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Question: The backpropagation (BP) algorithm is often thought to be biologically implausible in the brain. One of the main reasons is that BP requires symmetric weight matrices in the feedforward and feedback pathways. To address this “weight transport problem” , two biologically-plausible algorithms, proposed by and , relax BP’s weight symmetry requirements and demonstrate comparable learning capabilities to that of BP on small datasets. However, a recent study by finds that although feedback alignment (FA) and some variants of target-propagation (TP) perform well on MNIST and CIFAR, they perform significantly worse than BP on ImageNet. Here, we additionally evaluate the sign-symmetry (SS) algorithm , which differs from both BP and FA in that the feedback and feedforward weights do not share magnitudes but share signs. We examined the performance of sign-symmetry and feedback alignment on ImageNet and MS COCO datasets using different network architectures (ResNet-18 and AlexNet for ImageNet; RetinaNet for MS COCO). Surprisingly, networks trained with sign-symmetry can attain classification performance approaching that of BP-trained networks. These complement the study by and establish a new benchmark for future biologically-plausible learning algorithms on more difficult datasets and more complex architectures. Deep learning models today are highly successful in task performance, learning useful representations, and even matching representations in the brain BID26 BID24. However, it remains a contentious issue whether these models reflect how the brain learns. Core to the problem is the fact that backpropagation, the learning algorithm underlying most of today's deep networks, is difficult to implement in the brain given what we know about the brain's hardware BID2 however, see Hinton 2007). One main reason why backpropagation seems implausible in the brain is that it requires sharing of feedforward and feedback weights. Since synapses are unidirectional in the brain, feedforward and feedback connections are physically distinct. Requiring them to shared their weights, even as weights are adjusted during learning, seems highly implausible. One approach to addressing this issue is to relax the requirement for weight-symmetry in error backpropagation. Surprisingly, when the feedback weights share only the sign but not the magnitude of the feedforward weights BID16 or even when the feedback weights are random (but fixed) BID17, they can still guide useful learning in the network, with performance comparable to and sometimes even better than performance of backpropagation, on datasets such as MNIST and CIFAR. Here, we refer to these two algorithms, respectively, as "sign-symmetry" and "feedback alignment." Since weight symmetry in backpropagation is required for accurately propagating the derivative of the loss function through layers, the success of asymmetric feedback algorithms indicates that learning can be supported even by inaccurate estimation of the error derivative. In feedback alignment, the authors propose that the feedforward weights learn to align with the random feedback weights, thereby allowing feedback to provide approximate yet useful learning signals BID17.However, a recent paper by BID0 finds that feedback alignment and a few other biologically-plausible algorithms, including variants of target propagation, do not generalize to larger and more difficult problems such as ImageNet BID4 ) and perform much worse than backpropagation. Nevertheless, the specific conditions Bartunov et al. tested are somewhat restrictive. They only tested locally-connected networks (i.e., weight sharing is not allowed among convolution filters at different spatial locations), a choice that is motivated by biological plausibility but in practice limits the size of the network (without weight sharing, each convolutional layer needs much more memory to store its weights), making it unclear whether poor performance was attributable solely to the algorithm, or to the algorithm on those architectures.1 Second, Bartunov et al. did not test sign-symmetry, which may be more powerful than feedback alignment since signsymmetric feedback weights may carry more information about the feedforward weights than the random feedback weights used in feedback alignment. In this work, we re-examine the performance of sign-symmetry and feedback alignment on ImageNet and MS COCO datasets using standard ConvNet architectures (i.e., ResNet-18, AlexNet, and RetinaNet). We find that sign-symmetry can in fact train networks on both tasks, achieving similar performance to backpropagation on ImageNet and reasonable performance on MS COCO. In addition, we test the use of backpropagation exclusively in the last layer while otherwise using feedback alignment, hypothesizing that in the brain, the classifier layer may not be a fully-connected layer and may deliver the error signal through some other unspecified mechanism. Such partial feedback alignment can achieve better performance (relative to backpropagation) than in BID0. Taken together, these extend previous findings and indicate that existing biologicallyplausible learning algorithms remain viable options both for training artificial neural networks and for modeling how learning can occur in the brain. Consider a layer in a feedforward neural network. Let x i denote the input to the i th neuron in the layer and y j the output of the j th neuron. Let W denote the feedforward weight matrix and W ij the connection between input x i and output y j. Let f denote the activation function. Then, Equation 1 describes the computation in the feedforward step. Now, let B denote the feedback weight matrix and B ij the feedback connection between output y j and input x i, and let f denote the derivative of the activation function f. Given the objective function E, the error gradient ∂E ∂xi calculated in the feedback step is described by Equation 2. DISPLAYFORM0 DISPLAYFORM1 Standard backpropagation requires B = W. Sign-symmetry BID16 relaxes the above symmetry requirement by letting B = sign(W), where sign(·) is the (elementwise) sign function. Feedback alignment BID17 uses a fixed random matrix as the feedback weight matrix B. Lillicrap et al. showed that through training, W is adjusted such that on average, e T W Be > 0, where e is the error in the network's output. This condition implies that the error correction signal Be lies within 90• of e T W, the error calculated by standard backpropagation. We implement both algorithms in PyTorch for convolutional and fully-connected layers and post the code at https://github.com/willwx/sign-symmetry. We trained ResNet-18 BID6 on ImageNet using 5 different training settings: 1) backpropagation; 2) sign-symmetry for convolutional layers and backpropagation for the last, fully-connected layer; 3) sign-symmetry for all (convolutional and fully-connected) layers; 4) feedback alignment for convolutional layers and backpropagation for the fully-connected layer; and 5) feedback alignment for all (convolutional and fully-connected) layers. In sign-symmetry, at each backward step, feedback weights were taken as the signs of the feedforward weights, scaled by the same scale λ used to initialize that layer. 2 In feedback alignment, feedback weights were initialized once at the beginning as random variables from the same distribution used to initialize that layer. For backpropagation, standard training parameters were used (SGD with learning rate 0.1, momentum 0.9, and weight decay 10 −4). For ResNet-18 with other learning algorithms, we used SGD with learning rate 0.05 3, while momentum and weight decay remain unchanged. For AlexNet with all learning algorithms, standard training parameters were used (SGD with learning rate 0.01, momentum 0.9, and weight decay 5 × 10 −4). We used a version of AlexNet BID13, as used in torchvision) which we slightly modified to add batch normalization BID9 before every nonlinearity and consequently removed dropout. For all experiments, we used a batch size of 256, a learning rate decay of 10-fold every 10 epochs, and trained for 50 epochs. BID12. Sign-symmetry performed nearly as well as backpropagation, while feedback alignment performed better than previously reported when backpropagation was used to train the last layer. In all cases, the network was able to learn (FIG0, TAB0). Remarkably, sign-symmetry only slightly underperformed backpropagation in this benchmark large dataset, despite the fact that signsymmetry does not accurately propagate either the magnitude or the sign of the error gradient. Hence, this is not predicted by the performance of signSGD BID1, where weight updates use the sign of the gradients, but gradients are still calculate accurately; or XNORNet BID22, where both feedforward and feedback weights are binary but symmetrical, so error backpropagation is still accurate. An intuitive explanation for this performance is that the skip-connections in ResNet help prevent the degradation of the gradient being passed through many layers of