Question: Represent the article for finding a single-sentence TLDR

Question: We improve previous end-to-end differentiable neural networks (NNs) with fast weight memories. A gate mechanism updates fast weights at every time step of a sequence through two separate outer-product-based matrices generated by slow parts of the net. The system is trained on a complex sequence to sequence variation of the Associative Retrieval Problem with roughly 70 times more temporal memory (i.e. time-varying variables) than similar-sized standard recurrent NNs (RNNs). In terms of accuracy and number of parameters, our architecture outperforms a variety of RNNs, including Long Short-Term Memory, Hypernetworks, and related fast weight architectures. Recurrent Neural Networks (RNNs) are general parallel-sequential computers that can implement algorithms which map input sequences to output sequences. One variation of it, the Long ShortTerm Memory (LSTM), has achieved great success on a wide variety of Machine Learning tasks such as natural language translation, image caption generation, and speech recognition among others BID11; BID6; BID7. In practical applications, most RNNs are actually LSTM networks now used billions of times per day for automatic translation BID21, speech recognition Sak et al., and many other tasks BID13 BID17.However, plain RNNs but also LSTMs are known to have difficulty in performing memorization, like e.g. a simple copying task of outputting the same sequence as the input sequence BID22. But also other more high-level cognitive tasks have been shown to be difficult to master BID2.In this work, we explore a generalization of the Associative Retrieval problem. We follow a similar style as in BID2 but turned the task into a general sequence to sequence problem and also substantially increased its complexity. The underlying mechanism is essentially a dictionary with a certain number of key-value pairs which is controlled using a simple syntax of storage and query tokens. In order to overcome the limitation of current RNNs on this task, we propose a fast weight architecture that is able to learn and generalize using much fewer parameters. Our architecture consists of the two networks s and f which both operate on the input sequence in parallel. The small network f predicts the targets while the big network s generates on-the-fly weight-updates for f. The big network s is called the slow network because its weights change only after every mini-batch according to the gradient-based learning algorithm. f, on the other hand, is called the fast network because its weights can change after every time step. Our work is heavily influenced by early work in Neural Networks. Like many ideas at the time, the idea of fast-changing weights emerged out of biological evidence and the efforts of storing activation patterns in the weights of an associative network. Networks with non-differentiable fast weights or "dynamic links" have been published since 1981 von der BID19 BID5; BID10. Subsequent work showed that a slow network can use gradient descent learning to control fast weights of a separate network in end-to-end differentiable fashion BID14. A more recent work on fast weights, which provides a good overview of the physiological facts, is the work by BID0. In their work they aptly describe their own fast weight architecture as an attention mechanism which effectively increases the flexibility of the learned program, making it more adaptive through time and able to store temporally recent information but, by design, it is unable to store long-term knowledge using that same mechanism. Their decay of the modulation on top of the slow weights that long-term knowledge has to be learned using a learning algorithm like gradient descent and can't be learned using fast weights. Our approach follows more from BID14 which frames the fast weights idea as a means of program generation i.e. using the first network to produce context dependent weight updates for a second network. We also refer to the weights of a network as its program. An adaptive program, such as the one of f, and its benefits can be described through the concept of the ratio of time-varying variables BID15. Take as an example a plain RNN. The program is its recurrent weight matrix. Once trained, the number of time-varying variables in the computational graph is limited to the state variables and increasing the size of the recurrent weight matrix quadratically increases the number of weight variables available for the implementation of the during inference static program but only linearly increase the number of adaptive state variables. Fast weight architectures tend to relax this ratio allowing the program to also change through time in order to adapt to the context-specific needs. An adjacent approach to the relaxation of the ratio of time-varying variables is the Hypernetwork by BID9. Their initial idea consisted of an adaptive interpolation of several Long Short-Term Memory (LSTM) cells but then lead to a novel approximation which still allows for an adaptive program. Their sequential Hypernetwork was applied to language modelling and outperformed the LSTM on several datasets. As introduced, our fast weight architecture consists of a big network s and a small network f. For simplicity we chose both networks to be RNNs but they could be replaced by any other recurrent architecture, such as a LSTM. While the weights of s change after every batch, the weights of f change after every time step, hence the fast/slow dichotomy. The following formulas are for a single sample or a batch size of 1 and uppercase letters refer to matrices while lower case letters refer to vectors. The biases are omitted for simplicity. Both, the slow network s(x t, h S t) and the time-varying fast network f t (x t, h F t) use the same input embedding and the same output projection as it is common practice for RNNs. In this paper, we use in both cases a 2-layer transition RNN. Recall the basic RNN formulation, DISPLAYFORM0 where h is the hidden state, φ is a non-linear activation function such as tanh, x t is the current input from the input sequence X = (x 1, x 2, ..., x T), and the weights W ∈ R m×n are fixed parameters. Analogously, we define our fast network f t (x t, h DISPLAYFORM1 Where h F is the hidden state of the fast network, LN refers to layer normalization (LN) as introduced by BID1, and F ∈ R m×n, F ∈ R m×m are parameters which are not fixed and can change through time. Similarly, we define the the slow network s(x t, h S t): DISPLAYFORM2 DISPLAYFORM3 Where h S is the hidden state of the slow network, respectively. Every F t+1 is then calculated as follows: DISPLAYFORM4 DISPLAYFORM5 Where H and T are outer products to generate weight matricies in an Hebb-like manner as introduced by BID14 and have the same dimensionality as the respective F, is the element-wise product, 1 denotes a matrix of ones. We make use of LN in the fast network because it has shown to stabilize learning especially