[4] Emilio Parisotto, Francis Song, Jack Rae, Razvan Pascanu, Caglar Gulcehre, Siddhant Jayakumar, Max Jaderberg, Raphael Lopez Kaufman, Aidan Clark, Seb Noury, et al. Stabilizing transformers for reinforcement learning. 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parse/train/rJgqMRVYvr/rJgqMRVYvr.md

@@ -0,0 +1,319 @@
+# DIFFERENTIALLY PRIVATE META-LEARNING
+
+Jeffrey Li, Mikhail Khodak, Sebastian Caldas   
+Carnegie Mellon University   
+jwl3@cs.cmu.edu
+
+Ameet Talwalkar Carnegie Mellon University & Determined AI
+
+# ABSTRACT
+
+Parameter-transfer is a well-known and versatile approach for meta-learning, with applications including few-shot learning, federated learning, and reinforcement learning. However, parameter-transfer algorithms often require sharing models that have been trained on the samples from specific tasks, thus leaving the task-owners susceptible to breaches of privacy. We conduct the first formal study of privacy in this setting and formalize the notion of task-global differential privacy as a practical relaxation of more commonly studied threat models. We then propose a new differentially private algorithm for gradient-based parameter transfer that not only satisfies this privacy requirement but also retains provable transfer learning guarantees in convex settings. Empirically, we apply our analysis to the problems of federated learning with personalization and few-shot classification, showing that allowing the relaxation to task-global privacy from the more commonly studied notion of local privacy leads to dramatically increased performance in recurrent neural language modeling and image classification.
+
+# 1 INTRODUCTION
+
+The field of meta-learning offers promising directions for improving the performance and adaptability of machine learning methods. At a high level, the key assumption leveraged by these approaches is that the sharing of knowledge gained from individual learning tasks can help catalyze the learning of similar unseen tasks. However, the collaborative nature of this process, in which task-specific information must be sent to and used by a meta-learner, also introduces inherent data privacy risks.
+
+In this work, we focus on a popular and flexible meta-learning approach, parameter transfer via gradient-based meta-learning (GBML). This set of methods, which includes well-known algorithms such as MAML (Finn et al., 2017) and Reptile (Nichol et al., 2018), tries to learn a common initialization $\phi$ over a set of tasks $t = 1 , \dots , T$ such that a high-performance model can be learned in only a few gradient-steps on new tasks. Notably, information flows constantly between training tasks and the meta-learner as learning progresses; to make iterative updates, the meta-learner obtains feedback on the current $\phi$ by having task-specific models ${ \bar { \theta } } _ { t }$ trained with it.
+
+Meanwhile, in many settings amenable to meta-learning, it is crucial to ensure that sensitive information in each task’s dataset stays private. Examples of this include learning models for word prediction on cell phone data (McMahan et al., 2018), clinical predictions using hospital records (Zhang et al., 2019), and fraud detectors for competing credit card companies (Stolfo et al., 1997). In such cases, each data-owner can benefit from information learned from other tasks, but each also desires, or is legally required, to keep their raw data private. Thus, it is not sufficient to learn a well-performing $\phi$ ; it is equally imperative to ensure that a task’s sensitive information is not obtainable by anyone else.
+
+While parameter transfer algorithms can move towards this goal by peforming task-specific optimization locally, thus preventing direct access to private data, this provision is far from fail-safe in terms of privacy. A wealth of work has shown in the single-task setting that it is possible for an adversary with only access to the model to learn detailed information about the training set, such as the presence or absence of specific records (Shokri et al., 2017) or the identities of sensitive features given other covariates (Fredrikson et al., 2015). Furthermore, Carlini et al. (2018) showed that deep neural networks can effectively memorize user-unique training examples, which can be recovered even after only a single epoch of training. As such, in parameter-transfer methods, the meta-learner or any downstream participant can potentially recover data from a previous task.
+
+However, despite these serious risks, privacy-preserving meta-learning has remained largely an unstudied problem. Our work aims to address this issue by applying differential privacy (DP), a well-established definition of privacy with rich theoretical guarantees and consistent empirical success at preventing leakages of data (Carlini et al., 2018; Fredrikson et al., 2015; Jayaraman and Evans, 2019). Crucially, although there are various threat models and degrees of DP one could consider in the meta-learning setting (as we outline in Section 2), we balance the well-documented trade-off between privacy and model utility by formalizing and focusing on a setting that we call task-global DP. This setting provides a strong privacy guarantee for each task-owner that sharing $\widehat { \theta } _ { t }$ with the meta-learner will not reliably reveal anything about specific training examples to any downstream agent. It also allows us to use the framework of Khodak et al. (2019a) to provide a DP GBML algorithm that enjoys provable learning guarantees in convex settings.
+
+Finally, we show an application of our work by drawing connections to federated learning (FL) (Li et al., 2019). While standard methods for FL, such as FedAvg (McMahan et al., 2017), have inspired many works also concerning DP in a multi-user setup (Agarwal et al., 2018; Bhowmick et al., 2019; Geyer et al., 2018; McMahan et al., 2018; Truex et al., 2019), we are the first to consider task-global DP as a useful variation on standard DP settings. Moreover, these works fundamentally differ from ours in that they do not consider a task-based notion of learnability, instead focusing on the global federated learning problem to learn a single global model. That being said, a federated setting involving per-user personalization (Chen et al., 2018; Smith et al., 2017) is a natural meta-learning application.
+
+More specifically, our main contributions are:
+
+1. We are the first to provide a taxonomy for the different notions of DP possible for meta-learning. In particular, we formalize on a variant we call task-global DP, showing and arguing that it adds a useful option to commonly studied settings in terms of trading privacy and accuracy.   
+2. We propose the first DP GBML algorithm, which we construct to satisfy this privacy setting. Further, we show a straightforward extension for obtaining a group $D P$ version of our setting to protect multiple samples simultaneously.   
+3. While our privacy guarantees hold generally, we also prove learning-theoretic results in convex settings. Our learning guarantees scale with task-similarity, as measured by the closeness of the task-specific optimal parameters (Denevi et al., 2019; Khodak et al., 2019b).   
+4. We show that our algorithm, along with its theoretical guarantees, naturally carries over to federated learning with personalization. Compared to previous notions of privacy considered in works for DP federated learning (Agarwal et al., 2018; Bhowmick et al., 2019; Geyer et al., 2018; McMahan et al., 2018; Truex et al., 2019), we are, to the best of our knowledge, the first to simultaneously provide both privacy and learning guarantees.   
+5. Empirically, we demonstrate that our proposed privacy setting allows for strong performance on federated language-modeling and few-shot image classification tasks. For the former, we achieve close to the performance of non-private models and significantly improve upon the performance of models trained with local-DP guarantees, a previously studied notion that also provides protections against the meta-learner. Our setting reasonably relaxes this latter notion but can achieve roughly 1.7–2.3 times the accuracy on a modified version of the Shakespeare dataset (Caldas et al., 2018) and 1.6–1.7 times the accuracy on a modified version of Wiki-3029 (Arora et al., 2019) across various privacy budgets. For image-classification, we show that we show that we can still retain significant benefits of meta-learning while applying task-global DP on Omniglot (Lake et al., 2011) and Mini-ImageNet (Ravi and Larochelle, 2017).
+
+# 1.1 RELATED WORK
+
+DP Algorithms in Federated Learning Settings. Works most similar to ours focus on providing DP for federated learning. Specifically, Geyer et al. (2018) and McMahan et al. (2018) apply update clipping and the Gaussian Mechanism to achieve user-level global DP federated learning algorithms for language modeling and image classification tasks respectively. Their methods are shown to only suffer minor drops in accuracy compared to non-private training but they do not consider protections to inferences made by the meta-learner. Alternatively, Bhowmick et al. (2019) does achieve such protection by applying a theoretically rate-optimal local DP mechanism on the ${ \bar { \theta } } _ { t }$ ’s users send to the meta-learner. However, they sidestep hard minimax rates (Duchi et al., 2018) by assuming the central server has limited side-information and allowing for a large privacy budget. In this work, though we achieve a relaxation of the privacy of Bhowmick et al. (2019), we do not restrict the adversary’s power. Finally, Truex et al. (2019) does consider a setting that coincides with task-global DP, but they focus primarily on the added benefits of applying MPC (see below) rather than studying the merits of the setting in comparison to other potential settings.
+
+Secure Multiparty Computation (MPC). MPC is a cryptographic technique that allows parties to calculate a function of their inputs while also maintaining the privacy of each individual inputs. In GBML, sets of model updates may come in a batch from multiple tasks, and hence MPC can securely aggregate the batch before it is seen by the meta-learner. Though MPC itself gives no DP guarantees against future inference, it can combined with DP to increase privacy. This approach has been studied in the federated setting, e.g. by Agarwal et al. (2018), who apply MPC in the same difficult setting of Bhowmick et al. (2019), and Truex et al. (2019), who apply MPC similarly to a setting analogous to ours. On the other hand, MPC also comes with additional practical challenges such as peer-to-peer communication costs, drop outs, and vulnerability to collaborating participants. As such, combined with its applicability to multiple settings, including ours, we consider MPC to be an orthogonal direction.
+
+# 2 PRIVACY IN A META-LEARNING CONTEXT
+
+In this section, we first formalize the meta-learning setting that we consider. We then describe the various threat models that arise in the GBML setup, before presenting the different DP notions that can be achieved. Finally, we highlight the specific model and type of DP that we analyze.
+
+# 2.1 PARAMETER TRANSFER META-LEARNING
+
+In parameter transfer meta-learning, we assume that there is a set of learning tasks $t = 1 , \dots , T$ , each with its corresponding disjoint training set $D _ { t }$ . Each $D _ { t }$ contains $m _ { t }$ training examples $\{ z _ { t , i } \} _ { i = 1 } ^ { m _ { t } }$ where each $\boldsymbol { z } _ { t , i } \in \mathcal { X } \times \mathcal { Y }$ . The goal within each task is to learn a function $f _ { \hat { \theta } _ { t } } : \mathcal { X } \xrightarrow { } \mathcal { V }$ parameterized by $\widehat { \theta } _ { t } \in \Theta \subset \mathbb { R } ^ { d }$ that performs “well,” generally in the sense that it has low within-task population risk in the distributional setting. The meta-learner’s goal is to learn an initialization $\phi \in \Theta$ that leads to a well-performing $\widehat { \theta } _ { t }$ within-task. In GBML this $\phi$ is learned via an iterative process that alternates between the following two steps: (1) a within-task procedure where a batch of task-owners $B$ receives the current $\phi$ and each $t \in B$ uses $\phi$ as an initialization for running a within-task optimization procedure, obtaining $\bar { \theta } _ { t } ( D _ { t } , \phi )$ ; (2) a meta-level procedure where the meta-learner receives these model updates $\{ { \bar { \theta } } _ { t } \} _ { t \in B }$ and aggregates them to determine an updated $\phi$ . Note that we do not assume $\widehat { \theta } _ { t } = \bar { \theta } _ { t }$ , as the updates shared for the meta-learning procedure can be obtained from a different procedure than the refined model used for downstream within-task inference. This is especially the case when concerning the addition of noise for DP as part of the meta-learning procedure.
+
+# 2.2 THREAT MODELS FOR GBML
+
+As in any privacy endeavor, before discussing particular mechanisms, a key specification must be made in terms of what threat model is being considered. In particular, it must be specified both (1) who the potential adversaries are and (2) what information needs to be protected.
+
+Potential adversaries. For a single task-owner, adversaries may be either solely recipients of $\phi$ (i.e. other task-owners) or recipients of either $\phi$ or ${ \bar { \theta } } _ { t }$ (i.e. also the meta-learner). In the latter case, we consider only a honest-but-curious meta-learner, who does not deviate from the agreed upon algorithm but may try to make inferences from ${ \bar { \theta } } _ { t }$ . In both cases, concern is placed not only about these other participants’ intentions, but also their own security against access by malicious outsiders.
+
+Data to be protected. A system can choose either to protect information contained in single records $z _ { t , i }$ one-at-a-time or to protect entire datasets $D _ { t }$ simultaneously. This distinction between record-level and task-level privacy can be practically important. Multiple $z _ { t , i }$ within $D _ { t }$ may reveal the same secret (e.g., a cell-phone user has sent their SSN multiple times), or the entire distribution of $D _ { t }$ could reveal sensitive information (e.g., a user has sent all messages in a foreign language). In these cases, record-level privacy may not be sufficient. However, given that privacy and utility are often at odds, we often seek the weakest notion of privacy needed in order to best preserve utility.
+
+![](images/1af50ef442ecb2263b0ad5b33eec347594b127d79b6e088e27c64606c511143e.jpg)  
+Figure 1: Summary of the privacy protections guaranteed by local and global DP at the different levels of the meta-learning problem (with our notion in blue). On the right, we show what each specification would mean in two practical federated scenarios: mobile users and hospital networks.
+
+In related work, focus has primarily been placed on task-level protections. However, such approaches usually fall into two extremes, either obtaining strong learning but having to trust the meta-learner (McMahan et al., 2018; Geyer et al., 2018) or trusting nobody but also obtaining low performance (Bhowmick et al., 2019). In response, we try to bridge the gap between these threat models by considering a model that makes a relaxation from task-level to record-level privacy but retains protections for each task-owner against all other parties. This relaxation can be reasonably justified in practical situations, as while task-level guarantees are strictly stronger, they may also be unnecessary. In particular, record-level guarantees are likely to be sufficient whenever single records each pertain to different individuals. For example, for hospitals, what we care about is providing privacy to the individual patients and not aggregate hospital information. For cell-phones, if one can bound the number of texts that could reveal the same sensitive information, then a straightforward extension of our setting and methods, which protects up to $k$ records simultaneously, could also be sufficient.
+
+# 2.3 DIFFERENTIAL PRIVACY (DP) IN A SINGLE-TASK SETTING
+
+In terms of actually achieving privacy guarantees for machine learning, a de-facto standard has been to apply DP, a provision which strongly limits what one can infer about the examples a given model was trained on. Assuming a training set $D = \{ z _ { 1 } , \dots , z _ { m } \}$ , two common types of DP are considered.
+
+Differential Privacy (Global DP). A randomized mechanism $\mathcal { M }$ is $( \varepsilon , \delta )$ -differentially private if for all measurable $S \subseteq { \mathrm { R a n g e } } ( { \mathcal { M } } )$ and for all datasets $D , D ^ { \prime }$ that differ by at most one element:
+
+$$
+\mathbb { P } [ \mathcal { M } ( D ) \in \mathcal { S } ] \leq e ^ { \varepsilon } \mathbb { P } [ \mathcal { M } ( D ^ { \prime } ) \in \mathcal { S } ] + \delta
+$$
+
+If this holds for $D , D ^ { \prime }$ differing by at most $k$ elements, then $( \varepsilon , \delta ) \ k$ -group $D P$ is achieved.
+
+Local Differential Privacy. A randomized mechanism $\mathcal { M }$ is $( \varepsilon , \delta )$ -locally differentially private if for any two possible training examples $z , z ^ { \prime } \in \mathcal { X } \times \mathcal { Y }$ and measurable $S \subseteq \mathcal { X } \times \mathcal { Y }$ :
+
+$$
+\mathbb { P } [ \mathcal { M } ( z ) \in \mathcal { S } ] \leq e ^ { \varepsilon } \mathbb { P } [ \mathcal { M } ( z ^ { \prime } ) \in \mathcal { S } ] + \delta
+$$
+
+Global DP guarantees the difficulty of inferring the presence of a specific record in the training set by observing $\mathcal { M } ( D )$ . It assumes a trusted aggregator running $\mathcal { M }$ gets direct access to $D$ and privatizes the final output. Meanwhile, local DP assumes more strictly that the aggregator also cannot be trusted, thus requiring a random mechanism to be applied individually on each $z$ before training. However, it generally results in worse model performance, suffering from hard minimax rates (Duchi et al., 2018).
+
+Table 1: Broad categorization of the DP settings considered by our work in meta-learning and notable past works in the federated setting. Note that by using a de-centralized method for aggregation, Agarwal et al. (2018) can still protect against the meta-learner from making inferences on any individual $\widehat { \theta } _ { t }$ with what is only effectively a global DP mechanism.   
+
+<table><tr><td>Previous Work</td><td>Notion of DP</td><td>Privacy for Φ</td><td>Privacy for θt</td></tr><tr><td>McMahan et al. (2018)</td><td>Global</td><td>Task-level</td><td>1</td></tr><tr><td>Geyer et al. (2018)</td><td>Global</td><td>Task-level</td><td>=</td></tr><tr><td>Bhowmick et al. (2019)</td><td>Local, Global</td><td>Task-level</td><td>Task-level</td></tr><tr><td>Agarwal et al. (2018)</td><td>Global + MPC</td><td>Task-level</td><td>Task-level</td></tr><tr><td>Truex et al. (2019)</td><td>Task-Global + MPC</td><td>Record-level</td><td>Record-level</td></tr><tr><td>Our work</td><td>Task-Global</td><td>Record-level</td><td>Record-level</td></tr></table>
+
+# 2.4 DIFFERENTIAL PRIVACY FOR A GBML SETTING
+
+In meta-learning, there exists a hierarchy of agents and statistical queries, so we cannot as simply define global and local DP. Here, both the meta-level sub-procedure, $\{ \bar { \theta } _ { t } \} _ { t \in B }  \phi$ , and the withintask sub-procedure, $\{ z _ { t , i } \} _ { i = 1 } ^ { m _ { t } } \to \bar { \theta } _ { t }$ , can be considered individual queries and a DP algorithm can implement either to be DP. Further, for each query, the procedure may be altered to satisfy either local DP or global DP. Thus, there are four fundamental options that follow from standard DP definitions.
+
+(1) Global $D P$ : Releasing $\phi$ will at no point compromise information regarding any specific ${ \bar { \theta } } _ { t }$ .   
+(2) Local $D P$ : Additionally, each ${ \bar { \theta } } _ { t }$ is protected from being revealed to the meta-learner.   
+(3) Task-Global $D P$ : Releasing ${ \bar { \theta } } _ { t }$ will at no point compromise any specific $z _ { t , i }$ .   