sign-symmetric feedback. However, sign-symmetry also performed similarly well to backpropagation in a (modified) AlexNet architecture, which did not contain skip connections. Therefore, skip-connections alone do not explain the performance of sign-symmetry. In addition, although its performance was considerably worse, feedback alignment was still able to guide better learning in the network than reported by BID0 Figure 3 ) if we use backpropagation in the last layer. This condition is not unreasonable since, in the brain, the classifier layer is likely not a soft-max classifier and may deliver error signals by a different mechanism. We also tested using backpropagation exclusively for the last layer in a network otherwise trained with sign-symmetry, but the effect on the performance was minimal. One possibility why sign-symmetry performed better than feedback alignment is that in sign-symmetry, the feedback weight always tracks the sign of the feedforward weight, which may reduce the burden on the feedforward weight to learn to align with the feedback weight. Finally, in BID16, Batch-Manhattan (BM) SGD was proposed as a way to stabilize training with asymmetric feedback algorithms. In our experience, standard SGD consistently worked better than BM for sign-symmetry, but BM may improve for feedback alignment. We have not comprehensively characterized the effects of BM since many factors like learning rate can affect the outcome. Future experiments are needed to draw stronger . Besides the ImageNet classification task, we examined the performance of sign-symmetry on the MS COCO object detection task. Object detection is more complex than classification and might therefore require more complicated network architecture in order to achieve high accuracy. Thus, in this experiment we assessed the effectiveness of sign-symmetry in training networks that were more complicated and difficult to optimize. We trained the state-of-the-art object detection network RetinaNet proposed by BID18 on the COCO trainval35k split, which consists of 80k images from train and 35k random images from the 40k-image val set. RetinaNet comprises a ResNet-FPN backbone, a classification subnet, and a bounding box regressing subnet. The network was trained with three different training settings: 1) backpropagation for all layers; 2) backpropagation for the last layer in both subnets and sign-symmetry for rest of the layers; 3) backpropagation for the last layer in both subnets and feedback alignment for rest of the layers. We used a backbone ResNet-18 pretrained on ImageNet to initialize the network. In all the experiments, the network was trained with SGD with an initial learning rate of 0.01, momentum of 0.9, and weight decay of 0.0001. We trained the network for 40k iterations with 8 images in each minibatch. The learning rate was divided by 10 at iteration 20k. The on COCO are similar to those on ImageNet, although the performance gap between SS and BP on COCO is slightly more prominent FIG1. A number of factors could have potentially contributed to this . We followed the Feature Pyramid Network (FPN) architecture design choices, optimizers, and hyperparameters reported by BID18; these choices are all optimized for use with backpropagation instead of sign-symmetry. Hence, the here represent a lowerbound on the performance of sign-symmetry for training networks on the COCO dataset. We ran a number of analyses to understand how sign-symmetry guides learning. BID17 show that with feedback alignment, the alignment angles between feedforward and feedback weights gradually decrease because the feedforward weights learn to align with the feedback weights. We asked whether the same happens in sign-symmetry by computing alignment angles as in BID17: For every pair of feedforward and feedback weight matrices, we flattened the matrices into vectors and computed the angle between the vectors. Interestingly, we found that during training, the alignment angles decreased for the last 3 layers but increased for the other layers (Figure 3a). In comparison, in the backpropagation-trained network (where sign(W) was not used in any way), the analogous alignment angle between W and sign(W) increased for all layers. One possible explanation for the increasing trend is that as the training progresses, the feedforward weights tend to become sparse. Geometrically, this means that feedforward vectors become more aligned to the standard basis vectors and less aligned with the feedback weight vectors, which always lie on a diagonal by construction. This explanation is consistent with the similarly increasing trend of the average kurtosis of the feedforward weights (Figure 3b), which indicates that values of the weights became more dispersed during training. Since the magnitudes of the feedforward weights were discarded when calculating the error gradients, we also looked at how sign-symmetry affected the size of the trained weights. Sign-symmetry and backpropagation ed in weights with similar magnitudes (Figure 3c). More work is needed to elucidate how sign-symmetry guides efficient learning in the network. Our indicate that biologically-plausible learning algorithms, specifically sign-symmetry and feedback alignment, are able to learn on ImageNet. This finding seemingly conflicts with the findings by BID0. Why do we come to such different ?First, Bartunov et al. did not test sign-symmetry, which is expected to be more powerful than feedback alignment, because it is a special case of feedback alignment that allows feedback weights to have additional information about feedforward weights. Indeed, on ImageNet, the performance of sign-symmetry approached that of backpropagation and exceeded the performance of feedback alignment by a wide margin. Another reason may be that instead of using standard ConvNets on ImageNet, Bartunov et al. only tested locally-connected networks. While the later is a more biologically plausible architecture, in practice, it is limited in size by the need to store separate weights Figure 3: a, During training with sign-symmetry, alignment angles between feedforward weights W and feedback weights sign(W) decreased in the last 3 layers but increased in early layers, whereas during training with backpropagation, the analogous alignment angles increased for all layers and were overall larger. b, Kurtosis of the feedforward weight matrices increased during training. c, The magnitudes of weights trained by sign-symmetry were similar to those trained by backpropagation. Line and shading, mean ± std for epoch 50.for each spatial location. This reduced model capacity creates a bottleneck that may affect the performance of feedback alignment (see , Supplementary Note 9). Finally, the performance of feedback alignment also benefited from the use of backpropagation in the last layer in our conditions. A major reason why backpropagation is considered implausible in the brain is that it requires exact symmetry of physically distinct feedforward and feedback pathways. Sign-symmetry and feedback alignment address this problem by relaxing this tight coupling of weights between separate pathways. Feedback alignment requires no relation at all between feedforward and feedback weights and simply depends on learning to align the two. Hence, it can be easily realized in the brain (for example, see , Supplementary Figure 3). However, empirically, we and others have found its performance to be not ideal on relatively challenging problems. Sign-symmetry, on the other hand, introduces a mild constraint that feedforward and feedback connections be "antiparallel": They need to have opposite directions but consistent signs. This can be achieved in the brain with two additional yet plausible conditions: First, the feedforward and feedback pathways must be specifically wired in this antiparallel way. This can be achieved by using chemical signals to guide specific targeting of axons, similar to how known mechanisms for specific wiring operate in the brain BID20 BID8. One example scheme of how this can be achieved is shown in Figure 4. While the picture in Figure 4a is complex, most of the complexity comes from the fact that units in a ConvNet produce inconsistent outputs (i.e., both positive and negative). If the units are consistent (i.e., producing exclusively positive or negative outputs), the picture simplifies to Figure 4b. Neurons in the brain are observed to be consistent, as stated by the so-called "Dale's Law" BID3 BID25. Hence, this constraint would have to be incorporated at some point in any biologically plausible network, and remains an important direction for future work. We want to remark that Figure 4 is meant to indicate the relative ease of wiring sign-symmetry in the brain (compared to, e.g., wiring a network capable of weight transport), not that the brain is known to be wired this way. Nevertheless, it represents a hypothesis that is falsifiable by experimental data, potentially in the near future. Related, a second desideratum is that weights should not change sign during training. While our current setting for sign-symmetry removes weight magnitude transport, it still implicitly relies on "sign transport." However, in the brain, the sign of a connection weight depends on the type of the 4 A paper from last year examined connectivity patterns within tissue sizes of approx. 