Approaching Backpropagation Accuracy with Weight Perturbation on MNIST

Deep neural networks process data through a cascade of representations: input features, hidden activations, logits, and loss. While perturbations at the input, logit, and label levels have been systematically studied, the intermediate hidden activations, which constitute the bulk of the network's computation, have received no unified perturbation analysis. In this paper, we establish a unified framework for hidden activation perturbation, revealing that Dropout, Manifold Mixup, adversarial feature perturbation, and related methods all impose specific forms of activation perturbation but with class-agnostic or random strategies. We conjecture that expansive perturbation (increasing activation norm) acts as positive augmentation, while contractive perturbation (decreasing activation norm) acts as negative augmentation, and that the perturbation layer determines whether the effect resembles input-level augmentation (shallow layers) or logit-level manipulation (deep layers). We propose Learning to Perturb Activations (LPA), which adaptively perturbs activations at a selected hidden layer with class-level perturbations learned via PGD. We further provide theoretical analysis connecting activation perturbation to flat minima and perturbation amplification through layers. Experiments on balanced classification, long-tail classification, and domain generalization demonstrate that LPA consistently outperforms existing methods and provides complementary benefits to logit perturbation methods such as LPL.

We study classification dynamics under small initialization and identify a pronounced non-monotone behavior of accuracy that we call the accuracy well: accuracy rises rapidly early in training, subsequently drops, and only much later recovers. We observe this phenomenon in deep linear networks, fully connected ReLU networks on MNIST, deep convolutional networks on CIFAR-10, and controlled synthetic tasks, across different losses and optimizers and in both training and test accuracy. We propose a three-stage explanation in terms of target-driven organization, a degeneracy bottleneck, and subsequent recruitment and recovery. We develop a rigorous theory of these dynamics for shallow and deep linear networks, showing in particular that high accuracy can emerge while the predictor and the change in loss remain asymptotically vanishing, before continued optimization causes accuracy to deteriorate. For a simple synthetic classification task, we prove an extreme instance in which accuracy asymptotically rises to 100%, collapses to 0, and later recovers permanently to 100%. Our results show that classification accuracy can reveal aspects of learning dynamics that are not captured by the training loss alone.

Deep neural network training involves both forward propagation (from features through logits to loss) and backward propagation (from loss through gradients to parameter updates). While perturbations along the forward chain, including feature perturbation, logit perturbation, and label perturbation, have been extensively studied, the backward chain's gradient perturbation has received little systematic investigation. In this paper, we establish a unified framework for gradient perturbation, revealing that existing methods such as Sharpness-Aware Minimization (SAM), gradient clipping, and gradient noise injection can all be interpreted as imposing specific forms of gradient perturbation. Analogous to the recently proposed Logit Perturbation Learning (LPL), we conjecture that amplifying the gradient norm for a class acts as positive augmentation (enhancing learning), while dampening it acts as negative augmentation (suppressing overfitting). Based on these observations, we propose Learning to Perturb Gradients (LPG), which adaptively perturbs logit-level gradients at the class level to achieve category-aware training. We also establish theoretical connections between gradient perturbation bounds and generalization guarantees via PAC-Bayesian analysis. Experiments on balanced classification, long-tail classification, and noisy label learning demonstrate that LPG consistently outperforms existing methods and can be combined with them as a plug-in module.