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Register a new userDoubly Adaptive Scaled Algorithm for Machine Learning Using Second-Order Information
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Martin Tak\\'a\\v{c}
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We present a novel adaptive optimization algorithm for large-scale machine learning problems. Equipped with a low-cost estimate of local curvature and Lipschitz smoothness, our method dynamically adapts the search direction and step-size. The search direction contains gradient information preconditioned by a well-scaled diagonal preconditioning matrix that captures the local curvature information. Our methodology does not require the tedious task of learning rate tuning, as the learning rate is updated automatically without adding an extra hyperparameter. We provide convergence guarantees on a comprehensive collection of optimization problems, including convex, strongly convex, and nonconvex problems, in both deterministic and stochastic regimes. We also conduct an extensive empirical evaluation on standard machine learning problems, justifying our algorithms versatility and demonstrating its strong performance compared to other start-of-the-art first-order and second-order methods.
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We introduce ADAHESSIAN, a second order stochastic optimization algorithm which dynamically incorporates the curvature of the loss function via ADAptive estimates of the HESSIAN. Second order algorithms are among the most powerful optimization algorithms with superior convergence properties as compared to first order methods such as SGD and Adam. The main disadvantage of traditional second order methods is their heavier per iteration computation and poor accuracy as compared to first order methods. To address these, we incorporate several novel approaches in ADAHESSIAN, including: (i) a fast Hutchinson based method to approximate the curvature matrix with low computational overhead; (ii) a root-mean-square exponential moving average to smooth out variations of the Hessian diagonal across different iterations; and (iii) a block diagonal averaging to reduce the variance of Hessian diagonal elements. We show that ADAHESSIAN achieves new state-of-the-art results by a large margin as compared to other adaptive optimization methods, including variants of Adam. In particular, we perform extensive tests on CV, NLP, and recommendation system tasks and find that ADAHESSIAN: (i) achieves 1.80%/1.45% higher accuracy on ResNets20/32 on Cifar10, and 5.55% higher accuracy on ImageNet as compared to Adam; (ii) outperforms AdamW for transformers by 0.13/0.33 BLEU score on IWSLT14/WMT14 and 2.7/1.0 PPL on PTB/Wikitext-103; (iii) outperforms AdamW for SqueezeBert by 0.41 points on GLUE; and (iv) achieves 0.032% better score than Adagrad for DLRM on the Criteo Ad Kaggle dataset. Importantly, we show that the cost per iteration of ADAHESSIAN is comparable to first order methods, and that it exhibits robustness towards its hyperparameters.
Optimization in machine learning, both theoretical and applied, is presently dominated by first-order gradient methods such as stochastic gradient descent. Second-order optimization methods, that involve second derivatives and/or second order statistics of the data, are far less prevalent despite strong theoretical properties, due to their prohibitive computation, memory and communication costs. In an attempt to bridge this gap between theoretical and practical optimization, we present a scalable implementation of a second-order preconditioned method (concretely, a variant of full-matrix Adagrad), that along with several critical algorithmic and numerical improvements, provides significant convergence and wall-clock time improvements compared to conventional first-order methods on state-of-the-art deep models. Our novel design effectively utilizes the prevalent heterogeneous hardware architecture for training deep models, consisting of a multicore CPU coupled with multiple accelerator units. We demonstrate superior performance compared to state-of-the-art on very large learning tasks such as machine translation with Transformers, language modeling with BERT, click-through rate prediction on Criteo, and image classification on ImageNet with ResNet-50.
Bilevel optimization has become a powerful framework in various machine learning applications including meta-learning, hyperparameter optimization, and network architecture search. There are generally two classes of bilevel optimization formulations for machine learning: 1) problem-based bilevel optimization, whose inner-level problem is formulated as finding a minimizer of a given loss function; and 2) algorithm-based bilevel optimization, whose inner-level solution is an output of a fixed algorithm. For the first class, two popular types of gradient-based algorithms have been proposed for hypergradient estimation via approximate implicit differentiation (AID) and iterative differentiation (ITD). Algorithms for the second class include the popular model-agnostic meta-learning (MAML) and almost no inner loop (ANIL). However, the convergence rate and fundamental limitations of bilevel optimization algorithms have not been well explored. This thesis provides a comprehensive convergence rate analysis for bilevel algorithms in the aforementioned two classes. We further propose principled algorithm designs for bilevel optimization with higher efficiency and scalability. For the problem-based formulation, we provide a convergence rate analysis for AID- and ITD-based bilevel algorithms. We then develop acceleration bilevel algorithms, for which we provide shaper convergence analysis with relaxed assumptions. We also provide the first lower bounds for bilevel optimization, and establish the optimality by providing matching upper bounds under certain conditions. We finally propose new stochastic bilevel optimization algorithms with lower complexity and higher efficiency in practice. For the algorithm-based formulation, we develop a theoretical convergence for general multi-step MAML and ANIL, and characterize the impact of parameter selections and loss geometries on the their complexities.
We design an algorithm which finds an $epsilon$-approximate stationary point (with $| abla F(x)|le epsilon$) using $O(epsilon^{-3})$ stochastic gradient and Hessian-vector products, matching guarantees that were previously available only under a stronger assumption of access to multiple queries with the same random seed. We prove a lower bound which establishes that this rate is optimal and---surprisingly---that it cannot be improved using stochastic $p$th order methods for any $pge 2$, even when the first $p$ derivatives of the objective are Lipschitz. Together, these results characterize the complexity of non-convex stochastic optimization with second-order methods and beyond. Expanding our scope to the oracle complexity of finding $(epsilon,gamma)$-approximate second-order stationary points, we establish nearly matching upper and lower bounds for stochastic second-order methods. Our lower bounds here are novel even in the noiseless case.
Recurrent Neural Networks (RNNs) are powerful models that achieve exceptional performance on several pattern recognition problems. However, the training of RNNs is a computationally difficult task owing to the well-known vanishing/exploding gradient problem. Algorithms proposed for training RNNs either exploit no (or limited) curvature information and have cheap per-iteration complexity, or attempt to gain significant curvature information at the cost of increased per-iteration cost. The former set includes diagonally-scaled first-order methods such as ADAGRAD and ADAM, while the latter consists of second-order algorithms like Hessian-Free Newton and K-FAC. In this paper, we present adaQN, a stochastic quasi-Newton algorithm for training RNNs. Our approach retains a low per-iteration cost while allowing for non-diagonal scaling through a stochastic L-BFGS updating scheme. The method uses a novel L-BFGS scaling initialization scheme and is judicious in storing and retaining L-BFGS curvature pairs. We present numerical experiments on two language modeling tasks and show that adaQN is competitive with popular RNN training algorithms.
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