500 microns and axon lengths of approx. 250 microns BID23; recent progress (fueled by deep learning) can trace axons longer than 1 mm, although the imaging of large brain volumes is still limiting. In comparison, in mice, adjacent visual areas (corresponding to stages of visual processing) are 0.5-several mms apart BID19, while in primates it is tens of millimeters. Thus, testing the reality of sign-symmetric wiring is not quite possible today but potentially soon to be. presynaptic neuron-e.g., glutamatergic (excitatory) or GABAergic (inhibitory)-a quality that is intrinsic to and stable for each neuron given existing evidence. Hence, if sign-symmetry is satisfied initially-for example, through specific wiring as just described-it will be satisfied throughout learning, and "sign transport" will not be required. Thus, evaluating the capacity of sign-fixed networks to learn is another direction for future work. Figure 4: The specific wiring required for sign-symmetric feedback can be achieved using axonal guidance by specific receptor-ligand recognition. Assume that an axon carrying ligand L X will only synapse onto a downstream neuron carrying the corresponding receptor R X. By expressing receptors and ligands in an appropriate pattern, an antiparallel wiring pattern can be established that supports sign-symmetric feedback. a, An example scheme. In this scheme, one inconsistent unit (i.e., a unit that produce both positive and negative outputs) in the network is implemented by three consistent biological neurons, so that each synapse is exclusively positive or negative. n input neurons orthogonal ligand-receptor pairs is sufficient to implement all possible connection patterns. b, An example scheme for implementing a signsymmetric network with consistent units. Only 2 orthogonal ligand-receptor pairs are needed to implement all possible connectivities in this case. These schemes represent falsifiable hypotheses, although they do not exclude other possible implementations. Another element of unclear biological reality, common to feedback alignment and sign-symmetry, is that the update of a synaptic connection (i.e., weight) between two feedforward neurons (A to B) depends on the activity in a third, feedback neuron C, whose activation represents the error of neuron B. One way it can be implemented biologically is for neuron C to connect to B with a constant and fixed weight. When C changes its value due to error feedback, it will directly induce a change of B's electric potential and thus of the postsynaptic potential of the synapse between A and B, which might lead to either Long-term Potentiation (LTP) or Long-term Depression (LTD) of synapse A-B.Biological plausibility of ResNet has been previously discussed by BID14, claiming that ResNet corresponds to an unrolled recurrent network in the visual cortex. However, it is unclear yet how backpropagation through time can be implemented in the brain. Biological plausibility of batch normalization has been discussed in BID15, where they addressed the issues with online learning (i.e., one sample at a time, instead of minibatch), recurrent architecture and consistent training and testing normalization statistics. Other biological constraints include removing weight-sharing in convolutional layers as in BID0, incorporating temporal dynamics as in BID17, using realistic spiking neurons, addressing the sample inefficiency general to deep learning, etc. We believe that these are important yet independent issues to the problem of weight transport and that by removing the latter, we have taken a meaningful step toward biological plausibility. Nevertheless, many steps remain in the quest for a truly plausible, effective, and empirically-verified model of learning in the brain. Recent work shows that biologically-plausible learning algorithms do not scale to challenging problems such as ImageNet. We evaluated sign-symmetry and re-evaluated feedback alignment on their effectiveness training ResNet and AlexNet on ImageNet and RetinaNet on MS COCO. We find that 1) sign-symmetry performed nearly as well as backpropagation on ImageNet, 2) slightly modified feedback alignment performed better than previously reported, and 3) both algorithms had reasonable performance on MS COCO with minimal hyperparameter tuning. Taken together, these indicate that biologically-plausible learning algorithms, in particular sign-symmetry, remain promising options for training artificial neural networks and modeling learning in the brain.

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Biologically plausible learning algorithms, particularly sign-symmetry, work well on ImageNet

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For example, We learn an unsupervised learning algorithm that produces useful representations from a set of supervised tasks. At test-time, we apply this algorithm to new tasks without any supervision and show performance comparable to a VAE. should be similar to A major goal of unsupervised learning is to discover data representations that are useful for subsequent tasks, without access to supervised labels during training. Typically, this involves minimizing a surrogate objective, such as the negative log likelihood of a generative model, with the hope that representations useful for subsequent tasks will arise as a side effect. In this work, we propose instead to directly target later desired tasks by meta-learning an unsupervised learning rule which leads to representations useful for those tasks. Specifically, we target semi-supervised classification performance, and we meta-learn an algorithm -- an unsupervised weight update rule -- that produces representations useful for this task. Additionally, we constrain our unsupervised update rule to a be a biologically-motivated, neuron-local function, which enables it to generalize to different neural network architectures, datasets, and data modalities. We show that the meta-learned update rule produces useful features and sometimes outperforms existing unsupervised learning techniques. We further show that the meta-learned unsupervised update rule generalizes to train networks with different widths, depths, and nonlinearities. It also generalizes to train on data with randomly permuted input dimensions and even generalizes from image datasets to a text task. Supervised learning has proven extremely effective for many problems where large amounts of labeled training data are available. There is a common hope that unsupervised learning will prove similarly powerful in situations where labels are expensive, impractical to collect, or where the prediction target is unknown during training. Unsupervised learning however has yet to fulfill this promise. One explanation for this failure is that unsupervised representation learning algorithms are typically mismatched to the target task. Ideally, learned representations should linearly expose high level attributes of data (e.g. object identity) and perform well in semi-supervised settings. Many current unsupervised objectives, however, optimize for objectives such as log-likelihood of a generative model or reconstruction error, producing useful representations only as a side effect. Unsupervised representation learning seems uniquely suited for meta-learning BID0 ). Unlike most tasks where meta-learning is applied, unsupervised learning does not define an explicit objective, which makes it impossible to phrase the task as a standard optimization problem. It is possible, however, to directly express a meta-objective that captures the quality of representations produced by an unsupervised update rule by evaluating the usefulness of the representation for candidate tasks. In this work, we propose to meta-learn an unsupervised update rule by meta-training on a meta-objective that directly optimizes the utility of the unsupervised representation. Unlike hand-designed unsupervised learning rules, this meta-objective directly targets the usefulness of a representation generated from unlabeled data for later supervised tasks. By recasting unsupervised representation learning as meta-learning, we treat the creation of the unsupervised update rule as a transfer learning problem. Instead of learning transferable features, we learn a transferable learning rule which does not require access to labels and generalizes across both data domains and neural network architectures. Although we focus on the meta-objective of semi-supervised classification here, in principle a learning rule could be optimized to generate representations for any subsequent task. Unsupervised learning is a topic of broad and diverse interest. Here we briefly review several techniques that can lead to a useful latent representation of a dataset. In contrast to our work, each method imposes a manually defined training algorithm or loss