At the heart of deep learning we aim to use neural networks as function approxi-mators - training them to produce outputs from inputs in emulation of a ground truth function or data creation process.

Additional Feedback: As noted above, one of the biggest drawbacks of this very interesting work at present is the very limited scope of the demonstrations. I believe this should be easy to address, and were this done I would feel comfortable increasing my score. It would also be useful to see more detailed empirical study regarding the choice of the window (t1-t2) used to collect the data to inform the operator approximation.) There are a couple of details that I would like to see to help improve reproducibility. In terms of related work, there are a couple of more tangential directions that come to mind where connections could potentially be made / that may be interesting for the authors to consider. There may be connections related to initial stages of gradient descent identifying subspaces in which most of the parameter evolution will occur (i.e. containing the lottery ticket weights).

At the heart of deep learning we aim to use neural networks as function approximators - training them to produce outputs from inputs in emulation of a ground truth function or data creation process. In many cases we only have access to input-output pairs from the ground truth, however it is becoming more common to have access to derivatives of the target output with respect to the input - for example when the ground truth function is itself a neural network such as in network compression or distillation. Generally these target derivatives are not computed, or are ignored. This paper introduces Sobolev Training for neural networks, which is a method for incorporating these target derivatives in addition the to target values while training.

Training recurrent neural networks typically relies on backpropagation through time (BPTT). BPTT depends on forward and backward passes to be completed, rendering the network locked to these computations before loss gradients are available. Recently, Jaderberg et al. proposed synthetic gradients to alleviate the need for full BPTT. In their implementation synthetic gradients are learned through a mixture of backpropagated gradients and bootstrapped synthetic gradients, analogous to the temporal difference (TD) algorithm in Reinforcement Learning (RL). However, as in TD learning, heavy use of bootstrapping can result in bias which leads to poor synthetic gradient estimates. Inspired by the accumulate $\mathrm{TD}(\lambda)$ in RL, we propose a fully online method for learning synthetic gradients which avoids the use of BPTT altogether: accumulate $BP(\lambda)$. As in accumulate $\mathrm{TD}(\lambda)$, we show analytically that accumulate $\mathrm{BP}(\lambda)$ can control the level of bias by using a mixture of temporal difference errors and recursively defined eligibility traces. We next demonstrate empirically that our model outperforms the original implementation for learning synthetic gradients in a variety of tasks, and is particularly suited for capturing longer timescales. Finally, building on recent work we reflect on accumulate $\mathrm{BP}(\lambda)$ as a principle for learning in biological circuits. In summary, inspired by RL principles we introduce an algorithm capable of bias-free online learning via synthetic gradients.

RNN-like language models are getting renewed attention from NLP researchers in recent years and several models have made significant progress, which demonstrates performance comparable to traditional transformers. However, due to the recurrent nature of RNNs, this kind of language model can only store information in a set of fixed-length state vectors. As a consequence, they still suffer from forgetfulness though after a lot of improvements and optimizations, when given complex instructions or prompts. As the prompted generation is the main and most concerned function of LMs, solving the problem of forgetting in the process of generation is no wonder of vital importance. In this paper, focusing on easing the prompt forgetting during generation, we proposed an architecture to teach the model memorizing prompt during generation by synthetic gradient. To force the model to memorize the prompt, we derive the states that encode the prompt, then transform it into model parameter modification using low-rank gradient approximation, which hard-codes the prompt into model parameters temporarily. We construct a dataset for experiments, and the results have demonstrated the effectiveness of our method in solving the problem of forgetfulness in the process of prompted generation. We will release all the code upon acceptance.

