ICLR 2023 Papers

Recently the SP (Stochastic Polyak step size) method has emerged as a competitive adaptive method for setting the step sizes of SGD. SP can be interpreted as a method specialized to interpolated models, since it solves the interpolation equations. SP solves these equation by using local linearizations of the model. We take a step further and develop a method for solving the interpolation equations that uses the local second-order approximation of the model. Our resulting method SP2 uses Hessian-vector products to speed-up the convergence of SP. Furthermore, and rather uniquely among second-order methods, the design of SP2 in no way relies on positive definite Hessian matrices or convexity of the objective function. We show SP2 is competitive both in experiments and in theory. We show SP2 is very competitive on matrix completion, non-convex test problems and logistic regression. We also provide a convergence theory on sums-of-quadratics.

We study the theory of neural network (NN) from the lens of classical nonparametric regression problems with a focus on NN’s ability to adaptively estimate functions with heterogeneous smoothness — a property of functions in Besov or Bounded Variation (BV) classes. Existing work on this problem requires tuning the NN architecture based on the function spaces and sample sizes. We consider a “Parallel NN” variant of deep ReLU networks and show that the standard weight decay is equivalent to promoting the ℓp -sparsity (0 < p < 1) of the coefficient vector of an end-to-end learned function bases, i.e., a dictionary. Using this equivalence, we further establish that by tuning only the weight decay, such Parallel NN achieves an estimation error arbitrarily close to the minimax rates for both the Besov and BV classes. Notably, it gets exponentially closer to minimax optimal as the NN gets deeper. Our research sheds new lights on why depth matters and how NNs are more powerful than kernel methods

1-bit gradient compression and local steps are two representative techniques that enable drastic communication reduction in distributed SGD. Their benefits, however, remain an open question on Adam-based large model pre-training (e.g. BERT and GPT). In this paper, we demonstrate the non-linearity in Adam causes slow convergence even when 1-bit compression or local steps are individually applied. To alleviate this limitation, we propose \textbf{0/1 Adam} that linearizes each Adam step via approximating its optimizer states using their stale estimates and linear correlation. \textbf{0/1 Adam} performs an Adam-like step to preserve the adaptivity, while its linearity allows utilizing 1-bit compression and local steps simultaneously for wall-clock time speed up. We provide convergence guarantee for \textbf{0/1 Adam} on smooth non-convex objectives. On various large-scale benchmarks such as BERT-Base, BERT-Large, GPT-2 pre-training and ImageNet, we demonstrate on up to 128 GPUs that \textbf{0/1 Adam} is able to reduce up to 87\% of data volume, 54\% of communication rounds, and achieve up to 2$\times$ higher training throughput and end-to-end training time reduction compared to the state-of-the-art baseline 1-bit Adam; while enjoying the same statistical convergence speed and end task model accuracy on GLUE dataset and ImageNet validation set.