+(4) Task-Local $D P$ : Additionally, each $z _ { t , i }$ is protected from being revealed to task-owner.
+
+To form analogies to single-task DP, the examples in the meta-level procedure are the model updates and the aggregator is the meta-learner. For the within-task procedure, the examples are actually the individual records and the aggregator is the task-owner. As such, (1) is implemented by the metalearner, (2) and (3) are implemented by the task-owner, and (4) is implemented by record-owners. By immunity to post-processing, the guarantees for (3) and (4) also automatically apply to the release of any future iteration of $\phi$ , thus protecting against future task-owners as well. Meanwhile, though (1) and (2) by definition protect the identities of individual ${ \bar { \theta } } _ { t }$ , they actually satisfy a task-level threat model by doing so. Intuitively, not being able to reliably infer anything about ${ \bar { \theta } } _ { t }$ implies that nothing can be inferred about the $D _ { t }$ that was used to generate it.
+
+Using the terminology we introduce in Section 2.4, previous works for DP in federated settings can be categorized as in Table 1. While these works do not assume a multi-task setting, we can still naturally use the terms global/local and task-global/task-local to analogously refer to releasing the global model (by the central server in the case without MPC) and user-specific updates (by users’ devices) respectively.
+
+# 3 DIFFERENTIALLY PRIVATE PARAMETER-TRANSFER
+
+# 3.1 ALGORITHM
+
+We now present our DP GBML method, which is written out in its online (regret) form in Algorithm 1. Here, we observe that both within-task optimization and meta-optimization are done using some form of gradient descent. The key difference between this algorithm and traditional GBML is that since task-learners must send back privatized model updates, each now applies an DP gradient descent procedure to learn ${ \bar { \theta } } _ { t }$ when called. However, at meta-test time the task-learner will run a non-private descent algorithm to obtain the parameter $\widehat { \theta } _ { t }$ used for inference, as this parameter may remain locally. To obtain learning-theoretic guarantees, we use a variant of Algorithm 1 in which the DP algorithm is an SGD procedure (Bassily et al., 2019, Algorithm 1) that adds a properly scaled Gaussian noise vector at each iteration.
+
+Algorithm 1: Online version of our $( \varepsilon , \delta )$ -meta-private parameter-transfer algorithm.
+
+Meta-learner picks first meta-initialization $\phi _ { 1 } \in \Theta$ . for task $t \in [ T ]$ do
+
+Meta-learner sends meta-initialization $\phi _ { t }$ to task $t$ .   
+Task-learner runs OGD starting from $\theta _ { t , 1 } = \phi _ { t }$ on losses $\{ \ell _ { t , i } \} _ { i = 1 } ^ { m }$ to obtain $\widehat { \theta } _ { t }$ .   
+Task-learner $t$ runs $( \varepsilon , \delta )$ -DP algorithm (noisy-SGD) on losses $\{ \ell _ { t , i } \} _ { i = 1 } ^ { m }$ to get ${ \bar { \theta } } _ { t }$ .   
+Task-learner sends ${ \bar { \theta } } _ { t }$ to meta-learner.   
+Meta-learner constructs loss $\begin{array} { r } { \ell _ { t } ( \phi ) = \frac 1 2 \| \bar { \theta } _ { t } - \phi _ { t } \| _ { 2 } ^ { 2 } } \end{array}$ .   
+Meta-learner updates meta-initialization $\phi _ { t + 1 }$ using an OCO algorithm on $\ell _ { 1 } , \ldots , \ell _ { t }$ .
+
+Result: Meta-initialization $\begin{array} { r } { { \hat { \phi } } = { \frac { 1 } { T } } \sum _ { t = 1 } ^ { T } \phi _ { t } } \end{array}$ to use on test tasks.
+
+# 3.2 PRIVACY GUARANTEES
+
+We run a certified $( \varepsilon , \delta )$ -DP version of SGD (Bassily et al., 2019, Algorithm 1) within each task. Therefore, this guarantees that the contribution of each task-owner, a $\bar { \theta } _ { t }$ trained on their data, carries global DP guarantees with respect to the meta-learner. Additionally, since DP is preserved under post-processing, the release of any future calculation stemming from ${ \bar { \theta } } _ { t }$ also carries the same DP guarantee.
+
+# 3.3 LEARNING GUARANTEES
+
+Our learning result follows the setup of Baxter (2000), who formalized the LTL problem as using task-distribution samples $\mathcal { P } _ { 1 } , . . . , \mathcal { P } _ { T } \sim \mathcal { Q }$ from some meta-distribution $\mathcal { Q }$ and samples indexed by $i = 1 , \ldots , m$ from those tasks to improve performance when a new task $\mathcal { P }$ is sampled from $\mathcal { Q }$ and we draw $m$ samples from it. In the setting of parameter-transfer meta-learning we are learning functions parameterized by real-valued vectors $\overline { { \theta } } \in \dot { \Theta } \subset \mathbb { R } ^ { d }$ , so our goal will follow that of Denevi et al. (2019) and Khodak et al. (2019b) in seeking bounds on the transfer-risk – the distributional performance of a learned parameter on a new task from $\mathcal { Q }$ – that improve with task similarity.
+
+The specific task-similarity metric we consider is the average deviation of the risk-minimizing parameters of tasks sampled from the distribution $\mathcal { Q }$ are close together. This will be measured in-terms of the following quantity: $\begin{array} { r } { V ^ { 2 } = \operatorname* { m i n } _ { \phi \in \Theta } \frac { 1 } { 2 } \mathbb { E } _ { \mathcal { P } \sim \mathcal { Q } } \| \theta _ { \mathcal { P } } - \phi \| _ { 2 } ^ { 2 } } \end{array}$ , for $\theta _ { P } \in \arg \operatorname* { m i n } _ { \theta \in \Theta } \ell _ { \mathcal { P } } ( \theta )$ a risk-minimizer of task-distribution $\mathcal { P }$ . This quantity is roughly the variance of risk-minimizing task-parameters and is a standard quantifier of improvement due to meta-learning (Denevi et al., 2019; Khodak et al., 2019b). For example, Denevi et al. (2019) show excess transfer-risk guarantees of the form $\begin{array} { r } { \mathcal { O } \left( \frac { V } { \sqrt { m } } + \sqrt { \frac { \log T } { T } } \right) } \end{array}$ when $T$ tasks with $m$ samples are drawn from the distribution. This guarantee ensures that as we see more tasks our transfer risk becomes roughly $V / \sqrt { m }$ , which if the tasks are similar, i.e. $V$ is small, implies that LTL improves over single-task learning.
+
+In Algorithm 1, each user $t$ obtains a within-task parameter ${ \bar { \theta } } _ { t }$ by running (non-private) OGD on a sequence of losses $\ell _ { t , 1 } , \ldots , \ell _ { t , m }$ and averaging the iterates. The regret of this procedure, when averaged across the users, implies a bound on the expected excess transfer risk of new task from $\mathcal { Q }$ when running OGD from a learned initialization (Cesa-Bianchi et al., 2004). Thus our goal is to bound this regret in terms of $V$ ; here we follow the Average Regret-Upper-Bound Analysis (ARUBA) framework of Khodak et al. (2019b) and treat meta-update procedure itself as an online algorithm optimizing a bound on the performance measure (regret) of each within-task algorithm. As OGD’s regret depends on the squared distance $\frac { 1 } { 2 } \| \theta _ { t } ^ { * } - \phi _ { t } \| _ { 2 } ^ { 2 }$ of the optimal parameter from the initialization $\phi _ { t }$ , with no privacy concerns one could simply update $\phi _ { t }$ using $\begin{array} { r } { \theta _ { t } ^ { * } \in \arg \operatorname* { m i n } _ { \theta \in \Theta } \sum _ { i = 1 } ^ { m } \ell _ { t , i } ( \theta ) } \end{array}$ to recover guarantees similar to those in Denevi et al. (2019) and Khodak et al. (2019b).
+
+However, this approach requires sending $\theta _ { t } ^ { * }$ to the meta-learner, which is not private; instead in Algorithm 1 we send $\widehat { \theta } _ { t }$ , which is the output of noisy SGD. To apply ARUBA, we need an additional assumption – that the losses satisfy the following quadratic growth (QG) property: for some $\alpha > 0$ ,
+
+$$
+\frac { \alpha } { 2 } \| \theta - \theta _ { \mathcal { P } } \| _ { 2 } ^ { 2 } \le \ell _ { \mathcal { P } } ( \theta ) - \ell _ { \mathcal { P } } ( \theta _ { \mathcal { P } } ) \quad \forall \theta \in \Theta
+$$
+
+Here $\theta _ { \mathcal { P } }$ is the risk minimizer of $\ell _ { \mathcal { P } }$ . This assumption, which Khodak et al. (2019a) shows is reasonable in settings such as logistic regression, amounts to a statistical non-degeneracy assumption on the parameter-space – that parameters far away from the risk-minimizer do not have low-risk. Note that assuming the population risk is QG is significantly weaker than assuming strong convexity of the empirical risk, which previous work (Finn et al., 2019) has assumed to hold for task losses but does not hold for applicable cases such as few-shot least-squares or logistic regression if the number of task-samples is smaller than the data-dimension.
+
+We are now able to state our main theoretical result, a proof of which is given in Appendix A. The result follows from a bound on the task-averaged regret across all tasks of a simple online meta-learning procedure that treats the update ${ \bar { \theta } } _ { t }$ sent by each task as an approximation of the optimal parameter in hindsight $\theta _ { t } ^ { * }$ . Since this parameter determines regret on that task, by reducing the metaupdate procedure to OCO on this sequence of functions in a manner similar to (Khodak et al., 2019a), we are able to show a task-similarity-dependent bound. Following this the statistical guarantee stems from a nested online-to-batch conversion, a standard procedure to convert low-regret online-learning algorithms to low-risk distribution-learning algorithms (Cesa-Bianchi et al., 2004).
+
+Theorem 3.1. Suppose $\mathcal { Q }$ is a distribution over task-distributions $\mathcal { P }$ over $G$ -Lipschtz, $\beta$ -stronglysmooth, $^ { l }$ -bounded convex loss functions $\ell : \Theta \mapsto \mathbb { R }$ over parameter space $\Theta$ with diameter $D$ for $\begin{array} { r } { \beta \leq \frac { G } { D } \operatorname* { m i n } \left( \sqrt { \frac { m } { 2 } } , \frac { \varepsilon m } { 2 \sqrt { 2 d \log \frac { 1 } { \delta } } } \right) } \end{array}$ and let each $\mathcal { P }$ satisfy the quadratic growth property (1). Suppose the distribution $\mathcal { P } _ { t }$ of each task is sampled i.i.d. from $\mathcal { Q }$ and we run Algorithm 1 with the $( \varepsilon , \delta )$ -DP procedure of Bassily et al. (2019, Algorithm $I$ ) to obtain ${ \bar { \theta } } _ { t }$ as the average iterate for the meta-update step, using $n \geq 1$ steps and learning rate $\frac { \gamma } { G \sqrt { n } } f o r \gamma > 0$ . Letting $\begin{array} { r } { V ^ { 2 } = \operatorname* { m i n } _ { \phi \in \Theta } \frac { 1 } { 2 } \mathbb { E } _ { \mathcal { P } \sim \mathcal { Q } } \| \theta _ { \mathcal { P } } - \phi \| _ { 2 } ^ { 2 } } \end{array}$ , there exist settings of $n , \gamma , \eta$ such that we have the following bound on the expected transfer risk when a new task $\mathcal { P }$ is sampled from $\mathcal { Q }$ , m samples are drawn i.i.d. from $\mathcal { P }$ , and we run $O G D$ with learning rate $\eta$ starting from $\begin{array} { r } { { \hat { \phi } } = { \frac { 1 } { T } } \sum _ { t = 1 } ^ { T } \phi _ { t } } \end{array}$ and use the average $\hat { \theta }$ of the resulting iterates as the learned parameter:
+
+$$
+\mathbb { E } \underset { \mathcal { P } \sim \mathcal { Q } \ : \ell \sim \mathcal { P } } { \mathbb { E } } \ell ( \hat { \theta } ) \leq \underset { \mathcal { P } \sim \mathcal { Q } \ : \ell \sim \mathcal { P } } { \mathbb { E } } \ell ( \theta ^ { * } ) + \tilde { \mathcal { O } } \left( \frac { V } { \sqrt { m } } + \frac { \alpha D ^ { 2 } } { T } + \frac { 1 } { \alpha } \operatorname* { m a x } \left( \frac { d \log \frac { 1 } { \delta } } { \varepsilon ^ { 2 } m ^ { 2 } } , \frac { 1 } { m } \right) \right)
+$$
+
+Here $\theta ^ { * }$ is any element of $\Theta$ and the outer expectation is taken over $\ell _ { t , i } \sim \mathcal { P } _ { t } \sim \mathcal { Q }$ and the randomness of the within-task $D P$ mechanism. Note that this procedure is $( \varepsilon , \delta )$ -DP.
+
+Theorem 3.1 shows that one can usefully run a DP-algorithm as the within-task method in meta-√ learning and still obtain improvement due to task-similarity. Specifically, the standard term of $1 / \sqrt { m }$ is multiplied by $V$ , which is small if the tasks are related via the closeness of their risk minimizers. Thus we can use meta-learning to improve within-task performance relative to single-task learning. We also obtain a very fast convergence of $\tilde { \mathcal { O } } ( 1 / T )$ in the number of tasks. However, we do gain some $O ( 1 / m )$ terms due to the quadratic growth approximation and the privacy mechanism. Note that the assumption that both the functions and its gradients are Lipschitz-continuous are standard and required by the noisy SGD procedure of Bassily et al. (2019).
+
+This theorem also gives us a relatively straightforward extension if the desire is to provide $( \varepsilon , \delta )$ - group-DP. Since any privacy mechanism that provides $( \varepsilon , \delta )$ -DP also provides $( k \varepsilon , k e ^ { ( k - 1 ) \epsilon } \delta )$ -DP guarantees for groups of size $k$ (Dwork and Roth, 2014), we immediately have the following corollary.
+
+Corollary 3.1. Under the same assumptions and setting as Theorem 3.1, achieving $( \varepsilon , \delta )$ -group $D P$ is possible with the same guarantee except replacing $\frac { d \log { \frac { 1 } { \delta } } } { \varepsilon ^ { 2 } }$ wit h k 3 d $\begin{array} { r } { \frac { k ^ { 3 } d } { \varepsilon } + \frac { k ^ { 2 } d } { \varepsilon } \left[ \frac { 1 } { \varepsilon } \log \frac { k } { \delta } - 1 \right] } \end{array}$
+
+For constant $k$ , this allows us to enjoy the stronger guarantee while maintaining largely the same learning rates. This is a useful result given that in some settings, it may be desired to simultaneously protect small groups of size $k < < m _ { t }$ , such as protecting entire families for hospital records.
+
+# 4 EMPIRICAL RESULTS
+
+We present results that show it is possible to learn useful deep models in federated scenarios while still preserving privacy against all other participants. Specifically, we evaluate the performance of models that have been trained with a task-global DP algorithm in comparison to models that have been trained both non-privately and with local DP algorithms. We evaluate performance on federated language modeling and few-shot image classification, applying a practical batched variant of Algorithm 1.
+
+![](images/6bf7b551960969d39cb4270213f258dac4ce3fbf1628fb1f4cb2c1861206e1cf.jpg)  
+Figure 2: Performance of different versions of Reptile on a next-word-prediction task for two federated datasets. We report the test accuracy on unseen tasks and repeat each experiment 10 times. Solid lines correspond to means, colored bands indicate 1 standard deviation, and dotted lines are for comparing final accuracies (private algorithms can only be trained until privacy budget is met).
+
+Datasets: We train a LSTM-RNN for next word prediction on two federated datasets: (1) The Shakespeare dataset as preprocessed by (Caldas et al., 2018), and (2) a dataset constructed from 3, 000 Wikipedia articles drawn from the Wiki-3029 dataset (Arora et al., 2019), where each article is used as a different task. For each dataset, we set a fixed number of tokens per task, discard tasks with fewer tokens than the specified, and discard samples from those tasks with more. We set the number of tokens per task to 800 for Shakespeare and to 1, 600 for Wikipedia, divide tokens into sequences of length 10, and we refer to these modified datasets as Shakespeare-800 and Wiki-1600.
+
+For few-shot image classification, we use the Omniglot (Lake et al., 2011) and Mini-ImageNet (Ravi and Larochelle, 2017) datasets, both with 5-shot-5-way test tasks. As has been done for non-private Reptile (Nichol et al., 2018), we use more training shots at meta-training (trying $m = 1 0 , 2 0 , 3 0$ for Omniglot and $m = 1 5$ , 30, 45 for Mini-ImageNet) than at meta-test time. Though tasks could be sampled indefinitely, we set a fixed budget of tasks at $T = 1 0 ^ { 6 }$ to reflect to a setting in which a finite number of training tasks constrains our learning and privacy trade-off. For both local and task-global DP, the more tasks that can be grouped in a meta-batch means that less total noise can be added at each round. However, this also means fewer iterations can be taken. We note that for the smallest values of $m$ , this setting is enough for non-private training to essentially achieve the reported final accuracies from Nichol et al. (2018).
+
+Meta Learning Algorithm. We study the performance of our method when applied to the batched version of Reptile (Nichol et al., 2018) (which, in our setup, reduces to personalized Federated Averaging when the meta-learning rate is set to 1.0). For the language modelling tasks, we tune various configurations of task batch size for all methods. We also allow for multiple visits per client, though at the cost of more added noise per iteration for the private methods. Additionally, for language modeling, we implement gradient clipping and exponential decay on the meta learning rate. For Omniglot and Mini-ImageNet, we use largely the same parameters as Nichol et al. (2018) but we tune the parameters most directly related to privacy: the $L _ { 2 }$ clipping threshold, the Adam Learning Rate at meta-training time, the meta-batch size, and the within-task batch size. We defer a more complete discussion of hyperparameter tuning to Appendix B.