A wide range of modern science and engineering applications are formulated as optimization problems with a system of partial differential equations (PDEs) as constraints. These PDE-constrained optimization problems are typically solved in a standard discretize-then-optimize approach. In many industry applications that require high-resolution solutions, the discretized constraints can easily have millions or even billions of variables, making it very slow for the standard iterative optimizer to solve the exact gradients. In this work, we propose a general framework to speed up PDE-constrained optimization using online neural synthetic gradients (ONSG) with a novel two-scale optimization scheme. We successfully apply our ONSG framework to computational morphogenesis, a representative and challenging class of PDE-constrained optimization problems. Extensive experiments have demonstrated that our method can significantly speed up computational morphogenesis (also known as topology optimization), and meanwhile maintain the quality of final solution compared to the standard optimizer. On a large-scale 3D optimal design problem with around 1,400,000 design variables, our method achieves up to 7.5x speedup while producing optimized designs with comparable objectives.

Many applications in machine learning require optimizing a function whose true gradient is unknown, but where surrogate gradient information (directions that may be correlated with, but not necessarily identical to, the true gradient) is available instead. This arises when an approximate gradient is easier to compute than the full gradient (e.g. in meta-learning or unrolled optimization), or when a true gradient is intractable and is replaced with a surrogate (e.g. in certain reinforcement learning applications, or when using synthetic gradients). We propose Guided Evolutionary Strategies, a method for optimally using surrogate gradient directions along with random search. We define a search distribution for evolutionary strategies that is elongated along a guiding subspace spanned by the surrogate gradients. This allows us to estimate a descent direction which can then be passed to a first-order optimizer. We analytically and numerically characterize the tradeoffs that result from tuning how strongly the search distribution is stretched along the guiding subspace, and we use this to derive a setting of the hyperparameters that works well across problems. Finally, we apply our method to example problems including truncated unrolled optimization and a synthetic gradient problem, demonstrating improvement over both standard evolutionary strategies and first-order methods that directly follow the surrogate gradient.

Artifical Neural Networks are a particular class of learning systems modeled after biological neural functions with an interesting penchant for Hebbian learning, that is "neurons that fire together, wire together". However, unlike their natural counterparts, artificial neural networks have a close and stringent coupling between the modules of neurons in the network. This coupling or locking imposes upon the network a strict and inflexible structure that prevent layers in the network from updating their weights until a full feed-forward and backward pass has occurred. Such a constraint though may have sufficed for a while, is now no longer feasible in the era of very-large-scale machine learning, coupled with the increased desire for parallelization of the learning process across multiple computing infrastructures. To solve this problem, synthetic gradients (SG) with decoupled neural interfaces (DNI) are introduced as a viable alternative to the backpropagation algorithm. This paper performs a speed benchmark to compare the speed and accuracy capabilities of SG-DNI as opposed to a standard neural interface using multilayer perceptron MLP. SG-DNI shows good promise, in that it not only captures the learning problem, it is also over 3-fold faster due to it asynchronous learning capabilities.

At the heart of deep learning we aim to use neural networks as function approximators - training them to produce outputs from inputs in emulation of a ground truth function or data creation process. In many cases we only have access to input-output pairs from the ground truth, however it is becoming more common to have access to derivatives of the target output with respect to the input -- for example when the ground truth function is itself a neural network such as in network compression or distillation. Generally these target derivatives are not computed, or are ignored. This paper introduces Sobolev Training for neural networks, which is a method for incorporating these target derivatives in addition the to target values while training. By optimising neural networks to not only approximate the function’s outputs but also the function’s derivatives we encode additional information about the target function within the parameters of the neural network. Thereby we can improve the quality of our predictors, as well as the data-efficiency and generalization capabilities of our learned function approximation. We provide theoretical justifications for such an approach as well as examples of empirical evidence on three distinct domains: regression on classical optimisation datasets, distilling policies of an agent playing Atari, and on large-scale applications of synthetic gradients. In all three domains the use of Sobolev Training, employing target derivatives in addition to target values, results in models with higher accuracy and stronger generalisation.

As a wild stream after a wet season in African savanna diverges into many smaller streams forming lakes and puddles, so deep learning has diverged into a myriad of specialized architectures. Each architecture has a diagram. Here are some of them. Neural networks are conceptually simple, and that's their beauty. A bunch of homogenous, uniform units, arranged in layers, weighted connections between them, and that's all.