λ-GELU: Learning Gating Hardness for Controlled ReLU-ization in Deep Networks

Gaussian Error Linear Unit (GELU) is a widely used smooth alternative to Rectifier Linear Unit (ReLU), yet many deployment, compression, and analysis toolchains are most naturally expressed for piecewise-linear (ReLU-type) networks. We study a hardness-parameterized formulation of GELU, f(x;λ)=xΦ(λ x), where Φ is the Gaussian CDF and λ \in [1, infty) controls gate sharpness, with the goal of turning smooth gated training into a controlled path toward ReLU-compatible models. Learning λ is non-trivial: naive updates yield unstable dynamics and effective gradient attenuation, so we introduce a constrained reparameterization and an optimizer-aware update scheme. Empirically, across a diverse set of model--dataset pairs spanning MLPs, CNNs, and Transformers, we observe structured layerwise hardness profiles and assess their robustness under different initializations. We further study a deterministic ReLU-ization strategy in which the learned gates are progressively hardened toward a principled target, enabling a post-training substitution of λ-GELU by ReLU with reduced disruption. Overall, λ-GELU provides a minimal and interpretable knob to profile and control gating hardness, bridging smooth training with ReLU-centric downstream pipelines.

When Learning Rates Go Wrong: Early Structural Signals in PPO Actor-Critic

Deep Reinforcement Learning systems are highly sensitive to the learning rate (LR), and selecting stable and performant training runs often requires extensive hyperparameter search. In Proximal Policy Optimization (PPO) actor--critic methods, small LR values lead to slow convergence, whereas large LR values may induce instability or collapse. We analyse this phenomenon from the behavior of the hidden neurons in the network using the Overfitting-Underfitting Indicator (OUI), a metric that quantifies the balance of binary activation patterns over a fixed probe batch. We introduce an efficient batch-based formulation of OUI and derive a theoretical connection between LR and activation sign changes, clarifying how a correct evolution of the neuron's inner structure depends on the step size. Empirically, across three discrete-control environments and multiple seeds, we show that OUI measured at only 10\% of training already discriminates between LR regimes. We observe a consistent asymmetry: critic networks achieving highest return operate in an intermediate OUI band (avoiding saturation), whereas actor networks achieving highest return exhibit comparatively high OUI values. We then compare OUI-based screening rules against early return, clip-based, divergence-based, and flip-based criteria under matched recall over successful runs. In this setting, OUI provides the strongest early screening signal: OUI alone achieves the best precision at broader recall, while combining early return with OUI yields the highest precision in best-performing screening regimes, enabling aggressive pruning of unpromising runs without requiring full training.

Regime Change Hypothesis: Foundations for Decoupled Dynamics in Neural Network Training

Despite the empirical success of DNN, their internal training dynamics remain difficult to characterize. In ReLU-based models, the activation pattern induced by a given input determines the piecewise-linear region in which the network behaves affinely. Motivated by this geometry, we investigate whether training exhibits a two-timescale behavior: an early stage with substantial changes in activation patterns and a later stage where weight updates predominantly refine the model within largely stable activation regimes. We first prove a local stability property: outside measure-zero sets of parameters and inputs, sufficiently small parameter perturbations preserve the activation pattern of a fixed input, implying locally affine behavior within activation regions. We then empirically track per-iteration changes in weights and activation patterns across fully-connected and convolutional architectures, as well as Transformer-based models, where activation patterns are recorded in the ReLU feed-forward (MLP/FFN) submodules, using fixed validation subsets. Across the evaluated settings, activation-pattern changes decay 3 times earlier than weight-update magnitudes, showing that late-stage training often proceeds within relatively stable activation regimes. These findings provide a concrete, architecture-agnostic instrument for monitoring training dynamics and motivate further study of decoupled optimization strategies for piecewise-linear networks. For reproducibility, code and experiment configurations will be released upon acceptance.

Why Adam Works Better with $β_1 = β_2$: The Missing Gradient Scale Invariance Principle

Adam has been at the core of large-scale training for almost a decade, yet a simple empirical fact remains unaccounted for: both validation scores and the qualitative behaviour of the training runs improve when the momentum parameters satisfy $β_{1}=β_{2}$. Some recent studies have reported this pattern, but there is still no explanation for why this choice helps. We show that this choice is closely tied to a structural property that we refer to as \textit{gradient scale invariance}. We formalize this notion and prove that Adam becomes gradient scale invariant of first order if and only if $β_{1}=β_{2}$. This perspective places the balanced regime of Adam in direct alignment with the design principles underlying several recent optimizers that explicitly enforce scale-robust updates. The theory is supported by experiments across vision and language tasks, and across different architectural families, in which rescaling the gradient has a markedly smoother effect on the update when $β_{1}=β_{2}$. Overall, our results offer a coherent explanation for an open question in the behavior of Adam and provide a simple principle that helps guide the design of future optimizers.