+
+![](images/980bb10fd82738ffbdd39b856bb0072baf7ab0596be743b1d9753fed73b6ca3d.jpg)  
+Figure 3: Performance of task-global DP Reptile on 5-shot-5-way Omniglot and Mini-ImageNet. $1 0 ^ { \overline { { 5 } } }$ sampled test- tasks were used for evaluation and experiments were repeated 3 times. We do not show a line for Local DP since all hyperparameter settings tried for Local DP resulted in worse performance than the “No Meta-Learning" baseline, whose performance can always be recovered.
+
+Privacy Considerations. For the task-global DP models, we set on each task and we implement DP-SGD (for language model $\delta = 1 0 ^ { - 3 } < \frac { 1 } { m ^ { 1 . 1 } }$ by conventionm (for imageclassification) within-task using the tools provided by TensorFlow Privacy1, using the RDP accountant to track our privacy budgets. Although these algorithms differ from the one presented in Section 3, they still let us realistically explore the efficacy of considering task-global privacy. For the language modeling datasets, we try three different privacy budgets (as determined relative to each other by successively doubling the amount of noise added when the goal is to take 1 full gradient step per task) and make sure that all training tasks are sampled without replacement with a fixed batch size until all are seen. This is necessary since multiple visits to a single client results in degradation of the privacy guarantee for that client. We instead aim to provide the same guarantee for each client. For local-DP, though this notion of DP is stronger, we explore the same privacy budgets so as to obtain guarantees that are of the same confidence. Here, we essentially run the DP-FedAvg algorithm from (McMahan et al., 2018) with some key changes. First, to get local DP instead of global, we add Gaussian noise to each clipped set of model updates before returning them to the central server instead of after aggregation. Second, while additional gradient steps within-task do not increase the amount of noise required, we do again iterate through tasks without replacement. Unlike for global DP, we cannot hope to have any privacy boosts due to sub-sampling if the meta-learner knows who it is communicating with.
+
+Results. Figure 2 shows the performance of both the non-private and task-global private versions of Reptile (Nichol et al., 2018) for the language modelling tasks across three different privacy budgets. As expected, neither private algorithm reaches the same accuracy of the non-private version of the algorithm. Nonetheless, the task-global version still comes within $7 8 \%$ , $8 8 \%$ , and $9 2 \%$ of the nonprivate accuracy for Shakespeare-800 and within $7 2 \%$ , $8 2 \%$ , and $8 3 \%$ for Wiki-1600. Meanwhile achieving local DP results in only about $5 5 \%$ and $5 0 \%$ of the non-private accuracy on both datasets for the most generous privacy budget. In practice, these differences can be toggled by further changing the privacy budget or continuing to trade off more training iterations for larger noise multipliers.
+
+We display results for few-shot image classification on Omniglot and Mini-ImageNet in Figure 3. In this setting, not applying meta-learning results in meta-test accuracies of around $6 2 \%$ and $3 6 \%$ , respectively. Thus, while performance is indeed lower than non-private learning, applying task-global DP does result in meta-learning benefits for test-time tasks. In settings where privacy is a concern, this increase in performance is still significantly advantageous for the “task-owners”– test-time tasks (who hold less data). On average, they are able to obtain better models and are still guaranteed privacy at a single-digit $\varepsilon$ . Intuitively, larger training-task datasets make it easier to apply privacy within-task, and in accordance with our learning guarantees, adding training shots indeed closes the gap in performance between task-global DP Reptile and non-private Reptile. In comparison, applying local-DP for a similar hyperparameter range consistently decreases performance at test-time. However, the no-meta-learning baseline is a theoretical lower bound for local-DP, as one could set the clipping threshold or meta-learning rate close to 0 to recover the effects of no meta-learning.
+
+# 5 CONCLUSIONS
+
+In this work, we have outlined and studied the issue of privacy in the context of meta-learning. Focusing on the class of gradient-based parameter-transfer methods, we used differential privacy to address the privacy risks posed to task-owners by sharing task-specific models with a central meta-learner. To do so, we formalized and considered the notion of task-global differential privacy, which guarantees that individual examples from the tasks are protected from all downstream agents (and particularly the meta-learner). Working in this privacy model, we developed a differentially private algorithm that guarantees both this protection as well as learning-theoretic results in the convex setting. Finally, we demonstrate how this notion of privacy can translate into useful deep learning models for non-convex language modelling and image-classification tasks.
+
+# ACKNOWLEDGMENTS
+
+This work was supported in part by DARPA FA875017C0141, the National Science Foundation grants IIS1618714, IIS1705121, and IIS1838017, an Okawa Grant, a Google Faculty Award, an Amazon Web Services Award, a JP Morgan A.I. Research Faculty Award, and a Carnegie Bosch Institute Research Award. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of DARPA, the National Science Foundation, or any other funding agency.
+
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+
+# A PROOFS OF LEARNING GUARANTEES
+
+Setting A.1. We assume that at each time-step t an adversary chooses a task-distribution $\mathcal { P } _ { t }$ over loss-functions on $\Theta \subset \mathbb { R } ^ { d }$ and samples m loss functions $\ell _ { t , i }$ for $i \in [ m ]$ . At each time-step $t$ the tasklearner receives a parameter $\phi _ { t }$ from the meta-learner, runs online gradient descent with step-size $\eta > 0$ starting from $\phi _ { t }$ , and uses the average iterate $\widehat { \theta } _ { t }$ as its learned parameter. The task-learner also runs Algorithm $I$ of Bassily et al. (2019) for $\begin{array} { r } { n = \operatorname* { m i n } \left\{ \frac { m } { 8 } , \frac { \varepsilon ^ { 2 } m ^ { 2 } } { 3 2 d \log \frac { 1 } { \delta } } \right\} } \end{array}$ steps with learning rate $\frac { \gamma } { G \sqrt { n } } > 0$ on these loss functions and sends the result ${ \bar { \theta } } _ { t }$ to the meta-learner. The meta-learner updates $\phi _ { t + 1 } = ( 1 - 1 / t ) \phi _ { t } + \bar { \theta } _ { t } / t$ . We assume all loss functions are $G$ -Lipschitz w.r.t. $\| \cdot \| _ { 2 }$ and $\beta$ -strongly-smooth w.r.t. $\| \cdot \| _ { 2 }$ for some $\begin{array} { r } { \beta \leq \frac { G } { D } \operatorname* { m i n } \left\{ \sqrt { \frac { m } { 2 } } , \frac { \varepsilon n } { 2 \sqrt { 2 d \log \frac { 1 } { \delta } } } \right\} } \end{array}$ , where $D$ is the diameter of $\Theta$ . For each distribution $\mathcal { P } _ { t }$ let $\ell _ { t } ( \theta ) = \mathbb { E } _ { \ell \sim \mathcal { P } _ { t } } ( \theta )$ be its population risk, $\begin{array} { r } { \hat { \ell } _ { t } ( \theta ) = \frac { 1 } { m } \sum _ { i = 1 } ^ { m } \ell _ { t , i } ( \theta ) b e } \end{array}$ its empirical risk, and $\theta _ { t } ^ { * } \in \arg \operatorname* { m i n } _ { \theta \in \Theta } \ell _ { t } ( \theta )$ be the closest population risk minimizer to ${ \bar { \theta } } _ { t }$ .
+
+Lemma A.1. In Setting A.1 we have
+
+$$
+\mathbb { E } \ell _ { t } ( \hat { \theta } _ { t } ) - \ell _ { t } ( \theta _ { t } ^ { * } ) \leq 5 G \left( \frac { \| \phi _ { t } - \theta _ { t } ^ { * } \| _ { 2 } ^ { 2 } } { \gamma } + \gamma \right) \operatorname* { m a x } \left\{ \frac { \sqrt { d \log \frac { 1 } { \delta } } } { \varepsilon m } , \frac { 1 } { \sqrt { m } } \right\}
+$$
+
+Proof. Similarly to Lemma 3.3 in Bassily et al. (2019), applying standard OGD analysis (e.g. Lemmas 14.1 and 14.9 of Shalev-Shwartz and Ben-David (2014)) to noisy gradient vectors and taking expectations yields
+
+$$
+\mathbb { E } \left( \hat { \ell } _ { t } ( \hat { \theta } _ { t } ) - \hat { \ell } _ { t } ( \theta _ { t } ^ { * } ) \right) \leq \frac { \| \phi _ { t } - \theta _ { t } ^ { * } \| _ { 2 } ^ { 2 } } { 2 \eta n } + \frac { \eta G ^ { 2 } } { 2 } + \eta \sigma ^ { 2 } d
+$$
+
+where $n$ is the number of steps in noisy SGD and $\sigma ^ { 2 }$ is the variance of the noise added at each step. As in the proof of Theorem 3.2 of Bassily et al. (2019), substituting $\begin{array} { r } { \sigma ^ { 2 } = \frac { 8 n G ^ { 2 } \log \frac { 1 } { \delta } } { m ^ { 2 } \varepsilon ^ { 2 } } } \end{array}$ and applying the stability result in Lemma 3.4 of the same paper yields
+
+$$
+\mathbb { E } \ell _ { t } ( \hat { \theta } _ { t } ) - \ell _ { t } ( \theta _ { t } ^ { * } ) \leq \frac { \| \phi _ { t } - \theta _ { t } ^ { * } \| _ { 2 } ^ { 2 } } { 2 \eta n } + \frac { \eta G ^ { 2 } } { 2 } \left( \frac { 1 6 n d \log \frac { 1 } { \delta } } { m ^ { 2 } \varepsilon ^ { 2 } } + 1 \right) + \frac { \eta G ^ { 2 } n } { m }
+$$
+
+Substituting $\begin{array} { r } { n = \operatorname* { m i n } \left\{ \frac { m } { 8 } , \frac { \varepsilon ^ { 2 } m ^ { 2 } } { 3 2 d \log \frac { 1 } { \delta } } \right\} } \end{array}$ and $\begin{array} { r } { \eta = \frac { \gamma } { G \sqrt { n } } } \end{array}$ yields the result.
+
+Lemma A.2. In Setting A.1, fix some $\phi ^ { * } \in \Theta$ and define $\begin{array} { r } { \bar { V } ^ { 2 } = \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \mathbb { E } \| \phi ^ { * } - \theta _ { t } ^ { * } \| _ { 2 } ^ { 2 } } \end{array}$ . Then for $\begin{array} { r } { \gamma = \frac { 1 2 0 G } { \alpha } \operatorname* { m a x } \left\{ \frac { \sqrt { d \log \frac { 1 } { \delta } } } { \varepsilon m } , \frac { 1 } { \sqrt { m } } \right\} } \end{array}$ we have
+
+$$
+\mathbb { E } \sum _ { t = 1 } ^ { T } \frac { \| \phi _ { t } - \theta _ { t } ^ { * } \| _ { 2 } ^ { 2 } } { 2 \eta m } \leq \frac { D ^ { 2 } ( 1 + \log T ) + 4 \hat { V } ^ { 2 } T } { 2 \eta m } + \frac { 7 2 0 0 G ^ { 2 } } { \alpha ^ { 2 } \eta m } \operatorname* { m a x } \left\{ \frac { d \log \frac { 1 } { \delta } } { \varepsilon ^ { 2 } m ^ { 2 } } , \frac { 1 } { m } \right\} T
+$$
+
+Proof. We first bound the left-hand side without the denominator as
+
+$$
+\begin{array} { r l } { \varepsilon _ { \perp } \ge \frac { \varepsilon _ { \perp } } { \varepsilon _ { \perp } } \ge \frac { 1 } { \varepsilon _ { \perp } } \ge } & { \varepsilon _ { \perp } } \\ & { = 1 } \\ & { \le 2 \kappa \sum _ { i = 1 } ^ { \nu } \| \partial _ { i } - \partial _ { i } \xi \| ^ { 2 } + \| \partial _ { i } - \theta _ { i } \| ^ { 2 } } \\ & { \le D ^ { \perp } ( 1 - \log { \varepsilon _ { \perp } } ) + 2 \sum _ { i = 1 } ^ { \nu } \| \phi - \varepsilon _ { \perp } \| ^ { 2 } + \| \partial _ { i } - \partial _ { i } \xi \| ^ { 2 } } \\ & { \le D ^ { \perp } ( 1 - \log { \varepsilon _ { \perp } } ) + 2 \sum _ { i = 1 } ^ { \nu } \| \phi - \varepsilon _ { \perp } \| ^ { 2 } + \| \partial _ { i } - \partial _ { i } \xi \| ^ { 2 } } \\ & { \le p ^ { 2 } \varepsilon _ { \perp } ^ { 2 } + 1 - \log { \varepsilon _ { \perp } } ) + 2 \sum _ { i = 1 } ^ { \nu } 2 \| \phi - \varepsilon _ { \perp } \| ^ { 2 } + \| \partial _ { i } - \partial _ { i } \xi \| ^ { 2 } } \\ & { = p ^ { 2 } \varepsilon _ { \perp } ^ { 2 } ( 1 - \log { \varepsilon _ { \perp } } ) + 4 \kappa ^ { 2 } T ^ { 2 } + 4 \| \partial _ { i } ^ { 2 } \sum _ { i = 1 } ^ { \nu } \| \hat { \varepsilon } _ { i } - \partial _ { i } \xi \| ^ { 2 } } \\ & { = D ^ { \perp } ( 1 - \log { \varepsilon _ { \perp } } ) + 4 \kappa ^ { 2 } T ^ { 2 } + 4 \| \partial _ { i } ^ { 2 } \sum _ { i = 1 } ^ { \nu } 2 \hat { \varepsilon } _ { \perp } \phi _ { i } ^ { 2 } - \varepsilon _ { \perp } ( | \partial _ { i } \xi | ^ { 2 } ) } \\ & { \le D ^ { \perp } ( 1 - \log { \varepsilon _ { \perp } } ) + 4 \kappa ^ { 2 } T ^ { 2 } + \frac { 1 } { \omega _ { \perp } } \sum _ { i = 1 } ^ { \nu } 2 \hat { \varepsilon } _ { \perp } ( \rho _ { i } ) - 6 \hat { \varepsilon } ( | \hat { \varepsilon } _ { \perp } - \hat { \varepsilon } _ { \perp } | ^ { 2 } ) } \\ &  \le D ^ { \perp } ( 1 - \ \end{array}
+$$
+
+where in the last step we applied Lemma A.1. Substituting $\begin{array} { r } { \gamma = \frac { 1 2 0 G } { \alpha } \operatorname* { m a x } \left\{ \frac { \sqrt { d \log \frac { 1 } { \delta } } } { \varepsilon m } , \frac { 1 } { \sqrt { m } } \right\} } \end{array}$ yields
+
+$$
+\mathbb { E } \sum _ { t = 1 } ^ { T } \| \phi _ { t } - \theta _ { t } ^ { * } \| _ { 2 } ^ { 2 } \leq D ^ { 2 } ( 1 + \log T ) + 4 { \bar { V } } ^ { 2 } T + \frac { 1 4 4 0 0 G ^ { 2 } } { \alpha ^ { 2 } } \operatorname* { m a x } \left\{ \frac { d \log \frac { 1 } { \delta } } { \varepsilon ^ { 2 } m ^ { 2 } } , \frac { 1 } { m } \right\} T
+$$
+
+The result follows by dividing by 2ηm.
+
+Theorem A.1. In Setting A.1, suppose all distributions $\mathcal { P } _ { t }$ were drawn i.i.d. from some metadistribution $\mathcal { Q }$ and we used $\begin{array} { r } { \gamma = \frac { 1 2 0 G } { \alpha } \operatorname* { m a x } \left\{ \frac { \sqrt { d \log \frac { 1 } { \delta } } } { \varepsilon m } , \frac { 1 } { G \sqrt { m } } \right\} } \end{array}$ . Suppose we draw another taskdistribution $\mathcal { P } \sim \mathcal { Q }$ with population risk $\ell _ { \mathcal { P } }$ and minimizer $\theta _ { \mathcal { P } }$ , set $\begin{array} { r } { \hat { \phi } = \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \phi _ { t } } \end{array}$ , and run OGD with learning rate $\begin{array} { r } { \eta = \frac { V + \frac { 1 } { \alpha \sqrt { m } } } { \sqrt { m } } } \end{array}$ starting from $\hat { \phi }$ on m samples from $\mathcal { P }$ . Then the average iterate $\hat { \theta }$ satisfies
+
+$$
+\mathbb { E } ( \ell _ { \mathcal { P } } ( \hat { \theta } ) - \ell _ { \mathcal { P } } ( \theta ^ { * } ) ) \le \frac { 7 G V } { 2 \sqrt { m } } + \frac { 7 2 0 1 G } { \alpha } \operatorname* { m a x } \left\{ \frac { d \log \frac { 1 } { \delta } } { \varepsilon ^ { 2 } m ^ { 2 } } , \frac { 1 } { m } \right\} + \frac { \alpha G D ^ { 2 } } { 2 T } ( 1 + \log T )
+$$
+
+for $\begin{array} { r } { V ^ { 2 } = \operatorname* { m i n } _ { \phi \in \Theta } \mathbb { E } _ { \mathcal { P } \sim \mathcal { Q } } \operatorname* { m a x } _ { \theta _ { \mathcal { P } } } \| \phi - \theta _ { \mathcal { P } } \| _ { 2 } ^ { 2 } . } \end{array}$
+
+Proof. Applying online-to-batch conversion (e.g. Proposition A.1 in Khodak et al. (2019b)) twice and substituting Lemma A.2 yields
+
+$$
+\begin{array} { r l } & { \mathbb { E } ( \ell _ { \mathcal { P } } ( \hat { \theta } ) - \ell _ { \mathcal { P } } ( \theta ^ { * } ) ) } \\ & { \qquad \le \mathbb { E } \frac { \| \hat { \phi } - \theta ^ { * } \| _ { 2 } ^ { 2 } } { 2 \eta m } + \eta G ^ { 2 } } \\ & { \qquad \le \mathbb { E } \frac { \| \hat { \phi } ^ { * } - \theta ^ { * } \| _ { 2 } ^ { 2 } } { 2 \eta m } + \eta G ^ { 2 } + \frac { 1 } { 2 \eta m T } \displaystyle \sum _ { t = 1 } ^ { T } \mathbb { E } \| \phi _ { t } - \theta _ { t } ^ { * } \| _ { 2 } ^ { 2 } } \\ & { \qquad \le \mathbb { E } \frac { \| \hat { \phi } ^ { * } - \theta ^ { * } \| _ { 2 } ^ { 2 } } { 2 \eta m } + \eta G ^ { 2 } + \frac { D ^ { 2 } \frac { 1 + \log T } { T } } { 2 \eta m } + 4 \mathbb { E } \bar { V } ^ { 2 } + \frac { 7 2 0 0 G ^ { 2 } } { \alpha ^ { 2 } \eta m } \operatorname* { m a x } \left\{ \frac { d \log \frac { 1 } { \delta } } { \varepsilon ^ { 2 } m ^ { 2 } } , \frac { 1 } { m } \right\} } \\ & { \qquad = \frac { 5 V ^ { 2 } } { 2 \eta m } + \eta G ^ { 2 } + \frac { 7 2 0 0 G ^ { 2 } } { \alpha ^ { 2 } \eta m } \operatorname* { m a x } \left\{ \frac { d \log \frac { 1 } { \delta } } { \varepsilon ^ { 2 } m ^ { 2 } } , \frac { 1 } { m } \right\} + \frac { D ^ { 2 } } { 2 \eta m T } ( 1 + \log T ) } \end{array}
+$$
+
+where we have applied $\mathbb { E } \bar { V } ^ { 2 } \le V ^ { 2 }$ . Substituting $\begin{array} { r } { \eta = \frac { V + \frac { 1 } { \alpha \sqrt { m } } } { G \sqrt { m } } } \end{array}$ yields the result.