function whereas we learn the algorithm that creates useful representations as determined by a meta-objective. Autoencoders work by first compressing and optimizing reconstruction loss. Extensions have been made to de-noise data (; 2010), as well as compress information in an information theoretic way . further explored scaling up these unsupervised methods to large image datasets. Generative adversarial networks take another approach to unsupervised feature learning. Instead of a loss function, an explicit min-max optimization is defined to learn a generative model of a data distribution. Recent work has shown that this training procedure can learn unsupervised features useful for few shot learning (; ;).Other techniques rely on self-supervision where labels are easily generated to create a non-trivial'supervised' loss. Domain knowledge of the input is often necessary to define these losses. use unscrambling jigsaw-like crops of an image. Techniques used by and rely on using temporal ordering from videos. Another approach to unsupervised learning relies on feature space design such as clustering. showed that k-means can be used for feature learning. jointly learn features and cluster assignments. develop a scalable technique to cluster by predicting noise. Other techniques such as , , and define various desirable properties about the latent representation of the input, such as predictability, complexity of encoding mapping, independence, or sparsity, and optimize to achieve these properties. Most meta-learning algorithms consist of two levels of learning, or'loops' of computation: an inner loop, where some form of learning occurs (e.g. an optimization process), and an outer loop or metatraining loop, which optimizes some aspect of the inner loop, parameterized by meta-parameters. The performance of the inner loop computation for a given set of meta-parameters is quantified by a meta-objective. Meta-training is then the process of adjusting the meta-parameters so that the inner loop performs well on this meta-objective. Meta-learning approaches differ by the computation performed in the inner loop, the domain, the choice of meta-parameters, and the method of optimizing the outer loop. Some of the earliest work in meta-learning includes work by , which explores a variety of meta-learning and self-referential algorithms. Similarly to our algorithm, Bengio et al. (1990; 1992) propose to learn a neuron local learning rule, though their approach differs in task and problem formulation. meta-learn supervised learning rules which mix local and global network information. A number of papers propose meta-learning for few shot learning (; ; ; ;), though these do not take advantage of unlabeled data. Others make use of both labeled and unlabeld data . uses a task created with no supervision to then train few-shot detectors. use meta-learning for unsupervised learning, primarily in the context of clustering and with a small number of meta-parameters. Figure 1: Left: Schematic for meta-learning an unsupervised learning algorithm. The inner loop computation consists of iteratively applying the UnsupervisedUpdate to a base model. During metatraining the UnsupervisedUpdate (parameterized by θ) is itself updated by gradient descent on the MetaObjective. Right: Schematic of the base model and UnsupervisedUpdate. Unlabeled input data, x 0, is passed through the base model, which is parameterised by W and colored green. The goal of the UnsupervisedUpdate is to modify W to achieve a top layer representation x L which performs well at few-shot learning. In order to train the base model, information is propagated backwards by the UnsupervisedUpdate in a manner analogous to backprop. Unlike in backprop however, the backward weights V are decoupled from the forward weights W. Additionally, unlike backprop, there is no explicit error signal as there is no loss. Instead at each layer, and for each neuron, a learning signal is injected by a meta-learned MLP parameterized by θ, with hidden state h. Weight updates are again analogous to those in backprop, and depend on the hidden state of the pre-and postsynaptic neurons for each weight. To allow easy comparison against other existing approaches, we present a more extensive survey of previous work in meta-learning in table form in TAB1, highlighting differences in choice of task, structure of the meta-learning problem, choice of meta-architecture, and choice of domain. To our knowledge, we are the first meta-learning approach to tackle the problem of unsupervised representation learning, where the inner loop consists of unsupervised learning. This contrasts with transfer learning, where a neural network is instead trained on a similar dataset, and then fine tuned or otherwise post-processed on the target dataset. We additionally believe we are the first representation meta-learning approach to generalize across input data modalities as well as datasets, the first to generalize across permutation of the input dimensions, and the first to generalize across neural network architectures (e.g. layer width, network depth, activation function). We consider a multilayer perceptron (MLP) with parameters φ t as the base model. The inner loop of our meta-learning process trains this base model via iterative application of our learned update rule. See Figure 1 for a schematic illustration and Appendix A for a more detailed diagram. In standard supervised learning, the'learned' optimizer is stochastic gradient descent (SGD). A supervised loss l (x, y) is associated with this model, where x is a minibatch of inputs, and y are the corresponding labels. The parameters φ t of the base model are then updated iteratively by performing SGD using the gradient ∂l(x,y) ∂φt. This supervised update rule can be written as φ t+1 = SupervisedUpdate(φ t, x t, y t ; θ), where t denotes the inner-loop iteration or step. Here θ are the meta-parameters of the optimizer, which consist of hyper-parameters such as learning rate and momentum. In this work, our learned update is a parametric function which does not depend on label information, φ t+1 = UnsupervisedUpdate(φ t, x t ; θ). This form of the update rule is general, it encompasses many unsupervised learning algorithms and all methods in Section 2.1.In traditional learning algorithms, expert knowledge or a simple hyper-parameter search determines θ, which consists of a handful of meta-parameters such as learning rate and regularization constants. In contrast, our update rule will have orders of magnitude more meta-parameters, including the weights of a neural network. We train these meta-parameters by performing SGD on the sum of the MetaObjective over the course of (inner loop) training in order to find optimal parameters θ *, DISPLAYFORM0 that minimize the meta-objective over a distribution of training tasks. Note that φ t is a function of θ since θ affects the optimization trajectory. In the following sections, we briefly review the main components of this model: the base model, the UnsupervisedUpdate, and the MetaObjective. See the Appendix for a complete specification. Additionally, code and meta-trained parameters θ for our meta-learned UnsupervisedUpdate is available 1. Our base model consists of a standard fully connected multi-layer perceptron (MLP), with batch normalization BID6, and ReLU nonlinearities. We chose this as opposed to a convolutional model to limit the inductive bias of convolutions in favor of learned behavior from the UnsupervisedUpdate. We call the pre-nonlinearity activations z 1, · · ·, z L, and post-nonlinearity activations DISPLAYFORM0 where L is the total number of layers, and x 0 ≡ x is the network input (raw data). The parameters are DISPLAYFORM1 where W l and b l are the weights and biases (applied after batch norm) for layer l, and V l are the corresponding weights used in the backward pass. We wish for our update rule to generalize across architectures with different widths, depths, or even network topologies. To achieve this, we design our update rule to be neuron-local, so that updates are a function of pre-and post-synaptic neurons in the base model, and are defined for any base model architecture. This has the added benefit that it makes the weight updates more similar to synaptic updates in biological neurons, which depend almost exclusively on local pre-and post-synaptic neuronal activity BID7. In practice, we relax this constraint and incorporate some cross neuron information to decorrelate neurons (see Appendix G.5 for more information).To build these updates, each neuron i in every layer l in the base model has an MLP, referred to as an update network, associated with it, with output h DISPLAYFORM0; θ where b indexes the training minibatch. The inputs to the MLP are the feedforward activations (x l & z l) defined above, and feedback weights and an error signal (V l and δ l, respectively) which are defined below. All update networks share meta-parameters θ. Evaluating the statistics of unit activation over a batch of data has proven helpful in supervised learning BID6. It has similarly proven