Daily Papers

Analytic continual learning has emerged as a strong exemplar-free alternative to gradient-based class-incremental learning because it replaces iterative optimization with closed-form ridge updates. Yet the usual forgetting narrative, centered on stochastic gradient overwriting, does not explain why analytic methods still drift on old classes despite exact recursive solvers. We identify the culprit as spectral interference: the joint ridge classifier for all tasks shares the inverse autocorrelation operator (R+λI)^{-1}, so incoming task samples that load onto old dominant eigendirections dilute the spectrum and perturb old-class logits even when old labels are never revisited. Based on this view, we propose SPARCL, a spectral partitioned analytic continual learner that decomposes the running autocorrelation into a high-energy core and a residual complement, freezes old-class classifier components in the core subspace, and updates only the residual block through recursive least squares with an optional residual random-projection expansion. This yields a simple closed-form update with a provable invariance guarantee for the core contribution of old logits. Across CIFAR-100, CUB-200, ImageNet-R, and ImageNet-A under a frozen ViT-B/16 protocol, SPARCL closes most of the gap from classical analytic learners to strong representation matchers, while remaining complementary to sparse feature-decorrelation approaches such as Fly-CL.

One of the major limits of kernel ridge regression (KRR) is that storing and manipulating the kernel matrix K_n for n samples requires O(n^2) space, which rapidly becomes unfeasible for large n. Nystrom approximations reduce the space complexity to O(nm) by sampling m columns from K_n. Uniform sampling preserves KRR accuracy (up to epsilon) only when m is proportional to the maximum degree of freedom of K_n, which may require O(n) columns for datasets with high coherence. Sampling columns according to their ridge leverage scores (RLS) gives accurate Nystrom approximations with m proportional to the effective dimension, but computing exact RLS also requires O(n^2) space. (Calandriello et al. 2016) propose INK-Estimate, an algorithm that processes the dataset incrementally and updates RLS, effective dimension, and Nystrom approximations on-the-fly. Its space complexity scales with the effective dimension but introduces a dependency on the largest eigenvalue of K_n, which in the worst case is O(n). In this paper we introduce SQUEAK, a new algorithm that builds on INK-Estimate but uses unnormalized RLS. As a consequence, the algorithm is simpler, does not need to estimate the effective dimension for normalization, and achieves a space complexity that is only a constant factor worse than exact RLS sampling.

As efficient alternatives to softmax Attention, linear State-Space Models (SSMs) achieve constant memory and linear compute, but maintain only a lossy, fading summary of the past, often leading to inferior performance in recall-oriented tasks. We propose Gated KalmaNet (GKA), a layer that accounts for the full past while maintaining SSM-style efficiency. We ground our approach in the Kalman Filter (KF) framework, which provides a principled solution for optimal inference in dynamical systems. We show that several existing SSM layers (DeltaNet, Gated DeltaNet, and Kimi Delta Attention) are approximations to the KF recurrence that assume identity error covariance, thereby ignoring how past measurements (keys and values) should optimally influence state updates. In contrast, GKA computes the exact Kalman gain by maintaining the full error covariance. Under a steady-state assumption that enables parallelization, this reduces to solving an online ridge regression problem with constant memory and linear compute cost. A critical insight is that standard KF equations are numerically unstable in low-precision environments (like bfloat16) and hard to parallelize on modern hardware. We address this through: (1) adaptive regularization with input-dependent gating to control the condition number of the ridge regression for numerical stability, and (2) Chebyshev Iteration, which we show is more stable than conventional iterative solvers in low-precision settings. We further develop hardware-aware chunk-wise kernels to enable efficient training. Empirically, GKA outperforms existing SSM layers (like Mamba2 and Gated DeltaNet) on short-context tasks and achieves more than 10\% relative improvement on long-context RAG and LongQA tasks up to 128k tokens.

Large language models (LLMs) with extended context lengths rely on the key-value (KV) cache to support attention over prior tokens. However, maintaining the KV cache incurs substantial memory overhead, motivating KV-cache compression methods that enforce a fixed budget through eviction and merging. Modern eviction methods increasingly adopt span-based retention because preserving contiguous spans is empirically effective and better preserves semantic coherence. Yet, when combined with post-eviction merging, span-based retention concentrates merges onto a small set of span-boundary carrier tokens, producing a highly imbalanced merge pattern that exacerbates over-merging and increases information loss. To address this imbalance, we propose GRKV (Global Regression for KV Cache), a training-free KV-cache merging method that directly minimizes the discrepancy between compressed-cache and full-cache attention outputs. GRKV uses ridge-regression-based merge steps to distribute information from evicted tokens across retained tokens, while regularizing the updates to prevent over-smoothing. Across the LongBench and RULER long-context benchmarks, GRKV is the only merging method that improves overall performance with minimal overhead.

Most models in machine learning contain at least one hyperparameter to control for model complexity. Choosing an appropriate set of hyperparameters is both crucial in terms of model accuracy and computationally challenging. In this work we propose an algorithm for the optimization of continuous hyperparameters using inexact gradient information. An advantage of this method is that hyperparameters can be updated before model parameters have fully converged. We also give sufficient conditions for the global convergence of this method, based on regularity conditions of the involved functions and summability of errors. Finally, we validate the empirical performance of this method on the estimation of regularization constants of L2-regularized logistic regression and kernel Ridge regression. Empirical benchmarks indicate that our approach is highly competitive with respect to state of the art methods.