+
+# B EXPERIMENT DETAILS
+
+Datasets: We train a next word predictor for two federated datasets: (1) The Shakespeare dataset as preprocessed by (Caldas et al., 2018), and (2) a dataset constructed from Wikipedia articles, where each article is used as a different task. For each dataset, we set a fixed number of tokens per task, discard tasks with less tokens than the specified, and discard samples from those tasks with more. For Shakespeare, we set the number of tokens per task to 800 tokens, leaving 279 tasks for meta-training, 31 for meta-validation, and 35 for meta-testing. For Wikipedia, we set the number of tokens to 1, 600, which corresponds to having 2, 179 tasks for meta-training, 243 for meta-validation, and 606 for meta-testing. For the meta-validation and meta-test tasks, $7 5 \%$ of the tokens are used for local training, and the remaining $2 5 \%$ for local testing.
+
+For the few-shot image classification experiments, we follow the standard set-up by splitting labels into training and testing and forming training tasks by randomly drawing labels from the training set. At evaluation time, we draw from the test set.
+
+Model Structure: Our model first maps each token to an embedding of dimension 200 before passing it through an LSTM of two layers of 200 units each. The LSTM emits an output embedding, which is scored against all items of the vocabulary via dot product followed by a softmax. We build the vocabulary from the tokens in the meta-training set and fix its length to 10, 000. We use a sequence length of 10 for the LSTM and, just as (McMahan et al., 2018), we evaluate using AccuracyTop1 (i.e., we only consider the predicted word to which the model assigned the highest probability) and consider all predictions of the unknown token as incorrect. For Omniglot and Mini-ImageNet, we use the architectures from Nichol et al. (2018) to also match the ones from Finn et al. (2017). We evaluate in the standard transductive setting.
+
+Hyperparameters: For the language-modeling experiments, we tune the hyperparameters on the set of meta-validation tasks. For both datasets and all versions of the meta-learning algorithm, we tune hyperparameters in a two step process. We first tune all the parameters that are not related to refinement: the meta learning rate, the local (within-task) meta-training learning rate, the maximum gradient norm, and the decay constant. Then, we use the configuration with the best accuracy pre-refinement and then tune the refinement parameters: the refine learning rate, refine batch size, and refine epochs.
+
+All other hyperparameters are kept fixed for the sake of comparison: full batch steps were taken on within-task data, with the maximum number of microbatches used for the task-global DP model. The parameter search spaces from which we sample are given in Tables 2, 3, 4 while Tables 5 and 6 contain our final choices. Note that the space for Local DP, especially in terms of the clipping threshold, is distinctively different from the others, as we did not find that searching through ranges similar to those for non-private and task-global DP led to learning high-quality meta-initializations.
+
+For Omniglot, we largely based our hyperparameters on the choices of Nichol et al. (2018) for 5-way classification. We vary $m$ , the number of training shots, but we continue to take 5 SGD steps of expected size $m$ within task and we leave the test-time SGD procedure exactly the same. However, we do tune for privacy clipping thresholds $\{ 0 . 0 1 , 0 . 0 2 5 , 0 . 0 5 , 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 5 \}$ , Adam Learning Rates for meta-training tasks $\left\{ 1 0 ^ { - 4 } , 5 \times \mathrm { i } 0 ^ { - 4 } , 1 0 ^ { - 3 } , 5 \times 1 0 ^ { - 3 } \right\}$ , and meta-batch sizes of $\{ 5 , 1 5 , \bar { 2 5 } , 5 0 \}$ .
+
+For Mini-ImageNet, we perform a similar search except we also double the inner batch size to $2 m$ (trading off less privacy amplification due to subsampling). We continue to tune for privacy clipping thresholds $\{ 0 . 0 1 , \dot { 0 } . 0 2 \dot { 5 } , 0 . 0 5 , 0 . 1 , 0 . 3 , 0 . 5 , 0 . 7 , 0 . 9 , 1 . 1 , 1 . 3 , 1 . 5 \}$ , Adam Learning Rates $\{ 1 \bar { 0 } ^ { - 4 } , \bar { 5 } \times 1 0 ^ { - 4 } , 1 0 ^ { \dot { - } 3 } , 5 \times 1 0 ^ { - 3 } \}$ , and meta-batch sizes of $\{ 5 , 1 5 , 2 5 , 5 0 \}$ .
+
+Table 2: Hyperparameter Search Space for Non-Private Training   
+
+<table><tr><td colspan="2">Shakespeare-800</td><td>Wiki-1600</td></tr><tr><td>VisitsPerTask</td><td>{1,2,3,4,5,6,7,8,9}</td><td>{1,2,3}</td></tr><tr><td>Tasks Per Round</td><td>{5,10}</td><td>{5,10}</td></tr><tr><td>Within-Task Steps</td><td>{1,3,5,7,9}</td><td>{1,3,5,7,9}</td></tr><tr><td>Meta LR</td><td>{1,√2,2,2√2,4,4√2,8,8√2}</td><td>{1,√2,2,2√2,4,4√2,8,8√2}</td></tr><tr><td rowspan="3">Meta Decay Rate Within-Task LR</td><td>{0,0.001,0.005,0.01,0.025,0.05,0.1}</td><td>{0,0.001,0.005,0.01,0.025,0.05}</td></tr><tr><td>{1,√2,2,2√2,4,4√2,8}</td><td>{1,√2,2,2√2,4,4√2,8}</td></tr><tr><td>{0.4,0.5,0.6,0.7, 0.8, 0.9, 1.0}</td><td>{0.3, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0}</td></tr><tr><td rowspan="2">Refine LR Refine Batch Size</td><td>{0.1, 0.15,0.3,0.5,0.7, 0.8}</td><td>{0.1,0.15, 0.3,0.5, 0.7, 0.8}</td></tr><tr><td>{10,20,30,60}</td><td>{10,20,30,60,120}</td></tr><tr><td>Refine Epochs</td><td>{1,2,3}</td><td>{1,2,3}</td></tr></table>
+
+Table 3: Hyperparameter Search Space for Task-Global DP Training   
+
+<table><tr><td></td><td>Shakespeare-800</td><td>Wiki-1600</td></tr><tr><td>VisitsPerTask</td><td>{1,2,3}</td><td>{1,2}</td></tr><tr><td>TasksPerRound</td><td>{5,10}</td><td>{5,10}</td></tr><tr><td>Within-Task Steps</td><td>1</td><td>1</td></tr><tr><td>MetaLR</td><td>{1,√2,2,2√2,4,4√2,8,8√2}</td><td>{1,√2,2,2√2,4,4√2,8,8√2}</td></tr><tr><td>Meta Decay Rate</td><td>{0,0.001,0.005,0.01, 0.025, 0.05,0.1}</td><td>{0,0.001,0.005,0.01, 0.025, 0.05}</td></tr><tr><td>Within-Task LR</td><td>{1,√2,2√2,4,4√2,8}</td><td>{1,√2,2√2,4,4√2,8}</td></tr><tr><td>L2 Clipping</td><td>{0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0}</td><td>{0.3, 0.4, 0.5, 0.6, 0.7, 0.8,0.9,1.0}</td></tr><tr><td>Refine LR</td><td>{0.1,0.15,0.3,0.5,0.7, 0.8}</td><td>{0.1,0.15,0.3,0.5,0.7, 0.8}</td></tr><tr><td>Refine Batch Size</td><td>{10,20,30,60}</td><td>{10,20,30,60,120}</td></tr><tr><td>Refine Epochs</td><td>{1,2,3}</td><td>{1,2,3}</td></tr></table>
+
+Table 4: Hyperparameter Search Space for Local-DP Training   
+
+<table><tr><td colspan="2">Shakespeare-800</td><td>Wiki-1600</td></tr><tr><td>VisitsPerTask</td><td>{1,2,3}</td><td>{1,2}</td></tr><tr><td>Tasks Per Round</td><td>{5,10,20}</td><td>{10,20,40,80}</td></tr><tr><td>Within-Task Steps</td><td>{1,2,3}</td><td>{1,2,3}</td></tr><tr><td>Meta LR</td><td>{1,√2,2,2√2,4,4√2,8,8√2}</td><td>{1,√2,2,2√2,4,4√2,8,8√2}</td></tr><tr><td>Meta Decay Rate</td><td>{0,0.001,0.005,0.01,0.025,0.05,0.1}</td><td>{0,0.001,0.005,0.01,0.025,0.05}</td></tr><tr><td>Within-Task LR</td><td>{1,√2,2√2,4,4√2,8}</td><td>{1,√2,2√2,4,4√2,8}</td></tr><tr><td>L2 Clipping</td><td>{0.005,0.01,0.025,0.05,0.1,0.25,0.5}</td><td>{0.005,0.01,0.025,0.05,0.1,0.25}</td></tr><tr><td>Refine LR</td><td>{0.1,0.15,0.3,0.5,0.7,0.8}</td><td>{0.1,0.15,0.3,0.5, 0.7, 0.8}</td></tr><tr><td>Refine Batch Size</td><td>{10,20,30,60}</td><td>{10,20,30,60,120}</td></tr><tr><td>Refine Epochs</td><td>{1,2,3}</td><td>{1,2,3}</td></tr></table>
+
+Table 5: Final Hyperparameters for Shakespeare-800   
+
+<table><tr><td></td><td>Non- private</td><td>T-G £= 22.5</td><td>T-G £=9.2</td><td>T-G ε= 4.5</td><td>Local ε= 22.5</td><td>Local £=9.2</td><td>Local ε= 4.5</td></tr><tr><td>Visits Per Task</td><td>7</td><td>2</td><td>2</td><td>1</td><td>2</td><td>2</td><td>1</td></tr><tr><td>Tasks Per Round</td><td>5</td><td>5</td><td>5</td><td>5</td><td>20</td><td>5</td><td>20</td></tr><tr><td>Within-Task Steps</td><td>5</td><td>1</td><td>1</td><td>1</td><td>4</td><td>1</td><td>2</td></tr><tr><td>Meta LR</td><td>8</td><td>8√2</td><td>8</td><td>8√2</td><td>4√2</td><td>4√2</td><td>4</td></tr><tr><td>Meta Decay Rate</td><td>0.01</td><td>0.01</td><td>0.01</td><td>0.05</td><td>0.1</td><td>0</td><td>0</td></tr><tr><td>Within-Task LR</td><td>2</td><td>2√2</td><td>2</td><td>2</td><td>4√2</td><td>4√2</td><td>1</td></tr><tr><td>L2 Clipping</td><td>0.5</td><td>0.6</td><td>0.5</td><td>0.4</td><td>0.1</td><td>0.01</td><td>0.01</td></tr><tr><td>RefineLR</td><td>0.15</td><td>0.5</td><td>0.1</td><td>0.3</td><td>0.8</td><td>0.8</td><td>0.5</td></tr><tr><td>Refine Batch Size</td><td>30</td><td>10</td><td>60</td><td>30</td><td>10</td><td>10</td><td>10</td></tr><tr><td>Refine Epochs</td><td>1</td><td>1</td><td>1</td><td>3</td><td>3</td><td>3</td><td>3</td></tr></table>
+
+Table 6: Final Hyperparameters for Wiki-800   
+
+<table><tr><td></td><td>Non- private</td><td>T-G £= 22.5</td><td>T-G £=9.2</td><td>T-G ε= 4.5</td><td>Local £= 22.5</td><td>Local £=9.2</td><td>Local £= 4.5</td></tr><tr><td>VisitsPerTask</td><td>2</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Tasks Per Round</td><td>5</td><td>10</td><td>10</td><td>20</td><td>20</td><td>20</td><td>20</td></tr><tr><td>Within-Task Steps</td><td>3</td><td>1</td><td>1</td><td>1</td><td>2</td><td>2</td><td>2</td></tr><tr><td>Meta LR</td><td>2√2</td><td>2√2</td><td>4</td><td>4</td><td>8</td><td>4√2</td><td>8</td></tr><tr><td>Meta Decay Rate</td><td>0.001</td><td>0</td><td>0.001</td><td>0.005</td><td>0.005</td><td>0.025</td><td>0</td></tr><tr><td>Within-Task LR</td><td>2</td><td>8</td><td>4√2</td><td>8</td><td>2√2</td><td>2√2</td><td>2√2</td></tr><tr><td>L2 Clipping</td><td>1</td><td>0.8</td><td>0.7</td><td>0.8</td><td>0.025</td><td>0.05</td><td>0.005</td></tr><tr><td>Refine LR</td><td>0.1</td><td>0.8</td><td>0.5</td><td>0.7</td><td>0.8</td><td>0.8</td><td>0.8</td></tr><tr><td>Refine Batch Size</td><td>10</td><td>10</td><td>60</td><td>10</td><td>10</td><td>10</td><td>10</td></tr><tr><td>Refine Epochs</td><td>1</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>3</td></tr></table>

parse/train/rJgqMRVYvr/rJgqMRVYvr_content_list.json

@@ -0,0 +1,1495 @@
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+    {
+        "type": "text",
+        "text": "DIFFERENTIALLY PRIVATE META-LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Jeffrey Li, Mikhail Khodak, Sebastian Caldas   \nCarnegie Mellon University   \njwl3@cs.cmu.edu ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ameet Talwalkar Carnegie Mellon University & Determined AI ",
+        "bbox": [
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+            208,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            273,
+            544,
+            287
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Parameter-transfer is a well-known and versatile approach for meta-learning, with applications including few-shot learning, federated learning, and reinforcement learning. However, parameter-transfer algorithms often require sharing models that have been trained on the samples from specific tasks, thus leaving the task-owners susceptible to breaches of privacy. We conduct the first formal study of privacy in this setting and formalize the notion of task-global differential privacy as a practical relaxation of more commonly studied threat models. We then propose a new differentially private algorithm for gradient-based parameter transfer that not only satisfies this privacy requirement but also retains provable transfer learning guarantees in convex settings. Empirically, we apply our analysis to the problems of federated learning with personalization and few-shot classification, showing that allowing the relaxation to task-global privacy from the more commonly studied notion of local privacy leads to dramatically increased performance in recurrent neural language modeling and image classification. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+            523,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The field of meta-learning offers promising directions for improving the performance and adaptability of machine learning methods. At a high level, the key assumption leveraged by these approaches is that the sharing of knowledge gained from individual learning tasks can help catalyze the learning of similar unseen tasks. However, the collaborative nature of this process, in which task-specific information must be sent to and used by a meta-learner, also introduces inherent data privacy risks. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work, we focus on a popular and flexible meta-learning approach, parameter transfer via gradient-based meta-learning (GBML). This set of methods, which includes well-known algorithms such as MAML (Finn et al., 2017) and Reptile (Nichol et al., 2018), tries to learn a common initialization $\\phi$ over a set of tasks $t = 1 , \\dots , T$ such that a high-performance model can be learned in only a few gradient-steps on new tasks. Notably, information flows constantly between training tasks and the meta-learner as learning progresses; to make iterative updates, the meta-learner obtains feedback on the current $\\phi$ by having task-specific models ${ \\bar { \\theta } } _ { t }$ trained with it. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Meanwhile, in many settings amenable to meta-learning, it is crucial to ensure that sensitive information in each task’s dataset stays private. Examples of this include learning models for word prediction on cell phone data (McMahan et al., 2018), clinical predictions using hospital records (Zhang et al., 2019), and fraud detectors for competing credit card companies (Stolfo et al., 1997). In such cases, each data-owner can benefit from information learned from other tasks, but each also desires, or is legally required, to keep their raw data private. Thus, it is not sufficient to learn a well-performing $\\phi$ ; it is equally imperative to ensure that a task’s sensitive information is not obtainable by anyone else. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "While parameter transfer algorithms can move towards this goal by peforming task-specific optimization locally, thus preventing direct access to private data, this provision is far from fail-safe in terms of privacy. A wealth of work has shown in the single-task setting that it is possible for an adversary with only access to the model to learn detailed information about the training set, such as the presence or absence of specific records (Shokri et al., 2017) or the identities of sensitive features given other covariates (Fredrikson et al., 2015). Furthermore, Carlini et al. (2018) showed that deep neural networks can effectively memorize user-unique training examples, which can be recovered even after only a single epoch of training. As such, in parameter-transfer methods, the meta-learner or any downstream participant can potentially recover data from a previous task. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "However, despite these serious risks, privacy-preserving meta-learning has remained largely an unstudied problem. Our work aims to address this issue by applying differential privacy (DP), a well-established definition of privacy with rich theoretical guarantees and consistent empirical success at preventing leakages of data (Carlini et al., 2018; Fredrikson et al., 2015; Jayaraman and Evans, 2019). Crucially, although there are various threat models and degrees of DP one could consider in the meta-learning setting (as we outline in Section 2), we balance the well-documented trade-off between privacy and model utility by formalizing and focusing on a setting that we call task-global DP. This setting provides a strong privacy guarantee for each task-owner that sharing $\\widehat { \\theta } _ { t }$ with the meta-learner will not reliably reveal anything about specific training examples to any downstream agent. It also allows us to use the framework of Khodak et al. (2019a) to provide a DP GBML algorithm that enjoys provable learning guarantees in convex settings. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Finally, we show an application of our work by drawing connections to federated learning (FL) (Li et al., 2019). While standard methods for FL, such as FedAvg (McMahan et al., 2017), have inspired many works also concerning DP in a multi-user setup (Agarwal et al., 2018; Bhowmick et al., 2019; Geyer et al., 2018; McMahan et al., 2018; Truex et al., 2019), we are the first to consider task-global DP as a useful variation on standard DP settings. Moreover, these works fundamentally differ from ours in that they do not consider a task-based notion of learnability, instead focusing on the global federated learning problem to learn a single global model. That being said, a federated setting involving per-user personalization (Chen et al., 2018; Smith et al., 2017) is a natural meta-learning application. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "More specifically, our main contributions are: ",
+        "bbox": [
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+            446,
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+            462