in the beginning of the training but also slightly improved validation set performance. Because of the generated nature of the fast weights, we expected them to be far from good initial values when training begins. We use LN to overcome this problem. We tested our architecture without LN, with pre-activation LN, with post-activation LN, and with pre and post-activation LN and found that using a post-activation LN always works best no matter if we normalize pre-activation. In equation 8 we use a gating matrix to blend the current fast matrix with a matrix update. This is essentially equal to the gating mechanism for activations in a highway network BID18. We also evaluated other multiplicative and additive interactions using more than one matrix but achieved the best using this specific update mechanism. Furthermore, we'd like to point out that the slow network s is generating weights for the next time step. We do this to prevent s from fully learning a prediction on its own which bypasses f. We didn't experiment with update delays greater than one step. Figure 1: An informal pictogram which visualises the fast and slow network of our architecture at time step t. e(x t) refers to the embedding of x t and biases, activation functions, layernorm, and outer-products are not displayed. We created a challenging sequence to sequence task based on the Associative Retrieval problem. Our version employs of a simple syntax of storage and query tokens. Storage tokens are key-value pairs while query tokens come with only a key to which the network must output the respective value according to a previously seen storage token. All tokens are part of the inputs sequence while the generated answers to the query tokens are part of the output sequence. Whenever there is no query to respond to the network is supposed to output some default value which in our case is the empty space character. This means that non-trivial problem-relevant predictions are rather sparse. The keys are 2 to 4 characters long while the respective values are a single character. All characters are uniformly sampled with replacement from a set of 8 possible ASCII characters (a to h). Each query token has to be a valid key which the network must have seen after the previous query token. Preceding every query token there are between 1 and 10 storage tokens. We concatenated all query tokens and their respective storage tokens into one large sequence and use truncated Backpropagation Through Time (truncated BPTT) to train the network. The following is an example with only 2 queries with quotes to show the beginning and end of both sequences.x: "S(hgb,c),S(ceaf,e),S(df,g),S(hac,b),Q(ceaf)e. S(hf,h),S(cc,d),Q(cc)d." y: " e d "We generate and concatenate 100'000 queries for the training set and 5'000 queries for the test and validation set. This in a single training sequence of roughly 5.7 million characters and a test and validation sequence of roughly 288'000 characters each. Instead of measuring the accuracy of all predictions we use the partial accuracy which is the percentage of the correct predictions of nonspace characters. This is because all models learn very quickly to predict spaces at every step which very quickly yields a high accuracy without actually having learned much. However, the cost and respective gradient are computed using all predictions. For all models and experiments, we used the Nesterov accelerated Adam (Nadam) by BID4. We observed in our experiments that Adam achieves similar performance but tends to converge slower than Nadam. The Model Our final architecture uses an embedding size of 15. The fast network is defined such that h DISPLAYFORM0 ∈ R 40×40 and the slow network such that h S ∈ R 40, S ∈ R 55×100, S ∈ R 100×394 both using a shared embedding of R 15×15 and a shared output projection W ∈ R 40×15 summing up to 46'234 trainable parameters (bias not listed for simplicity). We aimed for a small and very context specific fast network and a slow network big enough to support such a fast-changing fast network while using as few weights as possible. Different configurations are possible and a larger s seems to help the model to converge faster but doesn't help performance. Preliminary experiments with different RNNs, like an RHN or LSTM as the fast and slow network, also showed promising but were not part of this work. Results We provide our best experimental for other architectures for which we performed a hyper parameter search using a mix of grid and random search over each architecture's individual parameters. We compare our architecture to the LSTM, fast weights as a means of attention to the recent past (AttentionFW) by BID0, and HyperNetworks by BID9. We also performed experiments on the feed-forward fast weights and recurrent fast weights as introduced by BID14 BID16 but we were unable to get them to predict anything else than the trivial output on this task. We trained all models using a sequence length of 32 and a batch size of 256. We didn't perform learning rate decay or similar strategies but included the learning rate in our hyper parameter search. In the end, a learning rate of 0.002 achieved over all architectures the best and was in a second phase fixed for all models to allow for a comparison of convergence qualities. Our model achieves the highest partial accuracy and the lowest partial bits per character (BPC) while using 83 times fewer parameters than the next best model. Again, "partial" refers only to the non-space prediction targets. It has been shown before how generalizing a memory mechanism, such as required in this task, is difficult for vanilla RNNs to learn. Several previous works focused on integrating differentiable Figure 2: The left figure represents the accuracy of non-trivial targets and the right figure the respective bits per character. These are the validation set of the best models of the four examined architectures due to our hyper parameter search. Green is the LSTM, dark blue is the fast weights architecture as attention to the recent past, red is the hypernetwork, and cyan is our novel fast weight architecture.computer-like memory into the graph structure of the architecture such that the model wouldn't need to learn the mechanism itself but mainly how to use it. Examples of such are differentiable stacks by BID3; BID12, but also related storage types like those in LSTM-controlled Neural Turing Machines BID8 or memory nets BID20.A basic argument against a memory approach inspired by the Turing Machine or the von Neumann architecture is its biological plausibility, as well as, the fact that we know how the human memory system often doesn't really behave as computer memory does. It is generally known to be much more nuanced and forcing an architecture to include a strong and possibly misleading bias would certainly limit its ability to learn and generalize to a more effective mechanism. We think that learning high-level cognitive functions (i.e. high-level programs implemented under the constraints of