helpful in hand-designed unsupervised learning rules, such as sparse coding and clustering. We therefore allow h l bi to accumulate statistics across examples in each training minibatch. During an unsupervised training step, the base model is first run in a standard feed-forward fashion, populating x l bi, z l bi. As in supervised learning, an error signal δ l bi is then propagated backwards through the network. Unlike in supervised backprop, however, this error signal is generated by the corresponding update network for each unit. It is read out by linear projection of the per-neuron hidden state h, δ l bi = lin h l bi, and propogated backward using a set of learned'backward weights' (V l) T, rather than the transpose of the forward weights (W l) T as would be the case in backprop (diagrammed in Figure 1). This is done to be more biologically plausible.Again as in supervised learning, the weight updates (∆W l) are a product of pre-and post-synaptic signals. Unlike in supervised learning however, these signals are generated using the per-neuron update networks: DISPLAYFORM1 bj, W ij. The full weight update (which involves normalization and decorrelation across neurons) is defined in Appendix G.5. The meta-objective determines the quality of the unsupervised representations. In order to meta-train via SGD, this loss must be differentiable. The meta-objective we use in this work is based on fitting a linear regression to labeled examples with a small number of data points. In order to encourage the learning of features that generalize well, we estimate the linear regression weights on one minibatch {x a, y a} of K data points, and evaluate the classification performance on a second minibatch {x b, y b} also with K datapoints, DISPLAYFORM0 are features extracted from the base model on data x a, x b, respectively. The target labels y a, y b consist of one hot encoded labels and potentially also regression targets from data augmentation (e.g. rotation angle, see Section 4.2). We found that using a cosine distance, CosDist, rather than unnormalized squared error improved stability. Note this meta-objective is only used during meta-training and not used when applying the learned update rule. The inner loop computation is performed without labels via the UnsupervisedUpdate. We choose to meta-optimize via SGD as opposed to reinforcement learning or other black box methods, due to the superior convergence properties of SGD in high dimensions, and the high dimensional nature of θ. Training and computing derivatives through long recurrent computation of this form is notoriously difficult BID9. To improve stability and reduce the computational cost we approximate the gradients DISPLAYFORM0 via truncated backprop through time BID10. Many additional design choices were also crucial to achieving stability and convergence in meta-learning, including the use of batch norm, and restricting the norm of the UnsupervisedUpdate update step (a full discussion of these and other choices is in Appendix B). Generalization in our learned optimizer comes from both the form of the UnsupervisedUpdate (Section 3.2), and from the meta-training distribution. Our meta-training distribution is composed of both datasets and base model architectures. We construct a set of training tasks consisting of CIFAR10 BID11 ) and multi-class classification from subsets of classes from Imagenet BID12 as well as from a dataset consisting of rendered fonts (Appendix H.1.1). We find that increased training dataset variation actually improves the meta-optimization process. To reduce computation we restrict the input data to 16x16 pixels or less during meta-training, and resize all datasets accordingly. For evaluation, we use MNIST BID13, Fashion MNIST BID14, IMDB BID15, and a hold-out set of Imagenet classes. We additionally sample the base model architecture. We sample number of layers uniformly between 2-5 and the number of units per layer logarithmically between 64 to 512.As part of preprocessing, we permute all inputs along the feature dimension, so that the UnsupervisedUpdate must learn a permutation invariant learning rule. Unlike other work, we focus explicitly on learning a learning algorithm as opposed to the discovery of fixed feature extractors that generalize across similar tasks. This makes the learning task much harder, as the UnsupervisedUpdate has to discover the relationship between pixels based solely on their joint statistics, and cannot "cheat" and memorize pixel identity. To provide further dataset variation, we additionally augment the data with shifts, rotations, and noise. We add these augmentation coefficients as additional regression targets for the meta-objective-e.g. rotate the image and predict the rotation angle as well as the image class. For additional details, see Appendix H.1.1. We implement the above models in distributed TensorFlow BID16. Training uses 512 workers, each of which performs a sequence of partial unrolls of the inner loop UnsupervisedUpdate, and computes gradients of the meta-objective asynchronously. Training takes ∼8 days, and consists of ∼200 thousand updates to θ with minibatch size 256. Additional details are in Appendix C. First, we examine limitations of existing unsupervised and meta learning methods. Then, we show meta-training and generalization properties of our learned optimizer and finally we conclude by visualizing how our learned update rule works. For details of the experimental setup, see Appendix H. To illustrate the negative consequences of objective function mismatch in unsupervised learnin algorithms, we train a variational autoencoder on 16x16 CIFAR10. Over the course of training we evaluate classification performance from few shot classification using the learned latent representations. Training curves can be seen in FIG0. Despite continuing to improve the VAE objective throughout training (not shown here), the classification accuracy decreases sharply later in training. To demonstrate the reduced generalization that from learning transferable features rather than an update algorithm, we train a prototypical network with and without the input shuffling described in Section 4.2. As the prototypical network primarily learns transferrable features, performance is significantly hampered by input shuffling. Results are in Continuing to optimize a variational auto-encoder (VAE) hurts few-shot accuracy after some number of steps (dashed line). Right: Prototypical networks transfer features rather than a learning algorithm, and perform poorly if tasks don't have consistent data structure. Training a prototypical network with a fully connected architecture (same as our base model) on a MiniImagenet 10-way classification task with either intact inputs (light purple) or by permuting the pixels before every training and testing task (dark purple). Performance with permuted inputs is greatly reduced (gray line). Our performance is invariant to pixel permutation. While training, we monitor a rolling average of the meta-objective averaged across all datasets, model architectures, and the number of unrolling steps performed. In FIG2 the training loss is continuing to decrease after 200 hours of training, which suggests that the approximate training techniques still produce effective learning. In addition to this global number, we measure performance obtained by rolling out the UnsupervisedUpdate on various meta-training and meta-testing datasets. We see that on held out image datasets, such as MNIST and Fashion Mnist, the evaluation loss is still decreasing. However, for datasets in a different domain, such as IMDB sentiment prediction BID15, we start to see meta-overfitting. For all remaining experimental , unless otherwise stated, we use meta-parameters, θ, for the UnsupervisedUpdate ing from 200 hours of meta-training. The goal of this work is to learn a general purpose unsupervised representation learning algorithm. As such, this algorithm must be able to generalize across a wide range of scenarios, including tasks that are not sampled i.i.d. from the meta-training distribution. In the following sections, we explore a subset of the factors we seek to generalize over. In FIG3, we compare performance on few shot classification with 10 examples per class. We evaluate test performance on holdout datasets of MNIST and Fashion MNIST at 2 resolutions: 14×14 and 28×28 (larger than any dataset experienced in meta-training). On the same base model architecture, our learned UnsupervisedUpdate leads to performance better than a variational autoencoder, supervised learning on the labeled examples, and random initialization with trained readout layer. The learned UnsupervisedUpdate generalizes to unseen datasets. Our learned update rule produces representations more suitable for few shot classification than those from random initialization or a variational autoecoder and outperforms fully supervised learning on the same labeled examples. Error bars show standard error. Right: Early in meta-training (purple), the UnsupervisedUpdate is able to learn useful features on a 2 way text