We analyze the dynamics of finite width effects in wide but finite feature learning neural networks. Unlike many prior analyses, our results, while perturbative in width, are non-perturbative in the strength of feature learning. Starting from a dynamical mean field theory (DMFT) description of infinite width deep neural network kernel and prediction dynamics, we provide a characterization of the $\mathcal{O}(1/\sqrt{\text{width}})$ fluctuations of the DMFT order parameters over random initialization of the network weights. In the lazy limit of network training, all kernels are random but static in time and the prediction variance has a universal form. However, in the rich, feature learning regime, the fluctuations of the kernels and predictions are dynamically coupled with variance that can be computed self-consistently. In two layer networks, we show how feature learning can dynamically reduce the variance of the final NTK and final network predictions. We also show how initialization variance can slow down online learning in wide but finite networks. In deeper networks, kernel variance can dramatically accumulate through subsequent layers at large feature learning strengths, but feature learning continues to improve the SNR of the feature kernels. In discrete time, we demonstrate that large learning rate phenomena such as edge of stability effects can be well captured by infinite width dynamics and that initialization variance can decrease dynamically. For CNNs trained on CIFAR-10, we empirically find significant corrections to both the bias and variance of network dynamics due to finite width.
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Sliding window sums are widely used for string indexing, hashing and time series analysis. We have developed a family of the generic vectorized sliding sum algorithms that provide speedup of O(P/w) for window size $w$ and number of processors P. For a sum with a commutative operator the speedup is improved to O(P/log(w)). Even more important, our algorithms exhibit efficient memory access patterns. In this paper we study the application of the sliding sum algorithms to the training and inference of the Deep Neural Networks. We demonstrate how both pooling and convolution primitives could be expressed as sliding sums and evaluated by the compute kernels with the shared structure. We show that the sliding sum convolution kernels are more efficient than the commonly used GEMM kernels on the CPU, and could even outperform their GPU counterparts.
We analyze the dynamics of finite width effects in wide but finite feature learning neural networks. Starting from a dynamical mean field theory description of infinite width deep neural network kernel and prediction dynamics, we provide a characterization of the $\mathcal{O}(1/\sqrt{\text{width}})$ fluctuations of the DMFT order parameters over random initializations of the network weights. Our results, while perturbative in width, unlike prior analyses, are non-perturbative in the strength of feature learning. In the lazy limit of network training, all kernels are random but static in time and the prediction variance has a universal form. However, in the rich, feature learning regime, the fluctuations of the kernels and predictions are dynamically coupled with a variance that can be computed self-consistently. In two layer networks, we show how feature learning can dynamically reduce the variance of the final tangent kernel and final network predictions. We also show how initialization variance can slow down online learning in wide but finite networks. In deeper networks, kernel variance can dramatically accumulate through subsequent layers at large feature learning strengths, but feature learning continues to improve the signal-to-noise ratio of the feature kernels. In discrete time, we demonstrate that large learning rate phenomena such as edge of stability effects can be well captured by infinite width dynamics and that initialization variance can decrease dynamically. For CNNs trained on CIFAR-10, we empirically find significant corrections to both the bias and variance of network dynamics due to finite width.
This paper investigates the relationship between the universal approximation property of deep neural networks and topological characteristics of datasets. Our primary contribution is to introduce data topology-dependent upper bounds on the network width. Specifically, we first show that a three-layer neural network, applying a ReLU activation function and max pooling, can be designed to approximate an indicator function over a compact set, one that is encompassed by a tight convex polytope. This is then extended to a simplicial complex, deriving width upper bounds based on its topological structure. Further, we calculate upper bounds in relation to the Betti numbers of select topological spaces. Finally, we prove the universal approximation property of three-layer ReLU networks using our topological approach. We also verify that gradient descent converges to the network structure proposed in our study.
While distributional reinforcement learning (RL) has demonstrated empirical success, the question of when and why it is beneficial has remained unanswered. In this work, we provide one explanation for the benefits of distributional RL through the lens of small-loss bounds, which scale with the instance-dependent optimal cost. If the optimal cost is small, our bounds are stronger than those from non-distributional approaches. As warmup, we show that learning the cost distribution leads to small-loss regret bounds in contextual bandits (CB), and we find that distributional CB empirically outperforms the state-of-the-art on three challenging tasks. For online RL, we propose a distributional version-space algorithm that constructs confidence sets using maximum likelihood estimation, and we prove that it achieves small-loss regret in the tabular MDPs and enjoys small-loss PAC bounds in latent variable models. Building on similar insights, we propose a distributional offline RL algorithm based on the pessimism principle and prove that it enjoys small-loss PAC bounds, which exhibit a novel robustness property. For both online and offline RL, our results provide the first theoretical benefits of learning distributions even when we only need the mean for making decisions.