some artificial neural substrate) is difficult and find the idea to search and reverse engineer every human capability necessary for intelligence in order to engineer it into an architecture to be undesirable. Instead, we favour an approach which focuses on improving the capabilities of the artificial neural substrate which allows for the emergence of higher-level functions through training. We think fast weights are such a component from which many models could benefit. Limitations Fast weights seem to have a positive effect when they are incorporated into an architecture but we experienced at least two practical limitations. While the calculation of the gradient through these fast weight dynamics remains rather cheap, the number of values to be stored in the backward pass encompasses now all time-varying variables (i.e. all fast weights) at each relevant time step. This quadratically increases the memory consumption compared to a similar sized RNN. At the moment, these memory limitations are the main reason why such fast weight networks remain rather small compared to state-of-the-art RNNs one some popular application like e.g. neural machine translation. Another noteworthy limitation is the wall-time necessary for computing a more complex architecture. Reshaping tensors and other simple operations in a significant increase of wall-time. However, over 20 years ago it was pointed out that an RNN can also use additional, soft, end-toend differentiable attention mechanisms to learn to control its own internal spotlights of attention BID15 to quickly associate self-defined patterns through fast weights (on connections between certain units) that can quickly and dramatically change from one time step to the next. This approach can essentially increase the number of time-varying variables massively while keeping the model relatively small. We improved the update mechanism through which the slow network learns to write into its fast weight memory. This allows us to construct a model with a small but memory expensive fast network in addition to the standard slow network. However, the fast weights are not just passive memory like the state but are more like active memory in the sense of a context-specific computation. We force the model to use this active memory at every step to predict the current output by delaying the weight updates from slow network by one step. Consider the model introduced in the previous section. While the slow network is technically bigger, it contains only 40 time-varying variables, namely the state vector h S. The fast network is much smaller but has 3840 time-varying variables (h F, F, and F ). Increasing the total number of time-varying variables significantly. In this paper, we introduce a complex sequence to sequence variation of the Associative Retrieval problem. In that problem, the model has to learn how to store a number of associations from the input sequence, retrieve them if necessary, and forget them to learn new associations. We use a standard RNN to generate weight updates for a fast weight RNN. This allows our model to store temporal information not only in the state of either RNN but also in the weights of the fast weight RNN. Our contribution is a new way of updating the weight matrices of the fast weight RNN where we use a gate and two generated matrices instead of one. Without our contribution the model has never shown to be able to learn any non-trivial predictions. We compare it with other architectures on this general task and show how it outperforms them in convergence, accuracy, and number of parameters.

represent this text to find a much much longer article about it with say hundreds-thousands of words.
For instance, <<We find movement in function space is not proportional to movement in parameter space during optimization. We propose a new natural-gradient style optimizer to address this.>>  to "To optimize a neural network one often thinks of optimizing its parameters, but it is ultimately a matter of optimizing the function that maps inputs to outputs. Since a change in the parameters might serve as a poor proxy for the change in the function, it is of some concern that primacy is given to parameters but that the correspondence has not been tested. Here, we show that it is simple and computationally feasible to calculate distances between functions in a $L^2$ Hilbert space. We examine how typical networks behave in this space, and compare how parameter $\ell^2$ distances compare to function $L^2$ distances between various points of an optimization trajectory. We find that the two distances are nontrivially related. In particular, the $L^2/\ell^2$ ratio decreases throughout optimization, reaching a steady value around when test error plateaus. We then investigate how the $L^2$ distance could be applied directly to optimization. We first propose that in multitask learning, one can avoid catastrophic forgetting by directly limiting how much the input/output function changes between tasks. Secondly, we propose a new learning rule that constrains the distance a network can travel through $L^2$-space in any one update. This allows new examples to be learned in a way that minimally interferes with what has previously been learned. These applications demonstrate how one can measure and regularize function distances directly, without relying on parameters or local approximations like loss curvature. A neural network's parameters collectively encode a function that maps inputs to outputs. The goal of learning is to converge upon a good input/output function. In analysis, then, a researcher should ideally consider how a network's input/output function changes relative to the space of possible functions. However, since this space is not often considered tractable, most techniques and analyses consider the parameters of neural networks. Most regularization techniques, for example, act directly on the parameters (e.g. weight decay, or the implicit constraints stochastic gradient descent (SGD) places upon movement). These techniques are valuable to the extent that parameter space can be taken as a proxy for function space. Since the two might not always be easily related, and since we ultimately care most about the input/output function, it is important to develop metrics that are directly applicable in function space. In this work we show that it is relatively straightforward to measure the distance between two networks in function space, at least if one chooses the right space. Here we examine L 2 -space, which is a Hilbert space. Distance in L 2 space is simply the expected 2 distance between the outputs of two functions when given the same inputs. This computation relies only on function inference. Using this idea of function space, we first focus on characterizing how networks move in function space during optimization with SGD. Do random initializations track similar trajectories? What happens in the overfitting regime? We are particularly interested in the relationship between