classification data set, IMDB, despite being meta-trained only from image datasets. Later in meta-training (red) performance drops due to the domain mismatch. We show inner-loop training, consisting of 5k applications of the UnsupervisedUpdate evaluating the MetaObjective each iteration. Error bars show standard error across 10 runs. To further explore generalization limits, we test our learned optimizer on data from a vastly different domain. We train on a binary text classification dataset: IMDB movie reviews BID15, encoded by computing a bag of words with 1K words. We evaluate using a model 30 hours and 200 hours into meta-training (see FIG3 . Despite being trained exclusively on image datasets, the 30 hour learned optimizer improves upon the random initialization by almost 10%. When meta-training for longer, however, the learned optimizer "meta-overfits" to the image domain ing in poor performance. This performance is quite low in an absolute sense, for this task. Nevertheless, we find this very exciting as we are unaware of any work showing this kind of transfer of learned rules from images to text. We train models of varying depths and unit counts with our learned optimizer and compare at different points in training ( FIG4). We find that despite only training on networks with 2 to 5 layers and 64 to 512 units per layer, the learned rule generalizes to 11 layers and 10,000 units per layer. ReLU Leaky ReLU ELU Softplus Swish Tanh The learned UnsupervisedUpdate is capable of optimizing base models with hidden sizes and depths outside the meta-training regime. As we increase the number of units per layer, the learned model can make use of this additional capacity despite never having experienced it during meta-training. Right: The learned UnsupervisedUpdate generalizes across many different activation functions not seen in training. We show accuracy over the course of training on 14x14 MNIST.Next we look at generalization over different activation functions. We apply our learned optimizer on base models with a variety of different activation functions. Performance evaluated at different points in training (FIG4). Despite training only on ReLU activations, our learned optimizer is able to improve on random initializations in all cases. For certain activations, leaky ReLU BID17 and Swish BID18, there is little to no decrease in performance. Another interesting case is the step activation function. These activations are traditionally challenging to train as there is no useful gradient signal. Despite this, our learned UnsupervisedUpdate is capable of optimizing as it does not use base model gradients, and achieves performance double that of random initialization. To analyze how our learned optimizer functions, we analyze the first layer filters over the course of meta-training. Despite the permutation invariant nature of our data (enforced by shuffling input image pixels before each unsupervised training run), the base model learns features such as those shown in Figure 6, which appear template-like for MNIST, and local-feature-like for CIFAR10. Early in training, there are coarse features, and a lot of noise. As the meta-training progresses, more interesting and local features emerge. In an effort to understand what our algorithm learns to do, we fed it data from the two moons dataset. We find that despite being a 2D dataset, dissimilar from the image datasets used in meta-training, the learned model is still capable of manipulating and partially separating the data manifold in a purely unsupervised manner (Figure 6). We also find that almost all the variance in the embedding space is dominated by a few dimensions. As a comparison, we do the same analysis on MNIST. In this setting, the explained variance is spread out over more of the principal components. This makes sense as the generative process contains many more latent dimensions -at least enough to express the 10 digits. Figure 6: Left: From left to right we show first layer base model receptive fields produced by our learned UnsupervisedUpdate rule over the course of meta-training. Each pane consists of first layer filters extracted from φ after 10k applications of UnsupervisedUpdate on MNIST (top) and CIFAR10 (bottom). For MNIST, the optimizer learns image-template-like features. For CIFAR10, low frequency features evolve into higher frequency and more spatially localized features. For more filters, see Appendix D. Center: Visualization of learned representations before (left) and after (right) training a base model with our learned UnsupervisedUpdate for two moons (top) and MNIST (bottom). The UnsupervisedUpdate is capable of manipulating the data manifold, without access to labels, to separate the data classes. Visualization shows a projection of the 32-dimensional representation of the base network onto the top three principal components. Right: Cumulative variance explained using principal components analysis (PCA) on the learned representations. The representation for two moons data (red) is much lower dimensional than MNIST (blue), although both occupy a fraction of the full 32-dimensional space. In this work we meta-learn an unsupervised representation learning update rule. We show performance that matches or exceeds existing unsupervised learning on held out tasks. Additionally, the update rule can train models of varying widths, depths, and activation functions. More broadly, we demonstrate an application of meta-learning for learning complex optimization tasks where no objective is explicitly defined. Analogously to how increased data and compute have powered supervised learning, we believe this work is a proof of principle that the same can be done with algorithm design-replacing hand designed techniques with architectures designed for learning and learned from data via metalearning. Samy Bengio, Yoshua Bengio, Jocelyn Cloutier, and Jan Gecsei. On the optimization of a synaptic learning rule. In Figure App.1: Schematic for meta-learning an unsupervised learning algorithm. We show the hierarchical nature of both the meta-training procedure and update rule. a) Meta-training, where the meta-parameters, θ, are updated via our meta-optimizer (SGD). b) The gradients of the MetaObjective with respect to θ are computed by backpropagation through the unrolled application of the UnsupervisedUpdate. c) UnsupervisedUpdate updates the base model parameters (φ) using a minibatch of unlabeled data. d) Each application of UnsupervisedUpdate involves computing a forward and "backward" pass through the base model. The base model itself is a fully connected network producing hidden states x l for each layer l. The "backward" pass through the base model uses an error signal from the layer above, δ, which is generated by a meta-learned function. e.) The weight updates ∆φ are computed using a convolutional network, using δ and x from the pre-and post-synaptic neurons, along with several other terms discussed in the text. Training and computing derivatives through recurrent computation of this form is notoriously difficult BID9. Training parameters of recurrent systems in general can lead to chaos. We used the usual techniques such as gradient clipping BID19, small learning rates, and adaptive learning rate methods (in our case Adam ), but in practice this was not enough to train most UnsupervisedUpdate architectures. In this section we address other techniques needed for stable convergence. When training with truncated backprop the problem shifts from pure optimization to something more like optimizing on a Markov Decision Process where the state space is the base-model weights, φ, and the'policy' is the learned optimizer. While traversing these states, the policy is constantly meta-optimized and changing, thus changing the distribution of states the optimizer reaches. This type of non-i.i.d training has been discussed at great length with respect to on and off-policy RL training algorithms BID21. Other works cast optimizer meta-learning as RL BID2 for this very reason, at a large cost in terms of gradient variance. In this work, we partially address this issue by training a large number of workers in parallel, to always maintain a diverse set of states when computing gradients. For similar reasons, the number of steps per truncation, and the total number of unsupervised training steps, are both sampled in an attempt to limit the bias introduced by truncation. We found restricting the maximum inner loop step size to be crucial for stability. BID22 studied the effect of learning rates with respect to the stability of optimization and showed that as the learning rate increases gradients become chaotic. This effect was later demonstrated with respect to neural network training in. If learning rates are not constrained, we found that they rapidly grew and entered this chaotic regime. Another technique we found useful in addressing these problems is the use of batch norm in both the base model and in the UnsupervisedUpdate rule. Multi-layer perceptron training traditionally requires very precise weight initialization for learning to