Labeling a training set is often expensive and susceptible to errors, making the design of robust loss functions for label noise an important problem. The symmetry condition provides theoretical guarantees for robustness to such noise. In this work, we study a symmetrization method arising from the unique decomposition of any multi-class loss function into a symmetric component and a class-insensitive term. In particular, symmetrizing the cross-entropy loss leads to a linear multi-class extension of the unhinged loss. Unlike in the binary case, the multi-class version must have specific coefficients in order to satisfy the symmetry condition. Under suitable assumptions, we show that this multi-class unhinged loss is the unique convex multi-class symmetric loss. We also show that it has a fundamental local role: the linear approximation of any symmetric loss around score vectors with equal components is equivalent to the multi-class unhinged loss. We then introduce SGCE and alpha-MAE, two loss functions that interpolate between the multi-class unhinged loss and the Mean Absolute Error while allowing control of the beta-smoothness of the loss. Experiments on standard noisy-label benchmarks show competitive performance compared with existing robust loss functions.

Large language models demonstrate remarkable ability in factual recall, yet the fundamental limits of storing and retrieving input--output associations with neural networks remain unclear. We study these limits in a minimal setting: a linear associative memory that maps $p$ input embeddings in $\mathbb{R}^d$ to their corresponding~$d$-dimensional targets via a single layer, requiring each mapped input to be well separated from all other targets. Unlike in supervised classification, this strict separation induces~$p$ constraints per association and produces strong correlations between constraints that make a direct characterisation of the storage capacity difficult. Here, we provide a precise characterisation of this capacity in the following way. We first introduce a decoupled model in which each input has its own independent set of competing outputs, and provide numerical and analytical evidence that this decoupled model is equivalent to the original model in terms of storage capacity, spectra of the learnt weights, and storage mechanism. Using tools from statistical physics, we show that the decoupled model can store up to $p_c \log p_c / d^2 = 1 / 2$ associations, and generalise the computation of $p_c$ to linear two-layer architectures. Our analysis also gives mechanistic insight into how the optimal solution improves over a naïve Hebbian learning rule: rather than boosting input-output alignments with broad fluctuations, the optimal solution raises the correct scores just above the extreme-value threshold set by the competing outputs. These findings give a sharp statistical-physics characterisation of factual storage in linear networks and provide a baseline for understanding the memory capacity of more realistic neural architectures.

We thank all the reviewers (R1,R2,R3) for their feedback and suggestions.1 Table A: Multi-task comparison across task weights. We have per-2 formed loss balancing with five different weights t3 in the multi-task loss Lm = t Lc +(1 t) Lr for4 the classification and regression losses. The results5 on OmniArt are reported in Table A. Our proposal6 is robust to the weight value, tuning the task weight7 is not vital. We obtain a moderate gain for both clas-8 sification and regression with a weight of t = 0.25.9 For the multi-task baseline, emphasizing regression10 reduces the regression error, as the gradient magnitude of the regression loss is much lower than the one for the11 classification loss.

Pc i=1 yi = 1is satisfied, otherwise f (y) = by duality. A.2 Experiments on Binary Classification with Exponential Loss Here we present the results on a binary classification task over a synthetic dataset of 100 dimensional gaussian clusters. For Σ, similar to [23], we sample a diagonal matrix D, where each entry is sampled uniformly from a specified range, and a rotation matrix U from a HAAR distribution, giving Σ = UDUT. For the source data, we sample µ 1s,µ+1s,Σ 1s,Σ+1sas specified above with k = 0. Now to create a distribution shifted data of various severity, we sample µ 1t,µ+1t,Σ 1t,Σ+1tas specified above with k = 1, which are then used to sample the shifted data as follows: Exponential Loss for Binary Classification Let z be the classification score hθ(x). For logistic training loss, conjugate adaptation loss would default to entropy with sigmoid probability.