trajectories in function space and parameter space. If the two are tightly coupled, then parameter change can be taken as a proxy for function change. This common assumption (e.g. Lipschitz bounds) might not always be the case. Next, we demonstrate two possibilities as to how a function space metric could assist optimization. In the first setting we consider multitask learning, and the phenomenon of catastrophic forgetting that makes it difficult. Many well-known methods prevent forgetting by regularizing how much the parameters are allowed to shift due to retraining (usually scaled by a precision matrix calculated on previous tasks). We show that one can instead directly regularize changes in the input/output function of early tasks. Though this requires a "working memory" of earlier examples, this scheme turns out to be quite data-efficient (and more so than actually retraining on examples from old tasks).In the second setting we propose a learning rule for supervised learning that constrains how much a network's function can change any one update. This rule, which we call Hilbert-constrained gradient descent (HCGD), penalizes each step of SGD to reduce the magnitude of the ing step in L 2 -space. This learning rule thus changes the course of learning to track a shorter path in function space. If SGD generalizes in part because large changes to the function are prohibited, then this rule will have advantages over SGD. Interestingly, HCGD is conceptually related to the natural gradient. As we derive in §3.2.1, the natural gradient can be viewed as ing from constrains changes in a function space measured by the Kullbeck-Leibler divergence. We propose to examine the trajectories of networks in the space of functions defined by the inner product f, g = X f (x)g(x)dµ(x), which yields the following norm: DISPLAYFORM0 Here µ is a measure and corresponds to the probability density of the input distribution X. Note that this norm is over an empirical distribution of data and not over the uniform distribution of all possible inputs. The | · | 2 operator refers to the 2-norm and can apply to vector-valued functions. While we refer to this space as a Hilbert space, we make no use of an inner product and can also speak of this as any normed vector space, e.g. a Banach space. This norm leads to a notion of distance between two functions f and g given by DISPLAYFORM1 Since µ is a density, X dµ = 1, and we can write DISPLAYFORM2 The expectation can be approximated as an empirical expectation over a batch of examples drawn from the input distribution: DISPLAYFORM3 The quality of the empirical distance, of course, will depend on the shape and variance of the distribution of data as well as the form of f and g. In section 2.3, we empirically investigate the quality of this estimator for reasonably sample sizes N. We wish to compare at high level how networks move through parameter and function space. Our first approach is to compare a low-dimensional embedding of the trajectories through these spaces. In FIG2, we take a convolutional neural network and train three random initializations on a 5000-image subset of CIFAR-10. By saving the parameters of the network at each epoch as well as the output on a single large validation batch, we can later compute the 2 parameter distance and the FIG2: Visualization of the trajectories of three random initializations of a network through function space, left, and parameter space, right. The network is a convolutional network trained on a 5,000 image subset of CIFAR-10. At each epoch, we compute the L 2 and 2 distances between all previous epochs, forming two distance matrices, and then recompute the 2D embedding from these matrices using multidimensional scaling. Each point on the plots represents the network at a new epoch of training. The black arrows represent the direction of movement. 2 function distance between the snapshots of network at each epoch. The ing distance matrix is then visualized as a two-dimensional embedding. In parameter space, the networks are initialized at very different points and proceed to diverge yet further from these points. Despite this divergence, each trajectory yields a network that has learned the training data perfectly and generalizes with ∼ 50% accuracy to a test set. This illustrates the wide range of parameter settings that can be used to represent a given neural network function. The behavior of the same initializations in function space is quite different. First, note that all three initializations begin at approximately the same point in function space. This is an intriguing property of random initializations that, rather than encoding entirely random functions, random sets of parameters lead on average to the same function (for related work, see e.g. BID5). The initializations then largely follow an identical path for the initial stage of learning. Different initializations thus learn in similar manners, even if the distance between their parameters diverges. During late-stage optimization, random initializations turn from a shared trajectory and begin to diverge in L 2 space. These differences underlie the general principle that L 2 distances behave differently than 2 distances, and that functional regularization could assist training and reduce overfitting. How well do parameter distances reflect function distances? The answer to this question is relevant for any method that directly considers the behavior of parameters. Certain theoretical analyses, furthermore, desire bounds on function distances but instead find bounds on parameter distances and relate the two with a Lipschitz constant (e.g. BID6). Thus, for theoretical analyses and optimization methods alike, it is important to empirically evaluate how well parameter distances correspond to function distances in typical situations. We can compare these two situations by plotting a change in parameters ∆θ against the corresponding change in the function f θ − f θ+∆θ. In FIG0 we display this relation for several relevant distance during the optimization of a CNN on CIFAR-10. There are three scales: the distance between individual updates, the distance between epochs, and the distance from initialization. What changes strikingly throughout optimization is the relationship between parameter and function space. There is a qualitative difference in the ratio of parameter distances to function distances that visible at all three distance scales. Early epochs generally see larger changes in L 2 space for a given change in parameters. Intriguingly, the ratio of the two distances appears to converge to a single value at late optimization, after test error saturates. This is not because the network ceases to move, as noted above. Rather, the loss landscape shifts such that this ratio become constant. It is also clear from these plots that there is not a consistent positive correlation between the parameter and function distances between any two points on the optimization trajectory. For example, the parameter