Qianyi Li

Qianyi Li, Haim Sompolinsky · harvard

Recently proposed Gated Linear Networks present a tractable nonlinear network architecture, and exhibit interesting capabilities such as learning with local error signals and reduced forgetting in sequential learning. In this work, we introduce a novel gating architecture, named Globally Gated Deep Linear Networks (GGDLNs) where gating units are shared among all processing units in each layer, thereby decoupling the architectures of the nonlinear but unlearned gatings and the learned linear processing motifs. We derive exact equations for the generalization properties in these networks in the finite-width thermodynamic limit, defined by $P,N\rightarrow\infty, P/N\sim O(1)$, where P and N are the training sample size and the network width respectively. We find that the statistics of the network predictor can be expressed in terms of kernels that undergo shape renormalization through a data-dependent matrix compared to the GP kernels. Our theory accurately captures the behavior of finite width GGDLNs trained with gradient descent dynamics. We show that kernel shape renormalization gives rise to rich generalization properties w.r.t. network width, depth and L2 regularization amplitude. Interestingly, networks with sufficient gating units behave similarly to standard ReLU networks. Although gatings in the model do not participate in supervised learning, we show the utility of unsupervised learning of the gating parameters. Additionally, our theory allows the evaluation of the network's ability for learning multiple tasks by incorporating task-relevant information into the gating units. In summary, our work is the first exact theoretical solution of learning in a family of nonlinear networks with finite width. The rich and diverse behavior of the GGDLNs suggests that they are helpful analytically tractable models of learning single and multiple tasks, in finite-width nonlinear deep networks.

Haozhe Shan, Qianyi Li, Haim Sompolinsky · harvard

Continual learning (CL) enables animals to learn new tasks without erasing prior knowledge. CL in artificial neural networks (NNs) is challenging due to catastrophic forgetting, where new learning degrades performance on older tasks. While various techniques exist to mitigate forgetting, theoretical insights into when and why CL fails in NNs are lacking. Here, we present a statistical-mechanics theory of CL in deep, wide NNs, which characterizes the network's input-output mapping as it learns a sequence of tasks. It gives rise to order parameters (OPs) that capture how task relations and network architecture influence forgetting and anterograde interference, as verified by numerical evaluations. For networks with a shared readout for all tasks (single-head CL), the relevant-feature and rule similarity between tasks, respectively measured by two OPs, are sufficient to predict a wide range of CL behaviors. In addition, the theory predicts that increasing the network depth can effectively reduce interference between tasks, thereby lowering forgetting. For networks with task-specific readouts (multi-head CL), the theory identifies a phase transition where CL performance shifts dramatically as tasks become less similar, as measured by another task-similarity OP. While forgetting is relatively mild compared to single-head CL across all tasks, sufficiently low similarity leads to catastrophic anterograde interference, where the network retains old tasks perfectly but completely fails to generalize new learning. Our results delineate important factors affecting CL performance and suggest strategies for mitigating forgetting.

Yehonatan Avidan, Qianyi Li, Haim Sompolinsky · harvard

Artificial neural networks have revolutionized machine learning in recent years, but a complete theoretical framework for their learning process is still lacking. Substantial advances were achieved for wide networks, within two disparate theoretical frameworks: the Neural Tangent Kernel (NTK), which assumes linearized gradient descent dynamics, and the Bayesian Neural Network Gaussian Process (NNGP). We unify these two theories using gradient descent learning with an additional noise in an ensemble of wide deep networks. We construct an analytical theory for the network input-output function and introduce a new time-dependent Neural Dynamical Kernel (NDK) from which both NTK and NNGP kernels are derived. We identify two learning phases: a gradient-driven learning phase, dominated by loss minimization, in which the time scale is governed by the initialization variance. It is followed by a slow diffusive learning stage, where the parameters sample the solution space, with a time constant decided by the noise and the Bayesian prior variance. The two variance parameters strongly affect the performance in the two regimes, especially in sigmoidal neurons. In contrast to the exponential convergence of the mean predictor in the initial phase, the convergence to the equilibrium is more complex and may behave nonmonotonically. By characterizing the diffusive phase, our work sheds light on representational drift in the brain, explaining how neural activity changes continuously without degrading performance, either by ongoing gradient signals that synchronize the drifts of different synapses or by architectural biases that generate task-relevant information that is robust against the drift process. This work closes the gap between the NTK and NNGP theories, providing a comprehensive framework for the learning process of deep wide neural networks and for analyzing dynamics in biological circuits.

Qianyi Li, Haim Sompolinsky

The success of deep learning in many real-world tasks has triggered an intense effort to understand the power and limitations of deep learning in the training and generalization of complex tasks, so far with limited progress. In this work, we study the statistical mechanics of learning in Deep Linear Neural Networks (DLNNs) in which the input-output function of an individual unit is linear. Despite the linearity of the units, learning in DLNNs is nonlinear, hence studying its properties reveals some of the features of nonlinear Deep Neural Networks (DNNs). Importantly, we solve exactly the network properties following supervised learning using an equilibrium Gibbs distribution in the weight space. To do this, we introduce the Back-Propagating Kernel Renormalization (BPKR), which allows for the incremental integration of the network weights starting from the network output layer and progressing backward until the first layer's weights are integrated out. This procedure allows us to evaluate important network properties, such as its generalization error, the role of network width and depth, the impact of the size of the training set, and the effects of weight regularization and learning stochasticity. BPKR does not assume specific statistics of the input or the task's output. Furthermore, by performing partial integration of the layers, the BPKR allows us to compute the properties of the neural representations across the different hidden layers. We have proposed an extension of the BPKR to nonlinear DNNs with ReLU. Surprisingly, our numerical simulations reveal that despite the nonlinearity, the predictions of our theory are largely shared by ReLU networks in a wide regime of parameters. Our work is the first exact statistical mechanical study of learning in a family of DNNs, and the first successful theory of learning through successive integration of DoFs in the learned weight space.