How AI settled the complexity of the oldest SGD algorithm

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).

Distributed stochastic gradient descent (SGD) is limited by communication rather than computation, since each iteration requires an AllReduce across processes. Communication-avoiding SGD (CA-SGD) amortizes communication over $s$ iterations by replacing $s$ consecutive AllReduces with a single AllReduce of an $sb\times sb$ Gram matrix, trading more computation and bandwidth for fewer synchronization points. Modern GPUs with matrix hardware and reduced-precision formats offset this by accelerating the Gram GEMM and shrinking BF16 traffic. We study mixed-precision CA-SGD for generalized linear models on NVIDIA GPUs. Our finite-precision analysis decomposes the local rounding error of one CA-SGD outer iteration into nine independent precision choices, depending on the hardware only through its low-precision unit roundoffs, so the resulting recipes transfer in principle across GPU generations. The recipe stores the input matrix and margin vector in low precision, computes the Gram matrix from low-precision inputs with high-precision accumulation, communicates it in high precision, and performs the inner recurrence and weight updates in high precision. On NERSC Perlmutter A100 GPUs, mixed-precision CA-SGD matches FP32 SGD loss within $0.5\%$ on logistic, linear, and Poisson problems and reaches $5.1$--$6.8\times$ speedup over FP32 SGD on epsilon, SUSY, HIGGS, synth, and Poisson-synth. Our software is available at https://doi.org/10.5281/zenodo.20448273