distance between successive epochs is negatively correlated with the L 2 distance for most of optimization FIG0. The distance from initialization shows a clean and positive relationship, but the relationship changes during optimization. Between successive batches, L 2 distance correlates with parameter distance at late epochs, but less so early in optimization when learning is quickest. Thus, at different stages of optimization, the L 2 / 2 ratio is often quite different. The usage of Batch Normalization (BN) and weight decay in this analysis somewhat affects the trends in the L 2 / 2 ratio. In Appendix A we reproduce these plots for networks trained without BN and without weight decay. The overall message that the L 2 / 2 ratio changes during optimization is unchanged. However, these methods both change the scale of updates, and appear to do so differently throughout optimization, and thus some trends are different. In Appendix B, we also isolate the effect of training data, by reproducing these plots for a CNN trained on MNIST and find similar trends. Overall, the correspondence between parameter and function distances depends strongly on the context. It might be worried that since function space is of infinite dimension, one would require prohibitively many examples to estimate a function distance. However, we find that one can compute a distance between two functions with a relatively small amount of examples. FIG4 shows how the estimated L 2 distance converges with an increasing number examples. In general, we find that only a few hundred examples are necessary to converge to an estimation within a few percent. If, after having been trained on a task, a neural network is retrained on a new task, it often forgets the first task. This phenomenon is termed'catastrophic forgetting'. It is the central difficulty of multitask training as well as applications requiring that learning be done online (especially in non-IID situations). Essentially, new information must be encoded in the network, but the the information pertinent to the previous task must not be overwritten. Most efforts to combat catastrophic forgetting rely on restricting how much parameters can change between tasks. Elastic Weight Consolidation (EWC; BID10), for example, adds a penalty to the loss on a new task B that is the distance from the weights after learning on an earlier FIG4: The variance of the the L 2 estimator is small enough that it can be reasonably estimated from a few hundred examples. In panels A and D, we reproduced L 2 distances seen in the panels of FIG0. As we increase the number of validation examples these distances are computed over, the estimations become more accurate. Panels B and E show the 95% confidence bounds for the estimation; on 95% of batches, the value will lie bewteen these bounds. These bounds can be obtained from the standard deviation of the L 2 distance on single examples. In panel C we show that the standard deviation scales linearly with the L 2 distance when measured between updates, meaning that a fixed batch size will often give similar percentage errors. This is not true for the distance from initialization, in panel F; early optimization has higher variance relative to magnitude, meaning that more examples are needed for the same uncertainty. In the Appendix, we also display the convergence of the L 2 distance estimator between epochs.task A, multiplied by the diagonal of the Fisher information matrix F (calculated on task A): DISPLAYFORM0 This idea is closely related to well-studied approaches to Bayesian online learning, if F is interpreted as a precision matrix BID7, BID15 ). Other similar approaches include that of BID21, who use a more accurate approximation of the Fisher, and Synaptic Intelligence (SI; BID28), which discounts parameter change via a diagonal matrix in which each entry reflects the sum contribution of that parameter to the loss. Each of these method discourages catastrophic forgetting by restricting movement in parameter space between tasks, scaled by a (perhaps diagonal) precision matrix calculated on previous tasks. Using a function space metric, it is not hard to ensure that the network's output function on previous tasks does not change during learning. In this case, the loss for a new task B is modified to be: DISPLAYFORM1 The regularization term is the L 2 distance between the current function f θ B and the function after training on task A, f θ A. Since our function space metric is defined over a domain of examples, we will store a small set of previously seen examples in a working memory, as well as the output on those examples. This memory set will be used to calculate the L 2 distance between the current iteration and the snapshot after training. This is a simple scheme, but novel, and we are not aware of direct precedence in the literature. A working memory approach is employed in related work BID12 BID20 ). Note, however, that storing old examples violates the rules of strict online learning. Nevertheless, for large networks it will be more memory-efficient. EWC, for example, requires storing a snapshot of each parameter at the end of the previous task, as well as a diagonal precision matrix with as many entries as parameters. For the 2 hidden layer network with 400 nodes each that was used in the MNIST task in BID10, this is 1,148,820 new parameters, or as many pixels as 1,465 MNIST images. When each layer has as many as 2,000 nodes, as in FIG4 of BID10, the extra stored parameters are comparable to 15,489 MNIST images. The working memory approach that is required to regularize function change from old tasks is thus comparable or cheaper in memory. We compared the performance of our approach at the benchmark task of permuted MNIST. This task requires a single MLP to learn to classify a sequence of MNIST tasks in which the pixels have been randomly permuted differently on each task. We trained an MLP with 2 hidden layers, 400 nodes each, for 10 epochs on each of 8 such permuted datasets. In FIG5, we display how the test accuracy on the first of 8 tasks degrades with subsequent learning. To build the working memory, we keep 1024 examples from previous tasks, making sure that the number of examples from each task is equal. We also remember the predictions on those examples at the end of training on their originating tasks. To calculate the L 2 distance, we simply re-infer on the examples in working memory, and regularize the distance from the current outputs to the remembered outputs. We chose λ = 1.3 as the regularizing hyperparameter from a logarithmic grid search. In FIG5, we compare this method to four comparison methods. The "ADAM" method is ADAM with a learning rate of 0.001, which nearly forgets the first task completely at the end of the 8 tasks. The "ADAM+retrain" method is augmented with a working memory of 1024 examples that are stored from previous tasks. Every n iterations (we found n = 10 to be best), a step is taken to decrease the loss on the memory cache. This