#01Nov 3, 2026
cs.LG
Universal Hardness From Masking Channels
Parikshit Gopalan, Manoj Prabhakaran, Pranay Tankala
We prove a broad learning-theoretic generalization of the celebrated hardcore lemma concerning the average-case complexity of computing a Boolean function (Impagliazzo, 1995). Our starting point is the observation that the classic notion of a hardcore measure, which reweights a function's input distribution, is equivalent to an input-dependent masking channel that injects randomness into the function's output distribution until it appears fully pseudorandom. The analogue of density of the measure is the fidelity of the channel. From this viewpoint, we develop a general framework for certifying hardness that allows for non-Boolean functions $f$ and a wide range of notions of hardness besides the classic 0-1 loss, by choosing the right notion of fidelity via convex duality. We show the difficulty of learning $f$ using a class $\mathcal{C}$ to minimize a convex loss is tightly characterized by the maximum fidelity possible for a masking channel, which measures the least amount of noise the channel must add to hide the structure of $f$ and make it $\mathcal{C}$-indistinguishable from random noise. Using our framework, we derive a number of multiclass generalizations of the hardcore lemma, and reveal some unexpected subtleties in going to larger alphabets. We construct a universal masking channel that applies to many natural notions of hardness simultaneously. Such channels are dual to omnipredictors, which optimize multiple losses at once (Gopalan, Kalai, Reingold, Sharan, and Wieder, 2022). We observe a symmetry between masking channels and generalizations of the hardcore lemma using pseudoentropy. We present a tight characterization of pseudoentropy using the MaxEnt algorithm from machine learning. A long line of research has shown that learning and hardness are two sides of the same coin; our framework gives a crisp and general formulation of this principle via convex duality.
#02Nov 3, 2026
cs.DS
Gaussian Partitioning for Reliable Learning of Multi-Index Models
Ilias Diakonikolas, Giannis Iakovidis, Daniel M. Kane and 1 more
We introduce Gaussian partitioning, an algorithmic primitive for distributions smooth with respect to a Gaussian. It partitions space into polyhedral regions so that almost all mass lies in thin strips restricting a few orthogonal directions. Within each strip, the conditional distribution approximately matches low-degree Gaussian moments in the unrestricted directions. Our construction builds on logconcave partitioning of Diakonikolas, Kane, and Ma (FOCS 2026). Under input marginals smooth with respect to the standard Gaussian, we obtain agnostic learners with error $(1+c)\mathrm{opt}+ε$ for broad classes of Boolean multi-index models, including functions of halfspaces, low-dimensional convex concepts, and low-dimensional polynomial threshold functions. For functions of $k$ halfspaces, both agnostic and testable agnostic learners achieve this error with dimension dependence $d^{\mathrm{poly}(k/c)}$, improving the previous exponential dependence on $k/c$ in the agnostic exponent. For learning from positive examples with imperfect unlabeled data (PUI), we give algorithms and Statistical Query (SQ) lower bounds that qualitatively characterize complexity through Gaussian moment matching. For fixed smoothness and positive-mass parameters, we learn intersections of $k$ halfspaces with dimension dependence $d^{O(\log(k/ε))}$ and matching SQ lower bounds, improving the previous $d^{\mathrm{poly}(1/ε)}$ upper bound even for one halfspace. A general reduction from agnostic reliable learning to PUI yields negative reliable learners for these intersections with the same dimension dependence, for fixed smoothness. Finally, for PQ learning and testable learning with distribution shift, we prove qualitatively optimal SQ lower bounds of $d^{Ω(1/ε^2)}$, even for intersections of two halfspaces with Gaussian training data and arbitrary test distributions.
#03Nov 3, 2026
cs.LG
LoopVQ: Sub 1-bit KV Cache Quantization for Looped Transformers
Minsoo Cheong, Sungjoo Yoo
Looped transformers are a parameter-efficient alternative to standard transformers for language modeling. By reapplying the same layers across multiple loop steps, recent looped language models match the performance of non-looped models several times their size. While this architecture effectively reduces parameter memory, each loop step still stores its own keys and values, so the key-value (KV) cache grows linearly with the number of loop steps and becomes the main memory bottleneck. We observe that the KV cache of looped models is highly redundant along the loop axis and that the loop steps differ substantially in their sensitivity to quantization. Based on these observations, we propose $\textbf{LoopVQ}$, a KV cache vector quantization method for looped transformers that codes channels of multiple loop steps jointly, and fine-tunes the codebooks to match the output of the full-precision model. At 0.5 bits per element, LoopVQ outperforms existing KV cache vector quantization methods for standard LLMs on general, long chain-of-thought reasoning, and long-context retrieval tasks with Ouro and Huginn models. Its design is orthogonal to these methods, and combined with TaSQ, it retains much of the BF16 accuracy even at 0.25 bits per element. We also implement LoopVQ in SGLang, where it nearly halves the decode latency at a 32K-token context and raises the peak serving throughput by up to $4.3\times$ and $6.1\times$ on the 1.4B and 2.6B Ouro models compared to the BF16 baseline.
#04Nov 3, 2026
cs.LG
Logarithmic Regret for Alternating Mirror Descent
Tianlong Nan, Christian Kroer
Alternating online mirror descent (AltOMD) is a simple learning rule for zero-sum games, which improves upon simultaneous OMD numerically and theoretically. Previous work obtained $O(T^{1/3})$ and $O(T^{1/5})$ regret bounds for AltOMD, by leveraging modified-energy-function-based analyses. However, settling whether AltOMD achieves near-constant regret has remained an open question. In this paper we resolve the question of whether AltOMD achieves near-constant regret in the affirmative. In particular, we show an $O(log(T))$ regret bound on the sum of the players' external regrets for any two-player zero-sum bilinear game with compact convex strategy sets, under mild regularity assumptions on the regularizers. We verify that our assumptions hold for the negative entropy regularizer, thereby obtaining logarithmic regret for the alternating multiplicative weights updates algorithm. The regret bound also yields an $O(log(T)/T)$ duality gap bound for the averaged strategies.
#05Nov 3, 2026
cs.LG
Going Deeper With Memories
Afshin Khadangi
Byte level language models avoid tokenizer dependence and provide universal coverage, but processing every byte at full resolution is expensive. We introduce DeepMem, a causal multiscale architecture that separates semantic compression from exact byte preservation. A learned invertible subband transform constructs a hierarchy of shorter sequences, while a lossless carrier and explicit detail sidebands preserve byte identity. Fast weight memory blocks operate along the encoder, with iterative shared state computation at the compressed bottleneck, while an attention based super resolution decoder restores byte resolution representations. DeepMem achieves stronger language modeling performance than a matched byte level Transformer on FineWeb-Edu and PG19, and obtains the best macro average performance across the evaluated FLORES languages. It also improves PIQA accuracy while remaining competitive on code modeling, multiple choice reasoning, and byte level corruption robustness. Bottleneck refinement requires approximately $7{,}600\times$ fewer counted FLOPs than equivalent full resolution reasoning, while maintaining competitive evaluation throughput.