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1. We are the first to provide a taxonomy for the different notions of DP possible for meta-learning. In particular, we formalize on a variant we call task-global DP, showing and arguing that it adds a useful option to commonly studied settings in terms of trading privacy and accuracy.   \n2. We propose the first DP GBML algorithm, which we construct to satisfy this privacy setting. Further, we show a straightforward extension for obtaining a group $D P$ version of our setting to protect multiple samples simultaneously.   \n3. While our privacy guarantees hold generally, we also prove learning-theoretic results in convex settings. Our learning guarantees scale with task-similarity, as measured by the closeness of the task-specific optimal parameters (Denevi et al., 2019; Khodak et al., 2019b).   \n4. We show that our algorithm, along with its theoretical guarantees, naturally carries over to federated learning with personalization. Compared to previous notions of privacy considered in works for DP federated learning (Agarwal et al., 2018; Bhowmick et al., 2019; Geyer et al., 2018; McMahan et al., 2018; Truex et al., 2019), we are, to the best of our knowledge, the first to simultaneously provide both privacy and learning guarantees.   \n5. Empirically, we demonstrate that our proposed privacy setting allows for strong performance on federated language-modeling and few-shot image classification tasks. For the former, we achieve close to the performance of non-private models and significantly improve upon the performance of models trained with local-DP guarantees, a previously studied notion that also provides protections against the meta-learner. Our setting reasonably relaxes this latter notion but can achieve roughly 1.7–2.3 times the accuracy on a modified version of the Shakespeare dataset (Caldas et al., 2018) and 1.6–1.7 times the accuracy on a modified version of Wiki-3029 (Arora et al., 2019) across various privacy budgets. For image-classification, we show that we show that we can still retain significant benefits of meta-learning while applying task-global DP on Omniglot (Lake et al., 2011) and Mini-ImageNet (Ravi and Larochelle, 2017). ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1.1 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            827,
+            330,
+            842
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "DP Algorithms in Federated Learning Settings. Works most similar to ours focus on providing DP for federated learning. Specifically, Geyer et al. (2018) and McMahan et al. (2018) apply update clipping and the Gaussian Mechanism to achieve user-level global DP federated learning algorithms for language modeling and image classification tasks respectively. Their methods are shown to only suffer minor drops in accuracy compared to non-private training but they do not consider protections to inferences made by the meta-learner. Alternatively, Bhowmick et al. (2019) does achieve such protection by applying a theoretically rate-optimal local DP mechanism on the ${ \\bar { \\theta } } _ { t }$ ’s users send to the meta-learner. However, they sidestep hard minimax rates (Duchi et al., 2018) by assuming the central server has limited side-information and allowing for a large privacy budget. In this work, though we achieve a relaxation of the privacy of Bhowmick et al. (2019), we do not restrict the adversary’s power. Finally, Truex et al. (2019) does consider a setting that coincides with task-global DP, but they focus primarily on the added benefits of applying MPC (see below) rather than studying the merits of the setting in comparison to other potential settings. ",
+        "bbox": [
+            174,
+            854,
+            823,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            215
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Secure Multiparty Computation (MPC). MPC is a cryptographic technique that allows parties to calculate a function of their inputs while also maintaining the privacy of each individual inputs. In GBML, sets of model updates may come in a batch from multiple tasks, and hence MPC can securely aggregate the batch before it is seen by the meta-learner. Though MPC itself gives no DP guarantees against future inference, it can combined with DP to increase privacy. This approach has been studied in the federated setting, e.g. by Agarwal et al. (2018), who apply MPC in the same difficult setting of Bhowmick et al. (2019), and Truex et al. (2019), who apply MPC similarly to a setting analogous to ours. On the other hand, MPC also comes with additional practical challenges such as peer-to-peer communication costs, drop outs, and vulnerability to collaborating participants. As such, combined with its applicability to multiple settings, including ours, we consider MPC to be an orthogonal direction. ",
+        "bbox": [
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+            222,
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+            375
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 PRIVACY IN A META-LEARNING CONTEXT ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            393,
+            560,
+            410
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we first formalize the meta-learning setting that we consider. We then describe the various threat models that arise in the GBML setup, before presenting the different DP notions that can be achieved. Finally, we highlight the specific model and type of DP that we analyze. ",
+        "bbox": [
+            174,
+            425,
+            825,
+            467
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 PARAMETER TRANSFER META-LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            483,
+            508,
+            497
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In parameter transfer meta-learning, we assume that there is a set of learning tasks $t = 1 , \\dots , T$ , each with its corresponding disjoint training set $D _ { t }$ . Each $D _ { t }$ contains $m _ { t }$ training examples $\\{ z _ { t , i } \\} _ { i = 1 } ^ { m _ { t } }$ where each $\\boldsymbol { z } _ { t , i } \\in \\mathcal { X } \\times \\mathcal { Y }$ . The goal within each task is to learn a function $f _ { \\hat { \\theta } _ { t } } : \\mathcal { X } \\xrightarrow { } \\mathcal { V }$ parameterized by $\\widehat { \\theta } _ { t } \\in \\Theta \\subset \\mathbb { R } ^ { d }$ that performs “well,” generally in the sense that it has low within-task population risk in the distributional setting. The meta-learner’s goal is to learn an initialization $\\phi \\in \\Theta$ that leads to a well-performing $\\widehat { \\theta } _ { t }$ within-task. In GBML this $\\phi$ is learned via an iterative process that alternates between the following two steps: (1) a within-task procedure where a batch of task-owners $B$ receives the current $\\phi$ and each $t \\in B$ uses $\\phi$ as an initialization for running a within-task optimization procedure, obtaining $\\bar { \\theta } _ { t } ( D _ { t } , \\phi )$ ; (2) a meta-level procedure where the meta-learner receives these model updates $\\{ { \\bar { \\theta } } _ { t } \\} _ { t \\in B }$ and aggregates them to determine an updated $\\phi$ . Note that we do not assume $\\widehat { \\theta } _ { t } = \\bar { \\theta } _ { t }$ , as the updates shared for the meta-learning procedure can be obtained from a different procedure than the refined model used for downstream within-task inference. This is especially the case when concerning the addition of noise for DP as part of the meta-learning procedure. ",
+        "bbox": [
+            173,
+            510,
+            825,
+            699
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 THREAT MODELS FOR GBML ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            715,
+            421,
+            729
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As in any privacy endeavor, before discussing particular mechanisms, a key specification must be made in terms of what threat model is being considered. In particular, it must be specified both (1) who the potential adversaries are and (2) what information needs to be protected. ",
+        "bbox": [
+            176,
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+            784
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Potential adversaries. For a single task-owner, adversaries may be either solely recipients of $\\phi$ (i.e. other task-owners) or recipients of either $\\phi$ or ${ \\bar { \\theta } } _ { t }$ (i.e. also the meta-learner). In the latter case, we consider only a honest-but-curious meta-learner, who does not deviate from the agreed upon algorithm but may try to make inferences from ${ \\bar { \\theta } } _ { t }$ . In both cases, concern is placed not only about these other participants’ intentions, but also their own security against access by malicious outsiders. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Data to be protected. A system can choose either to protect information contained in single records $z _ { t , i }$ one-at-a-time or to protect entire datasets $D _ { t }$ simultaneously. This distinction between record-level and task-level privacy can be practically important. Multiple $z _ { t , i }$ within $D _ { t }$ may reveal the same secret (e.g., a cell-phone user has sent their SSN multiple times), or the entire distribution of $D _ { t }$ could reveal sensitive information (e.g., a user has sent all messages in a foreign language). In these cases, record-level privacy may not be sufficient. However, given that privacy and utility are often at odds, we often seek the weakest notion of privacy needed in order to best preserve utility. ",
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+        "img_path": "images/1af50ef442ecb2263b0ad5b33eec347594b127d79b6e088e27c64606c511143e.jpg",
+        "image_caption": [
+            "Figure 1: Summary of the privacy protections guaranteed by local and global DP at the different levels of the meta-learning problem (with our notion in blue). On the right, we show what each specification would mean in two practical federated scenarios: mobile users and hospital networks. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "In related work, focus has primarily been placed on task-level protections. However, such approaches usually fall into two extremes, either obtaining strong learning but having to trust the meta-learner (McMahan et al., 2018; Geyer et al., 2018) or trusting nobody but also obtaining low performance (Bhowmick et al., 2019). In response, we try to bridge the gap between these threat models by considering a model that makes a relaxation from task-level to record-level privacy but retains protections for each task-owner against all other parties. This relaxation can be reasonably justified in practical situations, as while task-level guarantees are strictly stronger, they may also be unnecessary. In particular, record-level guarantees are likely to be sufficient whenever single records each pertain to different individuals. For example, for hospitals, what we care about is providing privacy to the individual patients and not aggregate hospital information. For cell-phones, if one can bound the number of texts that could reveal the same sensitive information, then a straightforward extension of our setting and methods, which protects up to $k$ records simultaneously, could also be sufficient. ",
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+    {
+        "type": "text",
+        "text": "2.3 DIFFERENTIAL PRIVACY (DP) IN A SINGLE-TASK SETTING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In terms of actually achieving privacy guarantees for machine learning, a de-facto standard has been to apply DP, a provision which strongly limits what one can infer about the examples a given model was trained on. Assuming a training set $D = \\{ z _ { 1 } , \\dots , z _ { m } \\}$ , two common types of DP are considered. ",
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+    {
+        "type": "text",
+        "text": "Differential Privacy (Global DP). A randomized mechanism $\\mathcal { M }$ is $( \\varepsilon , \\delta )$ -differentially private if for all measurable $S \\subseteq { \\mathrm { R a n g e } } ( { \\mathcal { M } } )$ and for all datasets $D , D ^ { \\prime }$ that differ by at most one element: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/227e41802c0bdbe30922e8f878db78e12e35bd9c16fd24435a8ca7854df7aa52.jpg",
+        "text": "$$\n\\mathbb { P } [ \\mathcal { M } ( D ) \\in \\mathcal { S } ] \\leq e ^ { \\varepsilon } \\mathbb { P } [ \\mathcal { M } ( D ^ { \\prime } ) \\in \\mathcal { S } ] + \\delta\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "If this holds for $D , D ^ { \\prime }$ differing by at most $k$ elements, then $( \\varepsilon , \\delta ) \\ k$ -group $D P$ is achieved. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Local Differential Privacy. A randomized mechanism $\\mathcal { M }$ is $( \\varepsilon , \\delta )$ -locally differentially private if for any two possible training examples $z , z ^ { \\prime } \\in \\mathcal { X } \\times \\mathcal { Y }$ and measurable $S \\subseteq \\mathcal { X } \\times \\mathcal { Y }$ : ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f6d946eaa3c7bd500bb7f46143be83961e7d709d3e5f5fec0ca412f7ce98de74.jpg",
+        "text": "$$\n\\mathbb { P } [ \\mathcal { M } ( z ) \\in \\mathcal { S } ] \\leq e ^ { \\varepsilon } \\mathbb { P } [ \\mathcal { M } ( z ^ { \\prime } ) \\in \\mathcal { S } ] + \\delta\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Global DP guarantees the difficulty of inferring the presence of a specific record in the training set by observing $\\mathcal { M } ( D )$ . It assumes a trusted aggregator running $\\mathcal { M }$ gets direct access to $D$ and privatizes the final output. Meanwhile, local DP assumes more strictly that the aggregator also cannot be trusted, thus requiring a random mechanism to be applied individually on each $z$ before training. However, it generally results in worse model performance, suffering from hard minimax rates (Duchi et al., 2018). ",
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+    {
+        "type": "table",
+        "img_path": "images/22a1b54d29ef402de29aeb9e26c4f0a5f7d5e07fb01932791fe24a5e16045dc5.jpg",
+        "table_caption": [
+            "Table 1: Broad categorization of the DP settings considered by our work in meta-learning and notable past works in the federated setting. Note that by using a de-centralized method for aggregation, Agarwal et al. (2018) can still protect against the meta-learner from making inferences on any individual $\\widehat { \\theta } _ { t }$ with what is only effectively a global DP mechanism. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Previous Work</td><td>Notion of DP</td><td>Privacy for Φ</td><td>Privacy for θt</td></tr><tr><td>McMahan et al. (2018)</td><td>Global</td><td>Task-level</td><td>1</td></tr><tr><td>Geyer et al. (2018)</td><td>Global</td><td>Task-level</td><td>=</td></tr><tr><td>Bhowmick et al. (2019)</td><td>Local, Global</td><td>Task-level</td><td>Task-level</td></tr><tr><td>Agarwal et al. (2018)</td><td>Global + MPC</td><td>Task-level</td><td>Task-level</td></tr><tr><td>Truex et al. (2019)</td><td>Task-Global + MPC</td><td>Record-level</td><td>Record-level</td></tr><tr><td>Our work</td><td>Task-Global</td><td>Record-level</td><td>Record-level</td></tr></table>",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.4 DIFFERENTIAL PRIVACY FOR A GBML SETTING ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In meta-learning, there exists a hierarchy of agents and statistical queries, so we cannot as simply define global and local DP. Here, both the meta-level sub-procedure, $\\{ \\bar { \\theta } _ { t } \\} _ { t \\in B }  \\phi$ , and the withintask sub-procedure, $\\{ z _ { t , i } \\} _ { i = 1 } ^ { m _ { t } } \\to \\bar { \\theta } _ { t }$ , can be considered individual queries and a DP algorithm can implement either to be DP. Further, for each query, the procedure may be altered to satisfy either local DP or global DP. Thus, there are four fundamental options that follow from standard DP definitions. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "(1) Global $D P$ : Releasing $\\phi$ will at no point compromise information regarding any specific ${ \\bar { \\theta } } _ { t }$ .   \n(2) Local $D P$ : Additionally, each ${ \\bar { \\theta } } _ { t }$ is protected from being revealed to the meta-learner.   \n(3) Task-Global $D P$ : Releasing ${ \\bar { \\theta } } _ { t }$ will at no point compromise any specific $z _ { t , i }$ .   \n(4) Task-Local $D P$ : Additionally, each $z _ { t , i }$ is protected from being revealed to task-owner. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To form analogies to single-task DP, the examples in the meta-level procedure are the model updates and the aggregator is the meta-learner. For the within-task procedure, the examples are actually the individual records and the aggregator is the task-owner. As such, (1) is implemented by the metalearner, (2) and (3) are implemented by the task-owner, and (4) is implemented by record-owners. By immunity to post-processing, the guarantees for (3) and (4) also automatically apply to the release of any future iteration of $\\phi$ , thus protecting against future task-owners as well. Meanwhile, though (1) and (2) by definition protect the identities of individual ${ \\bar { \\theta } } _ { t }$ , they actually satisfy a task-level threat model by doing so. Intuitively, not being able to reliably infer anything about ${ \\bar { \\theta } } _ { t }$ implies that nothing can be inferred about the $D _ { t }$ that was used to generate it. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Using the terminology we introduce in Section 2.4, previous works for DP in federated settings can be categorized as in Table 1. While these works do not assume a multi-task setting, we can still naturally use the terms global/local and task-global/task-local to analogously refer to releasing the global model (by the central server in the case without MPC) and user-specific updates (by users’ devices) respectively. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 DIFFERENTIALLY PRIVATE PARAMETER-TRANSFER ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.1 ALGORITHM ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We now present our DP GBML method, which is written out in its online (regret) form in Algorithm 1. Here, we observe that both within-task optimization and meta-optimization are done using some form of gradient descent. The key difference between this algorithm and traditional GBML is that since task-learners must send back privatized model updates, each now applies an DP gradient descent procedure to learn ${ \\bar { \\theta } } _ { t }$ when called. However, at meta-test time the task-learner will run a non-private descent algorithm to obtain the parameter $\\widehat { \\theta } _ { t }$ used for inference, as this parameter may remain locally. To obtain learning-theoretic guarantees, we use a variant of Algorithm 1 in which the DP algorithm is an SGD procedure (Bassily et al., 2019, Algorithm 1) that adds a properly scaled Gaussian noise vector at each iteration. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1: Online version of our $( \\varepsilon , \\delta )$ -meta-private parameter-transfer algorithm. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Meta-learner picks first meta-initialization $\\phi _ { 1 } \\in \\Theta$ . for task $t \\in [ T ]$ do ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Meta-learner sends meta-initialization $\\phi _ { t }$ to task $t$ .   \nTask-learner runs OGD starting from $\\theta _ { t , 1 } = \\phi _ { t }$ on losses $\\{ \\ell _ { t , i } \\} _ { i = 1 } ^ { m }$ to obtain $\\widehat { \\theta } _ { t }$ .   \nTask-learner $t$ runs $( \\varepsilon , \\delta )$ -DP algorithm (noisy-SGD) on losses $\\{ \\ell _ { t , i } \\} _ { i = 1 } ^ { m }$ to get ${ \\bar { \\theta } } _ { t }$ .   \nTask-learner sends ${ \\bar { \\theta } } _ { t }$ to meta-learner.   \nMeta-learner constructs loss $\\begin{array} { r } { \\ell _ { t } ( \\phi ) = \\frac 1 2 \\| \\bar { \\theta } _ { t } - \\phi _ { t } \\| _ { 2 } ^ { 2 } } \\end{array}$ .   \nMeta-learner updates meta-initialization $\\phi _ { t + 1 }$ using an OCO algorithm on $\\ell _ { 1 } , \\ldots , \\ell _ { t }$ . ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Result: Meta-initialization $\\begin{array} { r } { { \\hat { \\phi } } = { \\frac { 1 } { T } } \\sum _ { t = 1 } ^ { T } \\phi _ { t } } \\end{array}$ to use on test tasks. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.2 PRIVACY GUARANTEES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We run a certified $( \\varepsilon , \\delta )$ -DP version of SGD (Bassily et al., 2019, Algorithm 1) within each task. Therefore, this guarantees that the contribution of each task-owner, a $\\bar { \\theta } _ { t }$ trained on their data, carries global DP guarantees with respect to the meta-learner. Additionally, since DP is preserved under post-processing, the release of any future calculation stemming from ${ \\bar { \\theta } } _ { t }$ also carries the same DP guarantee. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.3 LEARNING GUARANTEES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Our learning result follows the setup of Baxter (2000), who formalized the LTL problem as using task-distribution samples $\\mathcal { P } _ { 1 } , . . . , \\mathcal { P } _ { T } \\sim \\mathcal { Q }$ from some meta-distribution $\\mathcal { Q }$ and samples indexed by $i = 1 , \\ldots , m$ from those tasks to improve performance when a new task $\\mathcal { P }$ is sampled from $\\mathcal { Q }$ and we draw $m$ samples from it. In the setting of parameter-transfer meta-learning we are learning functions parameterized by real-valued vectors $\\overline { { \\theta } } \\in \\dot { \\Theta } \\subset \\mathbb { R } ^ { d }$ , so our goal will follow that of Denevi et al. (2019) and Khodak et al. (2019b) in seeking bounds on the transfer-risk – the distributional performance of a learned parameter on a new task from $\\mathcal { Q }$ – that improve with task similarity. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The specific task-similarity metric we consider is the average deviation of the risk-minimizing parameters of tasks sampled from the distribution $\\mathcal { Q }$ are close together. This will be measured in-terms of the following quantity: $\\begin{array} { r } { V ^ { 2 } = \\operatorname* { m i n } _ { \\phi \\in \\Theta } \\frac { 1 } { 2 } \\mathbb { E } _ { \\mathcal { P } \\sim \\mathcal { Q } } \\| \\theta _ { \\mathcal { P } } - \\phi \\| _ { 2 } ^ { 2 } } \\end{array}$ , for $\\theta _ { P } \\in \\arg \\operatorname* { m i n } _ { \\theta \\in \\Theta } \\ell _ { \\mathcal { P } } ( \\theta )$ a risk-minimizer of task-distribution $\\mathcal { P }$ . This quantity is roughly the variance of risk-minimizing task-parameters and is a standard quantifier of improvement due to meta-learning (Denevi et al., 2019; Khodak et al., 2019b). For example, Denevi et al. (2019) show excess transfer-risk guarantees of the form $\\begin{array} { r } { \\mathcal { O } \\left( \\frac { V } { \\sqrt { m } } + \\sqrt { \\frac { \\log T } { T } } \\right) } \\end{array}$ when $T$ tasks with $m$ samples are drawn from the distribution. This guarantee ensures that as we see more tasks our transfer risk becomes roughly $V / \\sqrt { m }$ , which if the tasks are similar, i.e. $V$ is small, implies that LTL improves over single-task learning. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In Algorithm 1, each user $t$ obtains a within-task parameter ${ \\bar { \\theta } } _ { t }$ by running (non-private) OGD on a sequence of losses $\\ell _ { t , 1 } , \\ldots , \\ell _ { t , m }$ and averaging the iterates. The regret of this procedure, when averaged across the users, implies a bound on the expected excess transfer risk of new task from $\\mathcal { Q }$ when running OGD from a learned initialization (Cesa-Bianchi et al., 2004). Thus our goal is to bound this regret in terms of $V$ ; here we follow the Average Regret-Upper-Bound Analysis (ARUBA) framework of Khodak et al. (2019b) and treat meta-update procedure itself as an online algorithm optimizing a bound on the performance measure (regret) of each within-task algorithm. As OGD’s regret depends on the squared distance $\\frac { 1 } { 2 } \\| \\theta _ { t } ^ { * } - \\phi _ { t } \\| _ { 2 } ^ { 2 }$ of the optimal parameter from the initialization $\\phi _ { t }$ , with no privacy concerns one could simply update $\\phi _ { t }$ using $\\begin{array} { r } { \\theta _ { t } ^ { * } \\in \\arg \\operatorname* { m i n } _ { \\theta \\in \\Theta } \\sum _ { i = 1 } ^ { m } \\ell _ { t , i } ( \\theta ) } \\end{array}$ to recover guarantees similar to those in Denevi et al. (2019) and Khodak et al. (2019b). ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "However, this approach requires sending $\\theta _ { t } ^ { * }$ to the meta-learner, which is not private; instead in Algorithm 1 we send $\\widehat { \\theta } _ { t }$ , which is the output of noisy SGD. To apply ARUBA, we need an additional assumption – that the losses satisfy the following quadratic growth (QG) property: for some $\\alpha > 0$ , ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/19380e8f459dc2d42023b24186cc68d2564672921cd2f4e457ef35bab832df49.jpg",
+        "text": "$$\n\\frac { \\alpha } { 2 } \\| \\theta - \\theta _ { \\mathcal { P } } \\| _ { 2 } ^ { 2 } \\le \\ell _ { \\mathcal { P } } ( \\theta ) - \\ell _ { \\mathcal { P } } ( \\theta _ { \\mathcal { P } } ) \\quad \\forall \\theta \\in \\Theta\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Here $\\theta _ { \\mathcal { P } }$ is the risk minimizer of $\\ell _ { \\mathcal { P } }$ . This assumption, which Khodak et al. (2019a) shows is reasonable in settings such as logistic regression, amounts to a statistical non-degeneracy assumption on the parameter-space – that parameters far away from the risk-minimizer do not have low-risk. Note that assuming the population risk is QG is significantly weaker than assuming strong convexity of the empirical risk, which previous work (Finn et al., 2019) has assumed to hold for task losses but does not hold for applicable cases such as few-shot least-squares or logistic regression if the number of task-samples is smaller than the data-dimension. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We are now able to state our main theoretical result, a proof of which is given in Appendix A. The result follows from a bound on the task-averaged regret across all tasks of a simple online meta-learning procedure that treats the update ${ \\bar { \\theta } } _ { t }$ sent by each task as an approximation of the optimal parameter in hindsight $\\theta _ { t } ^ { * }$ . Since this parameter determines regret on that task, by reducing the metaupdate procedure to OCO on this sequence of functions in a manner similar to (Khodak et al., 2019a), we are able to show a task-similarity-dependent bound. Following this the statistical guarantee stems from a nested online-to-batch conversion, a standard procedure to convert low-regret online-learning algorithms to low-risk distribution-learning algorithms (Cesa-Bianchi et al., 2004). ",
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+    {
+        "type": "text",
+        "text": "Theorem 3.1. Suppose $\\mathcal { Q }$ is a distribution over task-distributions $\\mathcal { P }$ over $G$ -Lipschtz, $\\beta$ -stronglysmooth, $^ { l }$ -bounded convex loss functions $\\ell : \\Theta \\mapsto \\mathbb { R }$ over parameter space $\\Theta$ with diameter $D$ for $\\begin{array} { r } { \\beta \\leq \\frac { G } { D } \\operatorname* { m i n } \\left( \\sqrt { \\frac { m } { 2 } } , \\frac { \\varepsilon m } { 2 \\sqrt { 2 d \\log \\frac { 1 } { \\delta } } } \\right) } \\end{array}$ and let each $\\mathcal { P }$ satisfy the quadratic growth property (1). Suppose the distribution $\\mathcal { P } _ { t }$ of each task is sampled i.i.d. from $\\mathcal { Q }$ and we run Algorithm 1 with the $( \\varepsilon , \\delta )$ -DP procedure of Bassily et al. (2019, Algorithm $I$ ) to obtain ${ \\bar { \\theta } } _ { t }$ as the average iterate for the meta-update step, using $n \\geq 1$ steps and learning rate $\\frac { \\gamma } { G \\sqrt { n } } f o r \\gamma > 0$ . Letting $\\begin{array} { r } { V ^ { 2 } = \\operatorname* { m i n } _ { \\phi \\in \\Theta } \\frac { 1 } { 2 } \\mathbb { E } _ { \\mathcal { P } \\sim \\mathcal { Q } } \\| \\theta _ { \\mathcal { P } } - \\phi \\| _ { 2 } ^ { 2 } } \\end{array}$ , there exist settings of $n , \\gamma , \\eta$ such that we have the following bound on the expected transfer risk when a new task $\\mathcal { P }$ is sampled from $\\mathcal { Q }$ , m samples are drawn i.i.d. from $\\mathcal { P }$ , and we run $O G D$ with learning rate $\\eta$ starting from $\\begin{array} { r } { { \\hat { \\phi } } = { \\frac { 1 } { T } } \\sum _ { t = 1 } ^ { T } \\phi _ { t } } \\end{array}$ and use the average $\\hat { \\theta }$ of the resulting iterates as the learned parameter: ",
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+        "type": "equation",
+        "img_path": "images/ea9a55a6d4ab67ec186106ca696cd6258f5df0b174b487c7a7ab932e243dd614.jpg",
+        "text": "$$\n\\mathbb { E } \\underset { \\mathcal { P } \\sim \\mathcal { Q } \\ : \\ell \\sim \\mathcal { P } } { \\mathbb { E } } \\ell ( \\hat { \\theta } ) \\leq \\underset { \\mathcal { P } \\sim \\mathcal { Q } \\ : \\ell \\sim \\mathcal { P } } { \\mathbb { E } } \\ell ( \\theta ^ { * } ) + \\tilde { \\mathcal { O } } \\left( \\frac { V } { \\sqrt { m } } + \\frac { \\alpha D ^ { 2 } } { T } + \\frac { 1 } { \\alpha } \\operatorname* { m a x } \\left( \\frac { d \\log \\frac { 1 } { \\delta } } { \\varepsilon ^ { 2 } m ^ { 2 } } , \\frac { 1 } { m } \\right) \\right)\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Here $\\theta ^ { * }$ is any element of $\\Theta$ and the outer expectation is taken over $\\ell _ { t , i } \\sim \\mathcal { P } _ { t } \\sim \\mathcal { Q }$ and the randomness of the within-task $D P$ mechanism. Note that this procedure is $( \\varepsilon , \\delta )$ -DP. ",
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+    {
+        "type": "text",
+        "text": "Theorem 3.1 shows that one can usefully run a DP-algorithm as the within-task method in meta-√ learning and still obtain improvement due to task-similarity. Specifically, the standard term of $1 / \\sqrt { m }$ is multiplied by $V$ , which is small if the tasks are related via the closeness of their risk minimizers. Thus we can use meta-learning to improve within-task performance relative to single-task learning. We also obtain a very fast convergence of $\\tilde { \\mathcal { O } } ( 1 / T )$ in the number of tasks. However, we do gain some $O ( 1 / m )$ terms due to the quadratic growth approximation and the privacy mechanism. Note that the assumption that both the functions and its gradients are Lipschitz-continuous are standard and required by the noisy SGD procedure of Bassily et al. (2019). ",
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+        "type": "text",
+        "text": "This theorem also gives us a relatively straightforward extension if the desire is to provide $( \\varepsilon , \\delta )$ - group-DP. Since any privacy mechanism that provides $( \\varepsilon , \\delta )$ -DP also provides $( k \\varepsilon , k e ^ { ( k - 1 ) \\epsilon } \\delta )$ -DP guarantees for groups of size $k$ (Dwork and Roth, 2014), we immediately have the following corollary. ",
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+        "type": "text",
+        "text": "Corollary 3.1. Under the same assumptions and setting as Theorem 3.1, achieving $( \\varepsilon , \\delta )$ -group $D P$ is possible with the same guarantee except replacing $\\frac { d \\log { \\frac { 1 } { \\delta } } } { \\varepsilon ^ { 2 } }$ wit h k 3 d $\\begin{array} { r } { \\frac { k ^ { 3 } d } { \\varepsilon } + \\frac { k ^ { 2 } d } { \\varepsilon } \\left[ \\frac { 1 } { \\varepsilon } \\log \\frac { k } { \\delta } - 1 \\right] } \\end{array}$ ",
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+    {
+        "type": "text",
+        "text": "For constant $k$ , this allows us to enjoy the stronger guarantee while maintaining largely the same learning rates. This is a useful result given that in some settings, it may be desired to simultaneously protect small groups of size $k < < m _ { t }$ , such as protecting entire families for hospital records. ",
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+    {
+        "type": "text",
+        "text": "4 EMPIRICAL RESULTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We present results that show it is possible to learn useful deep models in federated scenarios while still preserving privacy against all other participants. Specifically, we evaluate the performance of models that have been trained with a task-global DP algorithm in comparison to models that have been trained both non-privately and with local DP algorithms. We evaluate performance on federated language modeling and few-shot image classification, applying a practical batched variant of Algorithm 1. ",
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+    {
+        "type": "image",
+        "img_path": "images/6bf7b551960969d39cb4270213f258dac4ce3fbf1628fb1f4cb2c1861206e1cf.jpg",
+        "image_caption": [
+            "Figure 2: Performance of different versions of Reptile on a next-word-prediction task for two federated datasets. We report the test accuracy on unseen tasks and repeat each experiment 10 times. Solid lines correspond to means, colored bands indicate 1 standard deviation, and dotted lines are for comparing final accuracies (private algorithms can only be trained until privacy budget is met). "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Datasets: We train a LSTM-RNN for next word prediction on two federated datasets: (1) The Shakespeare dataset as preprocessed by (Caldas et al., 2018), and (2) a dataset constructed from 3, 000 Wikipedia articles drawn from the Wiki-3029 dataset (Arora et al., 2019), where each article is used as a different task. For each dataset, we set a fixed number of tokens per task, discard tasks with fewer tokens than the specified, and discard samples from those tasks with more. We set the number of tokens per task to 800 for Shakespeare and to 1, 600 for Wikipedia, divide tokens into sequences of length 10, and we refer to these modified datasets as Shakespeare-800 and Wiki-1600. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For few-shot image classification, we use the Omniglot (Lake et al., 2011) and Mini-ImageNet (Ravi and Larochelle, 2017) datasets, both with 5-shot-5-way test tasks. As has been done for non-private Reptile (Nichol et al., 2018), we use more training shots at meta-training (trying $m = 1 0 , 2 0 , 3 0$ for Omniglot and $m = 1 5$ , 30, 45 for Mini-ImageNet) than at meta-test time. Though tasks could be sampled indefinitely, we set a fixed budget of tasks at $T = 1 0 ^ { 6 }$ to reflect to a setting in which a finite number of training tasks constrains our learning and privacy trade-off. For both local and task-global DP, the more tasks that can be grouped in a meta-batch means that less total noise can be added at each round. However, this also means fewer iterations can be taken. We note that for the smallest values of $m$ , this setting is enough for non-private training to essentially achieve the reported final accuracies from Nichol et al. (2018). ",