method serves as a control for the working memory concept. We also include EWC and SI as comparisons, using the hyperparameters used in their publications (λ = 500, = c = 0.1). Overall, we found that regularizing the L 2 distance on a working memory cache was more successful than simply retraining on the same cache. It also outperformed EWC, but not SI. Note that these methods store diagonal matrices and the old parameters, and in this circumstance these were larger in memory than the memory cache. Figure 4: Regularizing the L 2 distance from old tasks (calculated over a working memory cache of size 1024) can successfully prevent catastrophic forgetting. Here we display the test performance on the first task as 7 subsequent tasks are learned. Our method outperforms simply retraining on the same cache (ADAM+retrain), which potentially overfits to the cache. Also displayed are ADAM without modifications, EWC, and SI. In this section we propose that the L 2 distance can be used for regularization in a single supervised task. In the space of parameters, SGD is a strongly local update rule and large jumps are generally prohibited. SGD is thus more likely to find solutions that are close to the initialization, and furthermore to trace a path of limited length. This discourages the sampling a large volume of parameter space during optimization. If the mapping between parameter and function space is not already very tight, and locality is important for generalization, then additionally constricting changes in function space should help. On the basis of this logic, we propose a learning rule that directly constrains the path length of the optimization trajectory L 2 space. If a network would have been trained to adjust the parameters θ to minimize some cost C 0, we will instead minimize at each step t a new cost given by: DISPLAYFORM0 Like all regularization terms, this can also be viewed as a Langrangian that satisfies a constraint. Here, this constraint ensures that the change in L 2 -space does not exceed some constant value. To evaluate Equation 1, we can approximate the norm with an empirical expectation over X: DISPLAYFORM1 This cost function imposes a penalty upon the difference between the output of the current network at time t and the proposed network at t + 1. The data x i may derive from some validation batch but must pull from the same distribution X. It would also be possible to use unlabeled data. We can write an update rule to minimize Equation 1 that is a modification of gradient descent. We call the rule Hilbert-constrained gradient descent (HCGD). It minimizes C in Equation 1 via an inner loop of gradient descent. To optimize C via gradient descent, we first replace C 0 with its first order approximation J T ∆θ, where J is the Jacobian. Thus we seek to converge to a ∆θ at each update step, where DISPLAYFORM2 Minimization of the proper ∆θ can be performed in an inner loop by a first order method. We first propose some ∆θ 0 = − J = − ∇ θ C 0 (for learning rate) and then iteratively correct this proposal by gradient descent towards ∆θ. If only one correction is performed, we simply add the derivative of the Hilbert-constraining term after ∆θ 0 has been proposed. We found empirically that a single correction was often sufficient. In Appendix C, we demonstrate that this algorithm does actually decrease the distance traveled in function space, as expected. This algorithm is shown in Algorithm 1. DISPLAYFORM3 Calculate correction 9: DISPLAYFORM4 10: DISPLAYFORM5 return θ t Note that the "proposed update" is presented as an SGD step, but could be a step of another optimizer (e.g. ADAM). In the Appendix, we display an extended version of this algorithm. This version allows for multiple corrective iterations in each step. It also allows for a form of momentum. In standard momentum for SGD, one follows a "velocity" term v which is adjusted at each step with the rule v ← βv + J (e.g. see BID26). For HCGD, we also keep a velocity term but update it with the final Hilbert-constrained update ∆θ rather than J. The velocity is used to propose the initial ∆θ 0 in the next update step. We found that this modification of momentum both quickened optimization and lowered generalization error. The natural gradient turns out to carry a similar interpretation as HCGD, in that the natural gradient also regularizes the change in functions' output distributions. Specifically, the natural gradient can be derived from a penalty upon the change in a network's output distribution as measured by the Kullbeck-Leibler divergence (rather than the L 2 distance).To show this, we start with a similar goal of function regularization and will come upon the natural gradient. Let us seek to regularize the change in a network's output distribution P θ throughout optimization of the parameters θ, choosing the Kullbeck-Leibler (KL) divergence as a measure of similarity between any two distributions. To ensure the output distribution changes little throughout optimization, we define a new cost function DISPLAYFORM0 where C 0 is the original cost function and λ is a hyperparameter that controls the importance of this regularization term. Optimization would be performed with respect to the proposed update θ t+1.Evaluating the KL divergence directly is problematic because it is infeasible to define the output density P θ everywhere. One can obtain a more calculable form by expanding D KL (P θt+1 P θt) around θ t to second order with respect to θ. The Hessian of the KL divergence is the Fisher information metric F. With ∆θ ≡ (θ t+1 − θ t), we can rewrite our regularized cost function as DISPLAYFORM1 To optimize C via gradient descent we first replace C 0 with its first order approximation. DISPLAYFORM2 At each evaluation, J is evaluated before any step is made, and we seek the value of ∆θ that minimizes Equation FORMULA15. By setting the derivative with respect to ∆θ to be zero, we can see that this value is DISPLAYFORM3 When λ = 1 this update is equal to the natural gradient. Thus, the natural gradient emerges as the optimal update when one regularizes the change in the output distribution during learning. In Appendix E, we show how one can approximate the natural gradient with an inner first-order optimization loop, like in HCGD. We note that HCGD is computationally cheaper than the exact natural gradient. It does not require any matrix inversions, nor the calculation of separate per-example gradients. When the validation batch X V is drawn anew for each of n corrective iterations (step 8 in Algorithm 1), HCGD requires an additional two forward passes and one backwards pass for each correction, for a total of 2 + 3n passes each outer step. In addition to being seen as a regularizer of functional change, it in an interesting aside to note that variants of the natural gradient have appeared with many justifications. These include data efficiency, minimizing a regret bound during learning, speeding optimization, and the benefits of whitened gradients. Amari