This work introduces an empirical quadrature-based hyperreduction procedure and greedy training algorithm to effectively reduce the computational cost of solving convection-dominated problems with limited training. The proposed approach circumvents the slowly decaying $n$-width limitation of linear model reduction techniques applied to convection-dominated problems by using a nonlinear approximation manifold systematically defined by composing a low-dimensional affine space with bijections of the underlying domain. The reduced-order model is defined as the solution of a residual minimization problem over the nonlinear manifold. An online-efficient method is obtained by using empirical quadrature to approximate the optimality system such that it can be solved with mesh-independent operations. The proposed reduced-order model is trained using a greedy procedure to systematically sample the parameter domain. The effectiveness of the proposed approach is demonstrated on two shock-dominated computational fluid dynamics benchmarks.
PCA-Net is a recently proposed neural operator architecture which combines principal component analysis (PCA) with neural networks to approximate operators between infinite-dimensional function spaces. The present work develops approximation theory for this approach, improving and significantly extending previous work in this direction: First, a novel universal approximation result is derived, under minimal assumptions on the underlying operator and the data-generating distribution. Then, two potential obstacles to efficient operator learning with PCA-Net are identified, and made precise through lower complexity bounds; the first relates to the complexity of the output distribution, measured by a slow decay of the PCA eigenvalues. The other obstacle relates to the inherent complexity of the space of operators between infinite-dimensional input and output spaces, resulting in a rigorous and quantifiable statement of the curse of dimensionality. In addition to these lower bounds, upper complexity bounds are derived. A suitable smoothness criterion is shown to ensure an algebraic decay of the PCA eigenvalues. Furthermore, it is shown that PCA-Net can overcome the general curse of dimensionality for specific operators of interest, arising from the Darcy flow and the Navier-Stokes equations.
We study the mean field Langevin dynamics and the associated particle system. By assuming the functional convexity of the energy, we obtain the $L^p$-convergence of the marginal distributions towards the unique invariant measure for the mean field dynamics. Furthermore, we prove the uniform-in-time propagation of chaos in both the $L^2$-Wasserstein metric and relative entropy.
This book develops an effective theory approach to understanding deep neural networks of practical relevance. Beginning from a first-principles component-level picture of networks, we explain how to determine an accurate description of the output of trained networks by solving layer-to-layer iteration equations and nonlinear learning dynamics. A main result is that the predictions of networks are described by nearly-Gaussian distributions, with the depth-to-width aspect ratio of the network controlling the deviations from the infinite-width Gaussian description. We explain how these effectively-deep networks learn nontrivial representations from training and more broadly analyze the mechanism of representation learning for nonlinear models. From a nearly-kernel-methods perspective, we find that the dependence of such models' predictions on the underlying learning algorithm can be expressed in a simple and universal way. To obtain these results, we develop the notion of representation group flow (RG flow) to characterize the propagation of signals through the network. By tuning networks to criticality, we give a practical solution to the exploding and vanishing gradient problem. We further explain how RG flow leads to near-universal behavior and lets us categorize networks built from different activation functions into universality classes. Altogether, we show that the depth-to-width ratio governs the effective model complexity of the ensemble of trained networks. By using information-theoretic techniques, we estimate the optimal aspect ratio at which we expect the network to be practically most useful and show how residual connections can be used to push this scale to arbitrary depths. With these tools, we can learn in detail about the inductive bias of architectures, hyperparameters, and optimizers.
Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation or neither of these. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.
To address the sparsity and cold start problem of collaborative filtering, researchers usually make use of side information, such as social networks or item attributes, to improve recommendation performance. This paper considers the knowledge graph as the source of side information. To address the limitations of existing embedding-based and path-based methods for knowledge-graph-aware recommendation, we propose Ripple Network, an end-to-end framework that naturally incorporates the knowledge graph into recommender systems. Similar to actual ripples propagating on the surface of water, Ripple Network stimulates the propagation of user preferences over the set of knowledge entities by automatically and iteratively extending a user's potential interests along links in the knowledge graph. The multiple "ripples" activated by a user's historically clicked items are thus superposed to form the preference distribution of the user with respect to a candidate item, which could be used for predicting the final clicking probability. Through extensive experiments on real-world datasets, we demonstrate that Ripple Network achieves substantial gains in a variety of scenarios, including movie, book and news recommendation, over several state-of-the-art baselines.
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