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+    {
+        "type": "text",
+        "text": "Meta Learning Algorithm. We study the performance of our method when applied to the batched version of Reptile (Nichol et al., 2018) (which, in our setup, reduces to personalized Federated Averaging when the meta-learning rate is set to 1.0). For the language modelling tasks, we tune various configurations of task batch size for all methods. We also allow for multiple visits per client, though at the cost of more added noise per iteration for the private methods. Additionally, for language modeling, we implement gradient clipping and exponential decay on the meta learning rate. For Omniglot and Mini-ImageNet, we use largely the same parameters as Nichol et al. (2018) but we tune the parameters most directly related to privacy: the $L _ { 2 }$ clipping threshold, the Adam Learning Rate at meta-training time, the meta-batch size, and the within-task batch size. We defer a more complete discussion of hyperparameter tuning to Appendix B. ",
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+        "img_path": "images/980bb10fd82738ffbdd39b856bb0072baf7ab0596be743b1d9753fed73b6ca3d.jpg",
+        "image_caption": [
+            "Figure 3: Performance of task-global DP Reptile on 5-shot-5-way Omniglot and Mini-ImageNet. $1 0 ^ { \\overline { { 5 } } }$ sampled test- tasks were used for evaluation and experiments were repeated 3 times. We do not show a line for Local DP since all hyperparameter settings tried for Local DP resulted in worse performance than the “No Meta-Learning\" baseline, whose performance can always be recovered. "
+        ],
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Privacy Considerations. For the task-global DP models, we set on each task and we implement DP-SGD (for language model $\\delta = 1 0 ^ { - 3 } < \\frac { 1 } { m ^ { 1 . 1 } }$ by conventionm (for imageclassification) within-task using the tools provided by TensorFlow Privacy1, using the RDP accountant to track our privacy budgets. Although these algorithms differ from the one presented in Section 3, they still let us realistically explore the efficacy of considering task-global privacy. For the language modeling datasets, we try three different privacy budgets (as determined relative to each other by successively doubling the amount of noise added when the goal is to take 1 full gradient step per task) and make sure that all training tasks are sampled without replacement with a fixed batch size until all are seen. This is necessary since multiple visits to a single client results in degradation of the privacy guarantee for that client. We instead aim to provide the same guarantee for each client. For local-DP, though this notion of DP is stronger, we explore the same privacy budgets so as to obtain guarantees that are of the same confidence. Here, we essentially run the DP-FedAvg algorithm from (McMahan et al., 2018) with some key changes. First, to get local DP instead of global, we add Gaussian noise to each clipped set of model updates before returning them to the central server instead of after aggregation. Second, while additional gradient steps within-task do not increase the amount of noise required, we do again iterate through tasks without replacement. Unlike for global DP, we cannot hope to have any privacy boosts due to sub-sampling if the meta-learner knows who it is communicating with. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Results. Figure 2 shows the performance of both the non-private and task-global private versions of Reptile (Nichol et al., 2018) for the language modelling tasks across three different privacy budgets. As expected, neither private algorithm reaches the same accuracy of the non-private version of the algorithm. Nonetheless, the task-global version still comes within $7 8 \\%$ , $8 8 \\%$ , and $9 2 \\%$ of the nonprivate accuracy for Shakespeare-800 and within $7 2 \\%$ , $8 2 \\%$ , and $8 3 \\%$ for Wiki-1600. Meanwhile achieving local DP results in only about $5 5 \\%$ and $5 0 \\%$ of the non-private accuracy on both datasets for the most generous privacy budget. In practice, these differences can be toggled by further changing the privacy budget or continuing to trade off more training iterations for larger noise multipliers. ",
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "We display results for few-shot image classification on Omniglot and Mini-ImageNet in Figure 3. In this setting, not applying meta-learning results in meta-test accuracies of around $6 2 \\%$ and $3 6 \\%$ , respectively. Thus, while performance is indeed lower than non-private learning, applying task-global DP does result in meta-learning benefits for test-time tasks. In settings where privacy is a concern, this increase in performance is still significantly advantageous for the “task-owners”– test-time tasks (who hold less data). On average, they are able to obtain better models and are still guaranteed privacy at a single-digit $\\varepsilon$ . Intuitively, larger training-task datasets make it easier to apply privacy within-task, and in accordance with our learning guarantees, adding training shots indeed closes the gap in performance between task-global DP Reptile and non-private Reptile. In comparison, applying local-DP for a similar hyperparameter range consistently decreases performance at test-time. However, the no-meta-learning baseline is a theoretical lower bound for local-DP, as one could set the clipping threshold or meta-learning rate close to 0 to recover the effects of no meta-learning. ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this work, we have outlined and studied the issue of privacy in the context of meta-learning. Focusing on the class of gradient-based parameter-transfer methods, we used differential privacy to address the privacy risks posed to task-owners by sharing task-specific models with a central meta-learner. To do so, we formalized and considered the notion of task-global differential privacy, which guarantees that individual examples from the tasks are protected from all downstream agents (and particularly the meta-learner). Working in this privacy model, we developed a differentially private algorithm that guarantees both this protection as well as learning-theoretic results in the convex setting. Finally, we demonstrate how this notion of privacy can translate into useful deep learning models for non-convex language modelling and image-classification tasks. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "This work was supported in part by DARPA FA875017C0141, the National Science Foundation grants IIS1618714, IIS1705121, and IIS1838017, an Okawa Grant, a Google Faculty Award, an Amazon Web Services Award, a JP Morgan A.I. Research Faculty Award, and a Carnegie Bosch Institute Research Award. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of DARPA, the National Science Foundation, or any other funding agency. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+    {
+        "type": "text",
+        "text": "A PROOFS OF LEARNING GUARANTEES ",
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+    {
+        "type": "text",
+        "text": "Setting A.1. We assume that at each time-step t an adversary chooses a task-distribution $\\mathcal { P } _ { t }$ over loss-functions on $\\Theta \\subset \\mathbb { R } ^ { d }$ and samples m loss functions $\\ell _ { t , i }$ for $i \\in [ m ]$ . At each time-step $t$ the tasklearner receives a parameter $\\phi _ { t }$ from the meta-learner, runs online gradient descent with step-size $\\eta > 0$ starting from $\\phi _ { t }$ , and uses the average iterate $\\widehat { \\theta } _ { t }$ as its learned parameter. The task-learner also runs Algorithm $I$ of Bassily et al. (2019) for $\\begin{array} { r } { n = \\operatorname* { m i n } \\left\\{ \\frac { m } { 8 } , \\frac { \\varepsilon ^ { 2 } m ^ { 2 } } { 3 2 d \\log \\frac { 1 } { \\delta } } \\right\\} } \\end{array}$ steps with learning rate $\\frac { \\gamma } { G \\sqrt { n } } > 0$ on these loss functions and sends the result ${ \\bar { \\theta } } _ { t }$ to the meta-learner. The meta-learner updates $\\phi _ { t + 1 } = ( 1 - 1 / t ) \\phi _ { t } + \\bar { \\theta } _ { t } / t$ . We assume all loss functions are $G$ -Lipschitz w.r.t. $\\| \\cdot \\| _ { 2 }$ and $\\beta$ -strongly-smooth w.r.t. $\\| \\cdot \\| _ { 2 }$ for some $\\begin{array} { r } { \\beta \\leq \\frac { G } { D } \\operatorname* { m i n } \\left\\{ \\sqrt { \\frac { m } { 2 } } , \\frac { \\varepsilon n } { 2 \\sqrt { 2 d \\log \\frac { 1 } { \\delta } } } \\right\\} } \\end{array}$ , where $D$ is the diameter of $\\Theta$ . For each distribution $\\mathcal { P } _ { t }$ let $\\ell _ { t } ( \\theta ) = \\mathbb { E } _ { \\ell \\sim \\mathcal { P } _ { t } } ( \\theta )$ be its population risk, $\\begin{array} { r } { \\hat { \\ell } _ { t } ( \\theta ) = \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } \\ell _ { t , i } ( \\theta ) b e } \\end{array}$ its empirical risk, and $\\theta _ { t } ^ { * } \\in \\arg \\operatorname* { m i n } _ { \\theta \\in \\Theta } \\ell _ { t } ( \\theta )$ be the closest population risk minimizer to ${ \\bar { \\theta } } _ { t }$ . ",
+        "bbox": [
+            171,
+            132,
+            828,
+            314
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Lemma A.1. In Setting A.1 we have ",
+        "bbox": [
+            174,
+            316,
+            416,
+            332
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a93839b4cc7f3acf9015aa1a631b28dfd77d36ad5c7e3446c970a791f24d7e2a.jpg",
+        "text": "$$\n\\mathbb { E } \\ell _ { t } ( \\hat { \\theta } _ { t } ) - \\ell _ { t } ( \\theta _ { t } ^ { * } ) \\leq 5 G \\left( \\frac { \\| \\phi _ { t } - \\theta _ { t } ^ { * } \\| _ { 2 } ^ { 2 } } { \\gamma } + \\gamma \\right) \\operatorname* { m a x } \\left\\{ \\frac { \\sqrt { d \\log \\frac { 1 } { \\delta } } } { \\varepsilon m } , \\frac { 1 } { \\sqrt { m } } \\right\\}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            266,
+            338,
+            732,
+            390
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Proof. Similarly to Lemma 3.3 in Bassily et al. (2019), applying standard OGD analysis (e.g. Lemmas 14.1 and 14.9 of Shalev-Shwartz and Ben-David (2014)) to noisy gradient vectors and taking expectations yields ",
+        "bbox": [
+            173,
+            402,
+            825,
+            445
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2ce7e669ba7dceeee9f4d905edcab11d61b2c8e48d054168db2f1ec577c903f0.jpg",
+        "text": "$$\n\\mathbb { E } \\left( \\hat { \\ell } _ { t } ( \\hat { \\theta } _ { t } ) - \\hat { \\ell } _ { t } ( \\theta _ { t } ^ { * } ) \\right) \\leq \\frac { \\| \\phi _ { t } - \\theta _ { t } ^ { * } \\| _ { 2 } ^ { 2 } } { 2 \\eta n } + \\frac { \\eta G ^ { 2 } } { 2 } + \\eta \\sigma ^ { 2 } d\n$$",
+        "text_format": "latex",
+        "bbox": [
+            326,
+            452,
+            671,
+            486
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "where $n$ is the number of steps in noisy SGD and $\\sigma ^ { 2 }$ is the variance of the noise added at each step. As in the proof of Theorem 3.2 of Bassily et al. (2019), substituting $\\begin{array} { r } { \\sigma ^ { 2 } = \\frac { 8 n G ^ { 2 } \\log \\frac { 1 } { \\delta } } { m ^ { 2 } \\varepsilon ^ { 2 } } } \\end{array}$ and applying the stability result in Lemma 3.4 of the same paper yields ",
+        "bbox": [
+            174,
+            492,
+            826,
+            541
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4ecf47eff1aa7def416f37488b2a25004707042c6ae499a595adb55a0711687d.jpg",
+        "text": "$$\n\\mathbb { E } \\ell _ { t } ( \\hat { \\theta } _ { t } ) - \\ell _ { t } ( \\theta _ { t } ^ { * } ) \\leq \\frac { \\| \\phi _ { t } - \\theta _ { t } ^ { * } \\| _ { 2 } ^ { 2 } } { 2 \\eta n } + \\frac { \\eta G ^ { 2 } } { 2 } \\left( \\frac { 1 6 n d \\log \\frac { 1 } { \\delta } } { m ^ { 2 } \\varepsilon ^ { 2 } } + 1 \\right) + \\frac { \\eta G ^ { 2 } n } { m }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            266,
+            546,
+            730,
+            583
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Substituting $\\begin{array} { r } { n = \\operatorname* { m i n } \\left\\{ \\frac { m } { 8 } , \\frac { \\varepsilon ^ { 2 } m ^ { 2 } } { 3 2 d \\log \\frac { 1 } { \\delta } } \\right\\} } \\end{array}$ and $\\begin{array} { r } { \\eta = \\frac { \\gamma } { G \\sqrt { n } } } \\end{array}$ yields the result. ",
+        "bbox": [
+            174,
+            590,
+            625,
+            616
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Lemma A.2. In Setting A.1, fix some $\\phi ^ { * } \\in \\Theta$ and define $\\begin{array} { r } { \\bar { V } ^ { 2 } = \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\mathbb { E } \\| \\phi ^ { * } - \\theta _ { t } ^ { * } \\| _ { 2 } ^ { 2 } } \\end{array}$ . Then for $\\begin{array} { r } { \\gamma = \\frac { 1 2 0 G } { \\alpha } \\operatorname* { m a x } \\left\\{ \\frac { \\sqrt { d \\log \\frac { 1 } { \\delta } } } { \\varepsilon m } , \\frac { 1 } { \\sqrt { m } } \\right\\} } \\end{array}$ we have ",
+        "bbox": [
+            173,
+            99,
+            826,
+            150
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/404d6aaa4be9ee845c37dfa7dd83ecded0899300e1f04fe9026e11fbb7495c02.jpg",
+        "text": "$$\n\\mathbb { E } \\sum _ { t = 1 } ^ { T } \\frac { \\| \\phi _ { t } - \\theta _ { t } ^ { * } \\| _ { 2 } ^ { 2 } } { 2 \\eta m } \\leq \\frac { D ^ { 2 } ( 1 + \\log T ) + 4 \\hat { V } ^ { 2 } T } { 2 \\eta m } + \\frac { 7 2 0 0 G ^ { 2 } } { \\alpha ^ { 2 } \\eta m } \\operatorname* { m a x } \\left\\{ \\frac { d \\log \\frac { 1 } { \\delta } } { \\varepsilon ^ { 2 } m ^ { 2 } } , \\frac { 1 } { m } \\right\\} T\n$$",
+        "text_format": "latex",
+        "bbox": [
+            232,
+            159,
+            764,
+            202
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Proof. We first bound the left-hand side without the denominator as ",
+        "bbox": [
+            173,
+            213,
+            620,
+            229
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d99a5c25a01c9a3f8ea95d0843d30901612e53fa5a4899c1f4d3b9e29b8663be.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\varepsilon _ { \\perp } \\ge \\frac { \\varepsilon _ { \\perp } } { \\varepsilon _ { \\perp } } \\ge \\frac { 1 } { \\varepsilon _ { \\perp } } \\ge } & { \\varepsilon _ { \\perp } } \\\\ & { = 1 } \\\\ & { \\le 2 \\kappa \\sum _ { i = 1 } ^ { \\nu } \\| \\partial _ { i } - \\partial _ { i } \\xi \\| ^ { 2 } + \\| \\partial _ { i } - \\theta _ { i } \\| ^ { 2 } } \\\\ & { \\le D ^ { \\perp } ( 1 - \\log { \\varepsilon _ { \\perp } } ) + 2 \\sum _ { i = 1 } ^ { \\nu } \\| \\phi - \\varepsilon _ { \\perp } \\| ^ { 2 } + \\| \\partial _ { i } - \\partial _ { i } \\xi \\| ^ { 2 } } \\\\ & { \\le D ^ { \\perp } ( 1 - \\log { \\varepsilon _ { \\perp } } ) + 2 \\sum _ { i = 1 } ^ { \\nu } \\| \\phi - \\varepsilon _ { \\perp } \\| ^ { 2 } + \\| \\partial _ { i } - \\partial _ { i } \\xi \\| ^ { 2 } } \\\\ & { \\le p ^ { 2 } \\varepsilon _ { \\perp } ^ { 2 } + 1 - \\log { \\varepsilon _ { \\perp } } ) + 2 \\sum _ { i = 1 } ^ { \\nu } 2 \\| \\phi - \\varepsilon _ { \\perp } \\| ^ { 2 } + \\| \\partial _ { i } - \\partial _ { i } \\xi \\| ^ { 2 } } \\\\ & { = p ^ { 2 } \\varepsilon _ { \\perp } ^ { 2 } ( 1 - \\log { \\varepsilon _ { \\perp } } ) + 4 \\kappa ^ { 2 } T ^ { 2 } + 4 \\| \\partial _ { i } ^ { 2 } \\sum _ { i = 1 } ^ { \\nu } \\| \\hat { \\varepsilon } _ { i } - \\partial _ { i } \\xi \\| ^ { 2 } } \\\\ & { = D ^ { \\perp } ( 1 - \\log { \\varepsilon _ { \\perp } } ) + 4 \\kappa ^ { 2 } T ^ { 2 } + 4 \\| \\partial _ { i } ^ { 2 } \\sum _ { i = 1 } ^ { \\nu } 2 \\hat { \\varepsilon } _ { \\perp } \\phi _ { i } ^ { 2 } - \\varepsilon _ { \\perp } ( | \\partial _ { i } \\xi | ^ { 2 } ) } \\\\ & { \\le D ^ { \\perp } ( 1 - \\log { \\varepsilon _ { \\perp } } ) + 4 \\kappa ^ { 2 } T ^ { 2 } + \\frac { 1 } { \\omega _ { \\perp } } \\sum _ { i = 1 } ^ { \\nu } 2 \\hat { \\varepsilon } _ { \\perp } ( \\rho _ { i } ) - 6 \\hat { \\varepsilon } ( | \\hat { \\varepsilon } _ { \\perp } - \\hat { \\varepsilon } _ { \\perp } | ^ { 2 } ) } \\\\ &  \\le D ^ { \\perp } ( 1 - \\ \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            187,
+            252,
+            807,
+            553
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "where in the last step we applied Lemma A.1. Substituting $\\begin{array} { r } { \\gamma = \\frac { 1 2 0 G } { \\alpha } \\operatorname* { m a x } \\left\\{ \\frac { \\sqrt { d \\log \\frac { 1 } { \\delta } } } { \\varepsilon m } , \\frac { 1 } { \\sqrt { m } } \\right\\} } \\end{array}$ yields ",
+        "bbox": [
+            171,
+            559,
+            818,
+            590
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5cc98e36fccb330c01924bc7758649584c8a29142487ad4793dd92af09bb0c73.jpg",