originally developed the natural gradient in the light of information geometry and efficiency BID1; ). If some directions in parameter space are more informative of the network's outputs than others, then updates should be scaled by each dimension's informativeness. Equivalently, if not all examples carry equal information about a distribution, then the update step should be modified to make use of highly informative examples. That is, we wish to find a Fisherefficient algorithm (see BID2). The natural gradient uses the Fisher information matrix to scale the update by parameters' informativeness. There is also a connection between the natural gradient (and thus HCGD) and techniques that normalize and whiten gradients. The term F −1 J, after all, simply ensures that steps are made in a parameter space that is whitened by the covariance of the gradients. Whitening the gradients thus has the effect that SGD becomes more similar to the natural gradient. It appears that many approaches to normalize and whiten activations or gradients have been forwarded in the literature BID19; BID25; BID24; BID3 BID27; BID11; BID23; BID22 ). A similar effect is able to be learned with Batch Normalization, as well BID9 ). By normalizing and whitening the gradients, or by proxy, the activations, these various methods ensure that parameter space is a better proxy for function space. We compared HCGD and SGD on feedforward and recurrent architectures. If it is important that SGD limits changes in function space, and parameter and function space are loosely coupled, then HCGD should improve upon SGD. In all tests, we used a tuned learning rate for SGD, and then used the same learning rate for HCGD. We use values of λ = 0.5 and η = 0.02, generally about 10 times less than the principal learning rate. (For the n = 1 version, λ can be folded into the inner learning rate η. Values were chosen so that λη = 0.01.) We chose the batch size for the "validation" batch to be 256. While the examples in each "validation" batch were different than the training batch, they were also drawn from the train set. All models were implemented in PyTorch BID17 ).We tested HCGD as applied to the CIFAR-10 image classification problem. For reproducibility, we trained a Squeezenet v1.1, a convolutional neural network model with batch normalization optimized for parameter efficiency BID8 ). Overall HCGD does not outperform SGD in the final learning stage when trained with the same learning rate as SGD (initial = 0.1), though it does perform better in the early stage while the learning rate is high (FIG1). When we increase the initial learning rate to = 0.3 (red trace), the training accuracy decreases but the test accuracy is still marginally higher than SGD. Given the difference in relative performance between the high and low learning rate stages, it is possible that HCGD requires a different learning rate schedule to achieve the same level of gradient noise. HCGD thus decreases the test error at a given learning rate, but needs to be trained at a higher learning rate to achieve the same level of gradient noise. We next tested the performance of HCGD on a recurrent task. We trained an LSTM on the sequential MNIST task, in which pixels are input one at a time. The order of the pixels was permuted to further complicate the task. We found that HCGD outperformed SGD (Figure 6 . We used 1 correction step, as before, but found that using more correction steps yielded even better performance. However, HCGD underperformed ADAM. While not the ideal optimizer for this task, the fact that SGD can be improved indicates that SGD does not move as locally in function space as it should. Parameter space thus a poor proxy for function space in recurrent networks. HCGD first proposes an update by SGD, and then corrects it, but the first update step can also be other optimizers. Since Adam worked well for the sequential MNIST task, we tested if Adam could also be improved by taking a step to penalize the change in function space. We found that this is indeed the case, and show the as well in Figure 6 . To differentiate the SGD-and Adam-based methods, we refer to in the figure as SGD+HC and Adam+HC. This combination of Adam and L 2 functional regularization could help to achieve state-of-the-art performance on recurrent tasks. Neural networks encode functions, and it is important that analyses discuss the empirical relationship between function space and the more direct parameter space. Here, we argued that the L 2 Hilbert space defined over an input distribution is a tractable and useful space for analysis. We found that networks traverse this function space qualitatively differently than they do parameter space. Depending on the situation, a distance of parameters cannot be taken to represent a proportional distance between functions. We proposed two possibilities for how the L 2 distance could be used directly in applications. The first addresses multitask learning. By remembering enough examples in a working memory to accurately Figure 6 : Results of a singlelayer LSTM with 128 hidden units trained on the sequential MNIST task with permuted pixels. Shown are the traces for SGD and Adam (both with learning rate 0.01). We then take variants of the HCGD algorithm in which the first proposed step is taken to be an SGD step (SGD+HC) or an Adam step (Adam+HC). For SGD+HC we also show the effect of introducing more iterations n in the SGD+HC step.estimate an L 2 distance, we can ensure that the function (as defined on old tasks) does not change as a new task is learned. This regularization term is agnostic to the architecture or parameterization of the network. We found that this scheme outperforms simply retraining on the same number of stored examples. For large networks with millions of parameters, this approach may be more appealing than comparable methods like EWC and SI, which require storing large diagonal matrices. We also proposed a learning rule that reduces movement in function space during single-task optimization. Hilbert-constrained gradient descent (HCGD) constrains the change in L 2 space between successive updates. This approach limits the movement of the encoded function in a similar way as gradient descent limits movement of the parameters. It also carries a similar intuition as the forgetting application: to learn from current examples only in ways that will not affect what has already been learned from other examples. HCGD can increase test performance at image classification in recurrent situations, indicating both that the locality of function movement is important to SGD and that it can be improved upon. However, HCGD did not always improve , indicating either that SGD is stable in those regimes or that other principles are more important to generalization. This is by no means the only possibility for using an L 2 norm to improve optimization. It may be possible, for example, to use the norm to regularize the confidence of the output function (e.g. BID18). We are particularly