occur. Poorly scaled initialization can make learning impossible BID23. When applying a learned optimizer, especially early in meta-training of the learned optimizer, it is very easy for the learned optimizer to cause high variance weights in the base model, after which recovery is difficult. Batch norm helps solve this issues by making more of the weight space usable. We implement the described models in distributed Tensorflow BID16. We construct a cluster of 512 workers, each of which computes gradients of the meta-objective asynchronously. Each worker trains on one task by first sampling a dataset, architecture, and a number of training steps. Next, each worker samples k unrolling steps, does k applications of the UnsupervisedUpdate(·; θ), computes the MetaObjective on each new state, computes DISPLAYFORM0 and sends this gradient to a parameter server. The final base-model state, φ, is then used as the starting point for the next unroll until the specified number of steps is reached. These gradients from different workers are batched and θ is updated with asynchronous SGD. By batching gradients as workers complete unrolls, we eliminate most gradient staleness while retaining the compute efficiency of asynchronous workers, especially given heterogeneous workloads which arise from dataset and model size variation. An overview of our training can be seen in algorithm F. Due to the small base models and the sequential nature of our compute workloads, we use multi core CPUs as opposed to GPUs. Training occurs over the course of ∼8 days with ∼200 thousand updates to θ with minibatch size 256. In this section we describe the details of our base model and learned optimizer. First, we describe the inner loop, and then we describe the meta-training procedure in more depth. In the following sections, λ name denotes a hyper-parameter for the learned update rule. The design goals of this system are stated in the text body. At the time of writing, we were unaware of any similar systems, so as such the space of possible architectures was massive. Future work consists of simplifying and improving abstractions around algorithmic components. An open source implementation of the UnsupervisedUpdate can be found at https: //github.com/tensorflow/models/tree/master/research/learning_ unsupervised_learning. The inner loop computation consists of iterative application of the UnsupervisedUpdate on a Base Model parameterized by φ, DISPLAYFORM0 where φ consists of forward weights, W l, biases, b l, parameterizing a multi-layer perceptron as well as backward weights, V l used by UnsupervisedUpdate when inner-loop training. This computation can be broken down further as a forward pass on an unlabeled batch of data, DISPLAYFORM1 where z l, and x l are the pre-and post-activations on layer l, and x 0 is input data. We then compute weight updates: DISPLAYFORM2 Finally, the next set of φ forward weights (W l), biases (b l), and backward weights (V l), are computed. We use an SGD like update but with the addition of a decay term. Equivalently, this can be seen as setting the weights and biases to an exponential moving average of the ∆W l, ∆V l, and ∆b l terms. DISPLAYFORM3 We use λ φlr = 3e − 4 in our work. ∆W l, ∆V l, ∆b l are computed via meta-learned functions (parameterized by θ).In the following sections, we describe the functional form of the base model, f, as well as the functional form of ComputeDeltaWeight (·; θ). The base model, the model our learned update rule is training, is an L layer multi layer perception with batch norm. We define φ as this model's parameters, consisting of weights (W l) and biases (b l) as well as the backward weights (V l) used only during inner-loop training (applications of UnsupervisedUpdate). We define N 1.. N L to be the sizes of the layers of the base model and N 0 to be the size of the input data. The forward computation parameterized by φ, consumes batches of unlabeled data from a dataset D: DISPLAYFORM0 DISPLAYFORM1 We define f (x; φ) as a function that returns the set of internal pre-and post-activation function hidden states as well asf (x; φ) as the function returning the final hidden state: DISPLAYFORM2 We define the MetaObjective to be a few shot linear regression. To increase stability and avoid undesirable loss landscapes, we additionally center, as well as normalize the predicted target before doing the loss computation. The full computation is as follows: DISPLAYFORM3 where N classes is the number of classes, and y, y are one hot encoded labels. First, the inputs are converted to embeddings with the base model, DISPLAYFORM4 x L =f (x ; φ) (App.19) Next, we center and normalize the prediction targets. We show this for y, but y is processed identically.ȳ DISPLAYFORM5 We then solve for the linear regression weights in closed form with features: x L and targets:ŷ. We account for a bias by concatenating a 1's vector to the features. DISPLAYFORM6 We then use these inferred regression weights C to make a prediction on the second batch of data, normalize the ing prediction, and compute a final loss, DISPLAYFORM7 Note that due to the normalization of predictions and targets, this corresponds to a cosine distance (up to an offset and multiplicative factor). The learned update rule is parameterized by θ. In the following section, we denote all instances of θ with a subscript to be separate named learnable parameters of the UnsupervisedUpdate. θ is shared across all instantiations of the unsupervised learning rule (shared across layers). It is this weight sharing that lets us generalize across different network architectures. The computation is split into a few main components: first, there is the forward pass, defined in f (x; φ). Next, there is a "backward" pass, which operates on the hidden states in a reverse order to propagate an error signal δ l back down the network. In this process, a hidden state h l is created for each layer. These h are tensors with a batch, neuron and feature index: h l ∈ R B,N l,λ hdims where λ hdims = 64. Weight updates to W l and b l are then readout from these h l and other signals found both locally, and in aggregate along both batch and neuron. In backprop, there exists a single scalar error signal that gets backpropagated back down the network. In our work, this error signal does not exist as there is no loss being optimized. Instead we have a learned top-down signal, dL, at the top of the network. Because we are no longer restricted to the form of backprop, we make this quantity a vector for each unit in the network, rather than a scalar, DISPLAYFORM0 In this work we set λ deltadims = 32. The architecture of TopD is a neural network that operates along every dimension of x L. The specification can be found in G.7.We structure our error propagation similarly to the structure of the backpropagated error signal in a standard MLP with contributions to the error at every layer in the network, DISPLAYFORM1 where δ l has the same shape as d l, and θ errorP ropW ∈ R λ hdims,λ deltadims and θ errorP ropB ∈ R λ deltadims. Note both θ errorP ropW and θ errorP ropB are shared for all layers l. In a similar way to backprop, we move this signal down the network via multiplying by a backward weight matrix (V l). We do not use the previous weight matrix transpose as done in backprop, instead we learn a separate set of weights that are not tied to the forward weights and updated along with the forward weights as described in G.5. Additionally, we normalize the signal to have fixed second moment,d DISPLAYFORM2 The internal h l ∈ R B,N l,λ hdims vectors are computed via: DISPLAYFORM3 The architecture of ComputeH is a neural network that operates on every dimension of all the inputs. It can be found in G.8. These definitions are recursive, and are computed in order: DISPLAYFORM4 With these computed, weight updates can be read out (Section G.5). When the corresponding symbols are not defined (e.g. z 0) a zeros tensor with the correct shape is used instead. The following is the implementation of ComputeDeltaWeight(DISPLAYFORM0 For a given layer, l, our weight updates are a mixture of multiple low rank readouts from h l and h l−1 . These terms are then added together with a learnable weight, in θ to form a final update. The final update is then normalized mixed with the previous weights. We update both the forward, W l, and the backward, V l, using the same update rule parameters θ. We show the forward weight update rule here, and drop the backward for brevity. For convenience, we define a low rank readout function LowRR that takes h l like tensors, and outputs a single lower rank tensor. DISPLAYFORM1 Here, Θ, is a placehoder for the parameters of the given readout. LowRR is defined as: DISPLAYFORM2 where λ gradc = 4 and is the rank of the readout matrix (per batch element). This sequence of terms allow the weight to be adjusted as a function of state in the pre-and postsynaptic neurons. They should be viewed as a basis function representation of the way in which the weight changes as a function of pre-and post-synaptic neurons, and the current weight