How AI settled the complexity of the oldest SGD algorithm

Long-context language modeling remains central to modern sequence modeling, but the quadratic cost of Transformer attention makes scaling computationally prohibitive. Linear recurrent models address this bottleneck by compressing the context into a fixed-size state, making the rule that forgets, writes, and edits information a central design problem. To address state maintenance, Gated DeltaNet (GDN) combines gated state decay with delta-rule residual writes, using a learnable coefficient to balance forgetting and update magnitude. However, this coefficient is learned empirically rather than derived from the underlying objective, which can lead to suboptimal update magnitudes. We revisit the online-regression objective underlying GDN and, inspired by the Kaczmarz projection method, derive the key-norm-normalized dynamic step size $β_t = η_t / (\|k_t\|_2^2 + ε)$ for residual updates. We propose Kaczmarz Linear Attention (KLA), a one-scalar modification of GDN that preserves the state shape, gates, linear recurrence, and chunkwise parallel algorithm. At the 0.4B scale with a 1B-token budget, KLA achieves the lowest validation perplexity among evaluated linear-time baselines, 8.09 versus 8.50 for GDN, and remains stable up to 65K tokens. On controlled tasks, KLA reaches 100% on single-needle-in-a-haystack retrieval, improves 8x multi-query associative recall by 7.03 points over GDN, and delivers 2.1x higher decode throughput at 32K context. These results suggest that the key-norm-normalized Kaczmarz coefficient is a first-order design axis for delta-rule sequence models: it improves accuracy, extrapolation, and decoding efficiency without changing the recurrent state or hardware kernel.

Parallelization has played an instrumental role in the field of artificial intelligence (AI), drastically reducing the time taken to train and evaluate large AI models. In contrast to its impact in the broader field of AI, applying parallelization to computational game solving is relatively unexplored, despite its great potential. In this paper, we parallelize the family of counterfactual regret minimization (CFR) algorithms, which were central to important breakthroughs for solving large imperfect-information games. We present a generalized parallelization framework, reframing CFR as a series of linear algebra operations. Then, existing techniques for parallelizing linear algebra operations can be applied to accelerate CFR. We also describe how our technique can be applied to other tabular members of the CFR family of algorithms, including the state-of-the-art, such as CFR+, discounted CFR, and predictive variants of CFR. Experimentally, we show that our CFR implementation on a GPU is up to four orders of magnitude faster than Google DeepMind OpenSpiel's CFR implementations on a CPU.

Efficient computation of min-max problems is a central question in optimization, learning, games, and control. Arguably the most natural algorithm is gradient-descent-ascent (GDA). However, since the 1970s, conventional wisdom has argued that GDA fails to converge even on simple problems. This failure spurred an extensive literature on modifying GDA with additional building blocks such as extragradients, optimism, momentum, anchoring, etc. In contrast, we show that GDA converges in its original form by simply using a judicious choice of stepsizes. The key innovation is the proposal of unconventional stepsize schedules (dubbed slingshot stepsize schedules) that are time-varying, asymmetric, and periodically negative. We show that all three properties are necessary for convergence, and that altogether this enables GDA to converge on the classical counterexamples (e.g., unconstrained convex-concave problems). The core algorithmic intuition is that although negative stepsizes make backward progress, they de-synchronize the min and max variables (overcoming the cycling issue of GDA), and lead to a slingshot phenomenon in which the forward progress in the other iterations is overwhelmingly larger. This results in fast overall convergence. Geometrically, the slingshot dynamics leverage the non-reversibility of gradient flow: positive/negative steps cancel to first order, yielding a second-order net movement in a new direction that leads to convergence and is otherwise impossible for GDA to move in. We interpret this as a second-order finite-differencing algorithm and show that, intriguingly, it approximately implements consensus optimization, an empirically popular algorithm for min-max problems involving deep neural networks (e.g., training GANs).