Neural networks are known to be susceptible to over-reliance on spurious correlations. However, the precise mechanism by which models exploit shortcut features is not fully understood, and algorithms to mitigate this behavior rely on as yet unjustified assumptions about the learned representations. In this work, we provide the first end-to-end theoretical characterization of spurious feature learning for two-layer ReLU neural networks trained by online minibatch SGD on the logistic loss. We consider data drawn from the high-dimensional Boolean hypercube with a quadratic signal function (namely XOR) and a linear spurious correlation. We show that SGD learns the spurious feature first, and exponentially fast. Moreover, the optimization dynamics couple the spurious and signal features, with a stronger spurious component inhibiting signal feature learning. Our analysis reveals precise phase transitions in the learning dynamics. In the first phase, alignment between the signs of the spurious feature and second-layer weight drives rapid growth of the spurious feature. In the second phase, large majority group margin slows learning and the signal feature remains suppressed. When the spurious correlation is maximally strong, we show theoretically that the spurious feature dominates even at the sample complexity threshold where XOR would be learned in isolation (i.e., if the spurious feature was absent). In contrast, when the correlation strength is constant, we provide preliminary empirical evidence that the model can eventually learn the XOR signal, although the spurious feature is not forgotten.

Recent work suggests that large language models (LLMs) can perform multi-hop reasoning implicitly--producing correct answers without explicitly verbalizing intermediate steps--but the underlying mechanisms remain poorly understood. In this paper, we study how such implicit reasoning emerges by training transformers from scratch in a controlled symbolic environment. Our analysis reveals a threestage developmental trajectory: early memorization, followed by in-distribution generalization, and eventually cross-distribution generalization. We find that training with atomic triples is not necessary but accelerates learning, and that second-hop generalization relies on query-level exposure to specific compositional structures. To interpret these behaviors, we introduce two diagnostic tools: cross-query semantic patching, which identifies semantically reusable intermediate representations, and a cosine-based representational lens, which reveals that successful reasoning correlates with the cosine-base clustering in hidden space. This clustering phenomenon in turn provides a coherent explanation for the behavioral dynamics observed across training, linking representational structure to reasoning capability. These findings provide new insights into the interpretability of implicit multi-hop reasoning in LLMs, helping to clarify how complex reasoning processes unfold internally and offering pathways to enhance the transparency of such models.

Gradient Descent (GD) is a simple yet efficient baseline algorithm for solving nonconvex optimization problems.

In our comparativestudy, we choose 4 challenging benchmark datasets for feature selection evaluation.

Existing methods for label-deficient concealed object segmentation (LDCOS) either rely on consistency constraints or Segment Anything Model (SAM)-based pseudo-labeling. However, their performance remains limited due to the intrinsic concealment of targets and the scarcity of annotations. This study investigates two key questions: (1) Can consistency constraints and SAM-based supervision be jointly integrated to better exploit complementary information and enhance the segmenter? and (2) beyond that, can the segmenter in turn guide SAM through reciprocal supervision, enabling mutual improvement? To answer these questions, we present SCALER, a unified collaborative framework toward LDCOS that jointly optimizes a mean-teacher segmenter and a learnable SAM. SCALER operates in two alternating phases. In \textbf{Phase \uppercase\expandafter{\romannumeral1}}, the segmenter is optimized under fixed SAM supervision using entropy-based image-level and uncertainty-based pixel-level weighting to select reliable pseudo-label regions and emphasize harder examples. In \textbf{Phase \uppercase\expandafter{\romannumeral2}}, SAM is updated via augmentation invariance and noise resistance losses, leveraging its inherent robustness to perturbations. Experiments demonstrate that SCALER yields consistent performance gains across eight semi- and weakly-supervised COS tasks. The results further suggest that SCALER can serve as a general training paradigm to enhance both lightweight segmenters and large foundation models under label-scarce conditions. Code will be released.