value. We express each of these contributions to the weight update as a sequence of weight update planes, with the ith plane written ∆W DISPLAYFORM0 Each of these planes will be linearly summed, with coefficients generated as described in Equation App.57, in order to generate the eventual weight update. Ŵ DISPLAYFORM1 Additional weight update planes are designed to aid units in remaining decorrelated from each other's activity, and in decorrelating their receptive fields. Without terms like this, a common failure mode DISPLAYFORM2 We then normalize, re-weight, and merge each of these weight update planes into a single term, which will be used in the weight update step, DISPLAYFORM0 where θ mergeW ∈ R 10 (as we have 10 input planes).To prevent pathologies during training, we perform two post processing steps to prevent the learned optimizer from cheating, and increasing its effective learning rate, leading to instability. We only allow updates which do not decrease the weight matrix magnitude, DISPLAYFORM1 whereŴ l is W l scaled to have unit norm, and we normalize the length of the update, DISPLAYFORM2 To compute changes in the biases, we do a readout from h l. We put some constraints on this update to prevent the biases from pushing all units into the linear regime, and minimizing learning. We found this to be a possible pathology. DISPLAYFORM3 where θ Breadout ∈ R λ hdims.We then normalize the update via the second moment: DISPLAYFORM4 Finally, we define ComputeDeltaWeight as all layer's forward weight updates, and backward weight updates, and bias updates. DISPLAYFORM5 This function performs various convolutions over the batch dimension and data dimension. For ease of notation, we use m as an intermediate variable. Additionally, we drop all convolution and batch norm parameterizations. They are all separate elements of θ topD. We define two 1D convolution operators that act on rank 3 tensors: ConvBatch which performs convolutions over the zeroth index, and ConvUnit which performs convolutions along the firs index. We define the S argument to be size of hidden units, and the K argument to be the kernel size of the 1D convolutions. Additionally, we set the second argument of BatchNorm to be the axis normalized over. DISPLAYFORM6 First, we reshape to add another dimension to the end of x L, in pseudocode: DISPLAYFORM7 Next, a convolution is performed on the batch dimension with a batch norm and a ReLU non linearity. This starts to pull information from around the batch into these channels. DISPLAYFORM8 We set λ topdeltasize = 64. Next, a series of unit convolutions (convolutions over the last dimension) are performed. These act as compute over information composed from the nearby elements of the batch. These unit dimensions effectively rewrite the batch dimension. This restricts the model to operate on a fixed size batch. DISPLAYFORM9 Next a series of 1D convolutions are performed over the batch dimension for more compute capacity. Finally, we convert the representations to the desired dimensions and output. DISPLAYFORM10 With the inputs prepared, we next perform a series of convolutions on batch and unit dimensions,. DISPLAYFORM11 The is then output. ComputeH (·; θ computeH) = m 11 (App.104)We set λ computehsize = 64 which is the inner computation size. DISPLAYFORM12 We trained on a data distribution consisting of tasks sampled uniformly over the following datasets. Half of our training tasks where constructed off of a dataset consisting of 1000 font rendered characters. We resized these to 14x14 black and white images. We call this the glyph dataset. "Alphabet" is an example of such a dataset consisting of alphabet characters. We used a mixture 10, 13, 14, 17, 20, and 30 way classification problems randomly sampled, as well as sampling from three 10-way classification problems sampled from specific types of images: letters of the alphabet, math symbols, and currency symbols. For half of the random sampling and all of the specialized selection we apply additional augmentation. This augmentation consists of random rotations (up to 360 degrees) and shifts up to +-5 pixels in the x and y directions. The parameters of the augmentation were inserted into the regression target of the MetaObjective as a curriculum of sorts and to provide diverse training signal. In addition to the the glyph set, we additionally used Cifar10, resized to 16x16, as well as 10, 15, 20, and 25 way classification problems from imagenet. Once again we resized to 16x16 for compute reasons. With a dataset selected, we apply additional augmentation with some probability consisting of the following augmentations. A per task dropout mask (fixed mask across all images in that task). A per example dropout mask (a random mask per image). A permutation sampled from a fixed number of pre created permutations per class. A per image random shift in the x direction each of image. All of these additional augmentations help with larger domain transfer. We employ Adam as our meta-optimizer. We use a learning rate schedule of 3e-4 for the first 100k steps, then 1e-4 for next 50k steps, then 2e-5 for remainder of meta-training. We use gradient clipping of norm 5 on minibatchs of size 256.We compute our meta-objective by averaging 5 evaluation of the linear regression. We use a ridge penalty of 0.1 for all this work. When computing truncated gradients, we initially sample the number of unrolled applications of the UnsupervisedUpdate in a uniform distribution of. This is the number of steps gradients are backpropogated through. Over the course of 50k meta-training steps we uniformly increase this to. This increases meta-training speed and stability as large unrolls early in training can be unstable and don't seem to provide any value. For sampling the number of truncated steps (number of times the above unrolls are performed), we use a shifted normal distribution -a normal distribution with the same mean and standard deviation. We chose this based on the expected distribution of the training step, φ iteration number, across the cluster of workers. We initially set the standard deviation low, 20, but slowly increased it over the course of 5000 steps to 20k steps. This slow increase also improved stability and training speed. For each experimental figure, we document the details. The VAE we used consists of 3 layers, size 128, with ReLU activations and batch norm between each layer. We then learn a projection to mean and log std of size 32. We sample, and use the inverse architecture to decode back to images. We use a quantized normal distribution (once again parameterized as mean and log std) as a posterior. We train with Adam with a learning rate of 1e-4.To isolate the effects of objective function mismatch and overfitting, we both train on the unlabeled training set and evaluate on the labeled training set instead of a validation set. We use a 4 layer, size 128 unit architecture with a 32 layer embedding for all models. We select performance at 100k training steps for the VAE, and 3k for our learned optimizer. Our supervised learning baseline consists of the same architecture for the base model but with an additional layer that outputs log probabilities. We train with cross entropy loss on a dataset consisting of only 10 examples per class (to match the other numbers). Surprisingly, the noise from batch norm acts as a regularizer, and allowing us to avoid needing a complex early stopping scheme as test set performance simply plateaus over the course of 10k steps. We train with Adam with a learning rate of 3e-3, selected via a grid over learning rate on test set performance. In this setting, having a true validation set would dramatically lower the amount of labeled data available (only 100 labeled examples) and using the test set only aids in the this baseline's performance. For the IMDB experiments, we tokenized and selected the top 1K words with an additional set of tokens for unknown, sentence start, and sentence end. Our encoding consisted of a 1 if the word is present, otherwise 0. We used the same 4 layer, 128 hidden unit MLP with an addition layer outputting a 32 dimensional embedding. We used ReLU activations and 4 layers of size 128 with an additional layer to 32 units unless otherwise specified by the specific experiment. We trained prototypical networks on either intact or shuffled mini-Imagenet images. For shuffled images, we generated a fixed random permutation for the inputs independently for every instantiation of the base network (or 'episode' in the meta-learning literature ). The purpose of shuffling was to demonstrate the inductive bias of this type of meta-learning, namely that they do not generalize across data domain. Note that the base network trained was the same fully connected architecture like that used in this paper (3 layers, size 128, with ReLU activations and batch normalization between layers). Though the original paper used a convolutional architecture, here we swapped it with the fully connected architecture because the tied weights in a convolutional model do not make sense with shuffled pixels.