+        "text": "$$\n\\mathbb { E } \\sum _ { t = 1 } ^ { T } \\| \\phi _ { t } - \\theta _ { t } ^ { * } \\| _ { 2 } ^ { 2 } \\leq D ^ { 2 } ( 1 + \\log T ) + 4 { \\bar { V } } ^ { 2 } T + \\frac { 1 4 4 0 0 G ^ { 2 } } { \\alpha ^ { 2 } } \\operatorname* { m a x } \\left\\{ \\frac { d \\log \\frac { 1 } { \\delta } } { \\varepsilon ^ { 2 } m ^ { 2 } } , \\frac { 1 } { m } \\right\\} T\n$$",
+        "text_format": "latex",
+        "bbox": [
+            232,
+            597,
+            766,
+            640
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The result follows by dividing by 2ηm. ",
+        "bbox": [
+            173,
+            646,
+            431,
+            662
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Theorem A.1. In Setting A.1, suppose all distributions $\\mathcal { P } _ { t }$ were drawn i.i.d. from some metadistribution $\\mathcal { Q }$ and we used $\\begin{array} { r } { \\gamma = \\frac { 1 2 0 G } { \\alpha } \\operatorname* { m a x } \\left\\{ \\frac { \\sqrt { d \\log \\frac { 1 } { \\delta } } } { \\varepsilon m } , \\frac { 1 } { G \\sqrt { m } } \\right\\} } \\end{array}$ . Suppose we draw another taskdistribution $\\mathcal { P } \\sim \\mathcal { Q }$ with population risk $\\ell _ { \\mathcal { P } }$ and minimizer $\\theta _ { \\mathcal { P } }$ , set $\\begin{array} { r } { \\hat { \\phi } = \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\phi _ { t } } \\end{array}$ , and run OGD with learning rate $\\begin{array} { r } { \\eta = \\frac { V + \\frac { 1 } { \\alpha \\sqrt { m } } } { \\sqrt { m } } } \\end{array}$ starting from $\\hat { \\phi }$ on m samples from $\\mathcal { P }$ . Then the average iterate $\\hat { \\theta }$ satisfies ",
+        "bbox": [
+            171,
+            102,
+            828,
+            205
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1d30381becf20adce4d12192dbd2df83d0509565cdc19d1056ff2fabdd542446.jpg",
+        "text": "$$\n\\mathbb { E } ( \\ell _ { \\mathcal { P } } ( \\hat { \\theta } ) - \\ell _ { \\mathcal { P } } ( \\theta ^ { * } ) ) \\le \\frac { 7 G V } { 2 \\sqrt { m } } + \\frac { 7 2 0 1 G } { \\alpha } \\operatorname* { m a x } \\left\\{ \\frac { d \\log \\frac { 1 } { \\delta } } { \\varepsilon ^ { 2 } m ^ { 2 } } , \\frac { 1 } { m } \\right\\} + \\frac { \\alpha G D ^ { 2 } } { 2 T } ( 1 + \\log T )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            225,
+            210,
+            769,
+            247
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "for $\\begin{array} { r } { V ^ { 2 } = \\operatorname* { m i n } _ { \\phi \\in \\Theta } \\mathbb { E } _ { \\mathcal { P } \\sim \\mathcal { Q } } \\operatorname* { m a x } _ { \\theta _ { \\mathcal { P } } } \\| \\phi - \\theta _ { \\mathcal { P } } \\| _ { 2 } ^ { 2 } . } \\end{array}$ ",
+        "bbox": [
+            171,
+            253,
+            468,
+            271
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Proof. Applying online-to-batch conversion (e.g. Proposition A.1 in Khodak et al. (2019b)) twice and substituting Lemma A.2 yields ",
+        "bbox": [
+            173,
+            284,
+            823,
+            314
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8a0015a28f9324288ea0eac70ffe63f8e45a0eb0d4a1461ab29f83c62a4eda43.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } ( \\ell _ { \\mathcal { P } } ( \\hat { \\theta } ) - \\ell _ { \\mathcal { P } } ( \\theta ^ { * } ) ) } \\\\ & { \\qquad \\le \\mathbb { E } \\frac { \\| \\hat { \\phi } - \\theta ^ { * } \\| _ { 2 } ^ { 2 } } { 2 \\eta m } + \\eta G ^ { 2 } } \\\\ & { \\qquad \\le \\mathbb { E } \\frac { \\| \\hat { \\phi } ^ { * } - \\theta ^ { * } \\| _ { 2 } ^ { 2 } } { 2 \\eta m } + \\eta G ^ { 2 } + \\frac { 1 } { 2 \\eta m T } \\displaystyle \\sum _ { t = 1 } ^ { T } \\mathbb { E } \\| \\phi _ { t } - \\theta _ { t } ^ { * } \\| _ { 2 } ^ { 2 } } \\\\ & { \\qquad \\le \\mathbb { E } \\frac { \\| \\hat { \\phi } ^ { * } - \\theta ^ { * } \\| _ { 2 } ^ { 2 } } { 2 \\eta m } + \\eta G ^ { 2 } + \\frac { D ^ { 2 } \\frac { 1 + \\log T } { T } } { 2 \\eta m } + 4 \\mathbb { E } \\bar { V } ^ { 2 } + \\frac { 7 2 0 0 G ^ { 2 } } { \\alpha ^ { 2 } \\eta m } \\operatorname* { m a x } \\left\\{ \\frac { d \\log \\frac { 1 } { \\delta } } { \\varepsilon ^ { 2 } m ^ { 2 } } , \\frac { 1 } { m } \\right\\} } \\\\ & { \\qquad = \\frac { 5 V ^ { 2 } } { 2 \\eta m } + \\eta G ^ { 2 } + \\frac { 7 2 0 0 G ^ { 2 } } { \\alpha ^ { 2 } \\eta m } \\operatorname* { m a x } \\left\\{ \\frac { d \\log \\frac { 1 } { \\delta } } { \\varepsilon ^ { 2 } m ^ { 2 } } , \\frac { 1 } { m } \\right\\} + \\frac { D ^ { 2 } } { 2 \\eta m T } ( 1 + \\log T ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            192,
+            318,
+            807,
+            498
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "where we have applied $\\mathbb { E } \\bar { V } ^ { 2 } \\le V ^ { 2 }$ . Substituting $\\begin{array} { r } { \\eta = \\frac { V + \\frac { 1 } { \\alpha \\sqrt { m } } } { G \\sqrt { m } } } \\end{array}$ yields the result. ",
+        "bbox": [
+            171,
+            503,
+            692,
+            529
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "B EXPERIMENT DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            398,
+            118
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Datasets: We train a next word predictor for two federated datasets: (1) The Shakespeare dataset as preprocessed by (Caldas et al., 2018), and (2) a dataset constructed from Wikipedia articles, where each article is used as a different task. For each dataset, we set a fixed number of tokens per task, discard tasks with less tokens than the specified, and discard samples from those tasks with more. For Shakespeare, we set the number of tokens per task to 800 tokens, leaving 279 tasks for meta-training, 31 for meta-validation, and 35 for meta-testing. For Wikipedia, we set the number of tokens to 1, 600, which corresponds to having 2, 179 tasks for meta-training, 243 for meta-validation, and 606 for meta-testing. For the meta-validation and meta-test tasks, $7 5 \\%$ of the tokens are used for local training, and the remaining $2 5 \\%$ for local testing. ",
+        "bbox": [
+            174,
+            133,
+            825,
+            258
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "For the few-shot image classification experiments, we follow the standard set-up by splitting labels into training and testing and forming training tasks by randomly drawing labels from the training set. At evaluation time, we draw from the test set. ",
+        "bbox": [
+            176,
+            265,
+            823,
+            308
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Model Structure: Our model first maps each token to an embedding of dimension 200 before passing it through an LSTM of two layers of 200 units each. The LSTM emits an output embedding, which is scored against all items of the vocabulary via dot product followed by a softmax. We build the vocabulary from the tokens in the meta-training set and fix its length to 10, 000. We use a sequence length of 10 for the LSTM and, just as (McMahan et al., 2018), we evaluate using AccuracyTop1 (i.e., we only consider the predicted word to which the model assigned the highest probability) and consider all predictions of the unknown token as incorrect. For Omniglot and Mini-ImageNet, we use the architectures from Nichol et al. (2018) to also match the ones from Finn et al. (2017). We evaluate in the standard transductive setting. ",
+        "bbox": [
+            173,
+            323,
+            825,
+            449
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Hyperparameters: For the language-modeling experiments, we tune the hyperparameters on the set of meta-validation tasks. For both datasets and all versions of the meta-learning algorithm, we tune hyperparameters in a two step process. We first tune all the parameters that are not related to refinement: the meta learning rate, the local (within-task) meta-training learning rate, the maximum gradient norm, and the decay constant. Then, we use the configuration with the best accuracy pre-refinement and then tune the refinement parameters: the refine learning rate, refine batch size, and refine epochs. ",
+        "bbox": [
+            174,
+            463,
+            825,
+            561
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "All other hyperparameters are kept fixed for the sake of comparison: full batch steps were taken on within-task data, with the maximum number of microbatches used for the task-global DP model. The parameter search spaces from which we sample are given in Tables 2, 3, 4 while Tables 5 and 6 contain our final choices. Note that the space for Local DP, especially in terms of the clipping threshold, is distinctively different from the others, as we did not find that searching through ranges similar to those for non-private and task-global DP led to learning high-quality meta-initializations. ",
+        "bbox": [
+            174,
+            568,
+            825,
+            651
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "For Omniglot, we largely based our hyperparameters on the choices of Nichol et al. (2018) for 5-way classification. We vary $m$ , the number of training shots, but we continue to take 5 SGD steps of expected size $m$ within task and we leave the test-time SGD procedure exactly the same. However, we do tune for privacy clipping thresholds $\\{ 0 . 0 1 , 0 . 0 2 5 , 0 . 0 5 , 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 5 \\}$ , Adam Learning Rates for meta-training tasks $\\left\\{ 1 0 ^ { - 4 } , 5 \\times \\mathrm { i } 0 ^ { - 4 } , 1 0 ^ { - 3 } , 5 \\times 1 0 ^ { - 3 } \\right\\}$ , and meta-batch sizes of $\\{ 5 , 1 5 , \\bar { 2 5 } , 5 0 \\}$ . ",
+        "bbox": [
+            174,
+            659,
+            825,
+            742
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "For Mini-ImageNet, we perform a similar search except we also double the inner batch size to $2 m$ (trading off less privacy amplification due to subsampling). We continue to tune for privacy clipping thresholds $\\{ 0 . 0 1 , \\dot { 0 } . 0 2 \\dot { 5 } , 0 . 0 5 , 0 . 1 , 0 . 3 , 0 . 5 , 0 . 7 , 0 . 9 , 1 . 1 , 1 . 3 , 1 . 5 \\}$ , Adam Learning Rates $\\{ 1 \\bar { 0 } ^ { - 4 } , \\bar { 5 } \\times 1 0 ^ { - 4 } , 1 0 ^ { \\dot { - } 3 } , 5 \\times 1 0 ^ { - 3 } \\}$ , and meta-batch sizes of $\\{ 5 , 1 5 , 2 5 , 5 0 \\}$ . ",
+        "bbox": [
+            174,
+            750,
+            823,
+            806
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/1d56b84c65f561a9c4aeeba07940ef87d16cd3418dd2f7d8e6ce7074e418463c.jpg",
+        "table_caption": [
+            "Table 2: Hyperparameter Search Space for Non-Private Training "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Shakespeare-800</td><td>Wiki-1600</td></tr><tr><td>VisitsPerTask</td><td>{1,2,3,4,5,6,7,8,9}</td><td>{1,2,3}</td></tr><tr><td>Tasks Per Round</td><td>{5,10}</td><td>{5,10}</td></tr><tr><td>Within-Task Steps</td><td>{1,3,5,7,9}</td><td>{1,3,5,7,9}</td></tr><tr><td>Meta LR</td><td>{1,√2,2,2√2,4,4√2,8,8√2}</td><td>{1,√2,2,2√2,4,4√2,8,8√2}</td></tr><tr><td rowspan=\"3\">Meta Decay Rate Within-Task LR</td><td>{0,0.001,0.005,0.01,0.025,0.05,0.1}</td><td>{0,0.001,0.005,0.01,0.025,0.05}</td></tr><tr><td>{1,√2,2,2√2,4,4√2,8}</td><td>{1,√2,2,2√2,4,4√2,8}</td></tr><tr><td>{0.4,0.5,0.6,0.7, 0.8, 0.9, 1.0}</td><td>{0.3, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0}</td></tr><tr><td rowspan=\"2\">Refine LR Refine Batch Size</td><td>{0.1, 0.15,0.3,0.5,0.7, 0.8}</td><td>{0.1,0.15, 0.3,0.5, 0.7, 0.8}</td></tr><tr><td>{10,20,30,60}</td><td>{10,20,30,60,120}</td></tr><tr><td>Refine Epochs</td><td>{1,2,3}</td><td>{1,2,3}</td></tr></table>",
+        "bbox": [
+            173,
+            162,
+            825,
+            334
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/f843673a59d28a3905ef73acae1dd4bbe276e7a9dc88905acf92b468c4c53b86.jpg",
+        "table_caption": [
+            "Table 3: Hyperparameter Search Space for Task-Global DP Training "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Shakespeare-800</td><td>Wiki-1600</td></tr><tr><td>VisitsPerTask</td><td>{1,2,3}</td><td>{1,2}</td></tr><tr><td>TasksPerRound</td><td>{5,10}</td><td>{5,10}</td></tr><tr><td>Within-Task Steps</td><td>1</td><td>1</td></tr><tr><td>MetaLR</td><td>{1,√2,2,2√2,4,4√2,8,8√2}</td><td>{1,√2,2,2√2,4,4√2,8,8√2}</td></tr><tr><td>Meta Decay Rate</td><td>{0,0.001,0.005,0.01, 0.025, 0.05,0.1}</td><td>{0,0.001,0.005,0.01, 0.025, 0.05}</td></tr><tr><td>Within-Task LR</td><td>{1,√2,2√2,4,4√2,8}</td><td>{1,√2,2√2,4,4√2,8}</td></tr><tr><td>L2 Clipping</td><td>{0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0}</td><td>{0.3, 0.4, 0.5, 0.6, 0.7, 0.8,0.9,1.0}</td></tr><tr><td>Refine LR</td><td>{0.1,0.15,0.3,0.5,0.7, 0.8}</td><td>{0.1,0.15,0.3,0.5,0.7, 0.8}</td></tr><tr><td>Refine Batch Size</td><td>{10,20,30,60}</td><td>{10,20,30,60,120}</td></tr><tr><td>Refine Epochs</td><td>{1,2,3}</td><td>{1,2,3}</td></tr></table>",
+        "bbox": [
+            173,
+            439,
+            828,
+            611
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/ae60aad18f415c26be568babe577b84fe499d89ae0e9a544a3b5eff10299ef05.jpg",
+        "table_caption": [
+            "Table 4: Hyperparameter Search Space for Local-DP Training "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Shakespeare-800</td><td>Wiki-1600</td></tr><tr><td>VisitsPerTask</td><td>{1,2,3}</td><td>{1,2}</td></tr><tr><td>Tasks Per Round</td><td>{5,10,20}</td><td>{10,20,40,80}</td></tr><tr><td>Within-Task Steps</td><td>{1,2,3}</td><td>{1,2,3}</td></tr><tr><td>Meta LR</td><td>{1,√2,2,2√2,4,4√2,8,8√2}</td><td>{1,√2,2,2√2,4,4√2,8,8√2}</td></tr><tr><td>Meta Decay Rate</td><td>{0,0.001,0.005,0.01,0.025,0.05,0.1}</td><td>{0,0.001,0.005,0.01,0.025,0.05}</td></tr><tr><td>Within-Task LR</td><td>{1,√2,2√2,4,4√2,8}</td><td>{1,√2,2√2,4,4√2,8}</td></tr><tr><td>L2 Clipping</td><td>{0.005,0.01,0.025,0.05,0.1,0.25,0.5}</td><td>{0.005,0.01,0.025,0.05,0.1,0.25}</td></tr><tr><td>Refine LR</td><td>{0.1,0.15,0.3,0.5,0.7,0.8}</td><td>{0.1,0.15,0.3,0.5, 0.7, 0.8}</td></tr><tr><td>Refine Batch Size</td><td>{10,20,30,60}</td><td>{10,20,30,60,120}</td></tr><tr><td>Refine Epochs</td><td>{1,2,3}</td><td>{1,2,3}</td></tr></table>",
+        "bbox": [
+            173,
+            715,
+            833,
+            886
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/29bc13fcf63794b0d224d308bafc3e14eabe13883efc74bc538b98a25facdbae.jpg",
+        "table_caption": [
+            "Table 5: Final Hyperparameters for Shakespeare-800 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Non- private</td><td>T-G £= 22.5</td><td>T-G £=9.2</td><td>T-G ε= 4.5</td><td>Local ε= 22.5</td><td>Local £=9.2</td><td>Local ε= 4.5</td></tr><tr><td>Visits Per Task</td><td>7</td><td>2</td><td>2</td><td>1</td><td>2</td><td>2</td><td>1</td></tr><tr><td>Tasks Per Round</td><td>5</td><td>5</td><td>5</td><td>5</td><td>20</td><td>5</td><td>20</td></tr><tr><td>Within-Task Steps</td><td>5</td><td>1</td><td>1</td><td>1</td><td>4</td><td>1</td><td>2</td></tr><tr><td>Meta LR</td><td>8</td><td>8√2</td><td>8</td><td>8√2</td><td>4√2</td><td>4√2</td><td>4</td></tr><tr><td>Meta Decay Rate</td><td>0.01</td><td>0.01</td><td>0.01</td><td>0.05</td><td>0.1</td><td>0</td><td>0</td></tr><tr><td>Within-Task LR</td><td>2</td><td>2√2</td><td>2</td><td>2</td><td>4√2</td><td>4√2</td><td>1</td></tr><tr><td>L2 Clipping</td><td>0.5</td><td>0.6</td><td>0.5</td><td>0.4</td><td>0.1</td><td>0.01</td><td>0.01</td></tr><tr><td>RefineLR</td><td>0.15</td><td>0.5</td><td>0.1</td><td>0.3</td><td>0.8</td><td>0.8</td><td>0.5</td></tr><tr><td>Refine Batch Size</td><td>30</td><td>10</td><td>60</td><td>30</td><td>10</td><td>10</td><td>10</td></tr><tr><td>Refine Epochs</td><td>1</td><td>1</td><td>1</td><td>3</td><td>3</td><td>3</td><td>3</td></tr></table>",
+        "bbox": [
+            187,
+            227,
+            810,
+            407
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "table",
+        "img_path": "images/0714e27f83a5d91015438973490a4627510a5b29f9b6b7154b6b83435f9db32b.jpg",
+        "table_caption": [
+            "Table 6: Final Hyperparameters for Wiki-800 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Non- private</td><td>T-G £= 22.5</td><td>T-G £=9.2</td><td>T-G ε= 4.5</td><td>Local £= 22.5</td><td>Local £=9.2</td><td>Local £= 4.5</td></tr><tr><td>VisitsPerTask</td><td>2</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Tasks Per Round</td><td>5</td><td>10</td><td>10</td><td>20</td><td>20</td><td>20</td><td>20</td></tr><tr><td>Within-Task Steps</td><td>3</td><td>1</td><td>1</td><td>1</td><td>2</td><td>2</td><td>2</td></tr><tr><td>Meta LR</td><td>2√2</td><td>2√2</td><td>4</td><td>4</td><td>8</td><td>4√2</td><td>8</td></tr><tr><td>Meta Decay Rate</td><td>0.001</td><td>0</td><td>0.001</td><td>0.005</td><td>0.005</td><td>0.025</td><td>0</td></tr><tr><td>Within-Task LR</td><td>2</td><td>8</td><td>4√2</td><td>8</td><td>2√2</td><td>2√2</td><td>2√2</td></tr><tr><td>L2 Clipping</td><td>1</td><td>0.8</td><td>0.7</td><td>0.8</td><td>0.025</td><td>0.05</td><td>0.005</td></tr><tr><td>Refine LR</td><td>0.1</td><td>0.8</td><td>0.5</td><td>0.7</td><td>0.8</td><td>0.8</td><td>0.8</td></tr><tr><td>Refine Batch Size</td><td>10</td><td>10</td><td>60</td><td>10</td><td>10</td><td>10</td><td>10</td></tr><tr><td>Refine Epochs</td><td>1</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>3</td></tr></table>",
+        "bbox": [
+            187,
+            640,
+            810,
+            821
+        ],
+        "page_idx": 17
+    }
+]