interested in exploring if more implicit, architectural methods, like normalization layers, could be designed with the L 2 norm in mind. It interesting to ask if there is support in neuroscience for learning rules that diminish the size of changes when that change would have a large effect on other tasks. One otherwise perplexing finding is that behavioral learning rates in motor tasks are dependent on the direction of an error but independent of the magnitude of that error BID4. This is not expected by most models of gradient descent, but would be expected if the size of the change in the output distribution (i.e. behavior) were regulated to be constant. Regularization upon behavioral change (rather than synaptic change) would predict that neurons central to many actions, like neurons in motor pools of the spinal cord, would learn very slowly after early development, despite the fact that their gradient to the error on any one task (if indeed it is calculated) is likely to be quite large. Given our general resistance to overfitting during learning, and the great variety of roles of neurons, it is likely that some type of regularization of behavioral and perceptual change is at play. Figure A.5: Same as above, but for a network trained without Batch Normalization and also without weight decay. Weight decay has a strong effect. The main effect is that decreases the 2 distance traveled at all three scales (from last update, last epoch, and initialization), especially at late optimization. This explains the left column, and some of the middle and right columns. (It is helpful to look at the "white point" on the color scale, which indicates the point halfway through training. Note that parameter distances continue to change after the white point when WD is not used). An additional and counterintuitive property is that the L 2 distance from the last epoch increases in scale during optimization when WD is not used, but decreases if it is. These comparisons show that WD has a strong effect on the L 2 / 2 ratio, but that this ratio still changes considerable throughout training. This is in line with this paper's motivation to consider L 2 distances directly. Figure B.6: Here we reproduce the of FIG0 and FIG4 for the MNIST task, again using a CNN with batch normalization trained with SGD with momentum. It can be seen first that the majority of function space movement occurs very early in optimization, mostly within the first epoch. The standard deviation of the L 2 estimator, which sets the number of examples needed to accurately estimate a consistent value, is somewhat higher than for CIFAR-10. Finally, at right, it can be seen that the relationship between parameter distance traveled and function distance is similar to that of a CNN on CIFAR-10, include the qualitative change after test error converges (which here is around epoch 1). 2 SPACE Figure C.7: The HCGD algorithm is designed to reduce motion through L2-space. To confirm this, here we plot the cumulative squared distance traveled during optimization for a simple MLP trained on MNIST. This is calculated by the simple cumulative sum of the squared distances between consecutive updates. (The squared distance is nice because Brownian motion will present as a linear increase in its cumulative sum). It can be seen that SGD continues to drift in L2-space during the overfitting regime (around epoch 15, which is when test error saturates), while HCGD plateaus. This indicates that the function has converged to a single location; it ceases to change. With SGD, on the other hand, the network continues to cahnge even long after test error saturates. It is interesting to note that HCGD allows the parameters to continue to drift even though the function has generally converged. This version of the algorithm includes momentum. It also allows for multiple corrections. J ← ∇ θ C 0 (X) Calculate gradients 8: DISPLAYFORM0 ∆θ 0 ← −v Proposed update 10: DISPLAYFORM1 12: DISPLAYFORM2 Update buffer 14:for 1 < j < n do Optional additional corrections 15: ∆θ j ← ∆θ j−1 − η(g L 2) DISPLAYFORM3 18: DISPLAYFORM4 19:θ t ← θ t−1 + ∆θ 20:return θ t In order to better compare the natural gradient to the Hilbert-constrained gradient, we propose a natural gradient algorithm of a similar style. Previous work on the natural gradient has aimed to approximate F −1 as best and as cheaply as possible. This is equivalent to minimizing Equation 2 (i.e. J∆θ + λ 2 ∆θ T F ∆θ) with a single iteration of a second-order optimizer. For very large neural networks, however, it is much cheaper to calculate matrix-vector products than to approximately invert a large matrix. It is possible that the natural gradient may be more accessible via an inner gradient descent, which would be performed during each update step as an inner loop. We describe this idea at high level in Algorithm 2. After an update step is proposed by a standard optimizer, the algorithm iteratively corrects this update step towards the natural gradient. To start with a good initial proposed update, it is better to use a fast diagonal approximation of the natural gradient (such as Adagrad or RMSprop) as the main optimizer. Each additional correction requires just one matrix-vector product after the gradients are calculated. Depending on the quality of the proposed update, the number of iterations required is likely to be small, and even a small number of iterations will improve the update. ∆θ i+1 = ∆θ i − η(J + λF ∆θ i)Step towards DISPLAYFORM0 θ ← θ + ∆θ 8:return θ tSince the Fisher matrix F can be calculated from the covariance of gradients, it never needs to be fully stored. Instead, for an array of gradients G of size (# parameters, # examples), we can write DISPLAYFORM1 The choice of G is an important one. It cannot be a vector of aggregated gradients (i.e. J), as that would destroy covariance structure and would in a rank-1 Fisher matrix. Thus, we must calculate the gradients on a per-example basis. To compute G efficiently it is required that a deep learning framework implement forward-mode differentiation, which is currently not supported in popular frameworks. If we choose G to be the array of per-example gradients on the minibatch, F is known as the'empirical Fisher'. As explained in BID13 and in BID16, the proper method is to calculate G from the predictive (output) distribution of the network, P θ (y|x). This can be done as in BID14 by sampling randomly from the output distribution and re-running backpropagation on these fictitious targets, using (by necessity) the activations from the minibatch. Alternatively, as done in BID16, one may also use unlabeled or validation data to calculate G on each batch."

An improved Fast Weight network which shows better results on a general toy task.

Represent this text to find a much much longer article about it with say hundreds-thousands of words

Information about whether a neural network's output will be correct or incorrect is somewhat present in the outputs of the network's intermediate layers.