Achraf Bahamou

Achraf Bahamou, Omar Besbes, Omar Mouchtaki

Problem definition: We study a data-driven pricing problem in which a seller sets a price for a single item based on demand observed at a limited number of historical prices. Our goal is to quantify the value of such information and to guide efficient price experimentation under practical constraints. Methodology/results: Our main methodological contribution is an exact reduction that characterizes the maximin revenue ratio, defined as the worst-case revenue achievable using only past data relative to the optimal revenue under full information. This reduction transforms an infinite-dimensional problem into a tractable one-dimensional optimization problem, allowing us to compute near-optimal pricing policies with explicit guarantees and to precisely quantify the value of historical data. Managerial implications: Motivated by practical constraints that limit price changes, we first evaluate the value of local information and show that the sign of the revenue gradient at a single price can provide significant guidance. We then use our framework to design efficient price experiments: we develop a method to select the next price to test so as to maximize future robust performance, and show how to substantially reduce the number of experiments needed to achieve target revenue guarantees in dynamic pricing. Finally, we show that our approach remains effective with noisy demand data, achieving near-optimal performance with as few as 25 to 100 samples per price.

Achraf Bahamou, Donald Goldfarb

We propose a new per-layer adaptive step-size procedure for stochastic first-order optimization methods for minimizing empirical loss functions in deep learning, eliminating the need for the user to tune the learning rate (LR). The proposed approach exploits the layer-wise stochastic curvature information contained in the diagonal blocks of the Hessian in deep neural networks (DNNs) to compute adaptive step-sizes (i.e., LRs) for each layer. The method has memory requirements that are comparable to those of first-order methods, while its per-iteration time complexity is only increased by an amount that is roughly equivalent to an additional gradient computation. Numerical experiments show that SGD with momentum and AdamW combined with the proposed per-layer step-sizes are able to choose effective LR schedules and outperform fine-tuned LR versions of these methods as well as popular first-order and second-order algorithms for training DNNs on Autoencoder, Convolutional Neural Network (CNN) and Graph Convolutional Network (GCN) models. Finally, it is proved that an idealized version of SGD with the layer-wise step sizes converges linearly when using full-batch gradients.

Achraf Bahamou, Donald Goldfarb, Yi Ren

Deep neural networks (DNNs) are currently predominantly trained using first-order methods. Some of these methods (e.g., Adam, AdaGrad, and RMSprop, and their variants) incorporate a small amount of curvature information by using a diagonal matrix to precondition the stochastic gradient. Recently, effective second-order methods, such as KFAC, K-BFGS, Shampoo, and TNT, have been developed for training DNNs, by preconditioning the stochastic gradient by layer-wise block-diagonal matrices. Here we propose a "mini-block Fisher (MBF)" preconditioned gradient method, that lies in between these two classes of methods. Specifically, our method uses a block-diagonal approximation to the empirical Fisher matrix, where for each layer in the DNN, whether it is convolutional or feed-forward and fully connected, the associated diagonal block is itself block-diagonal and is composed of a large number of mini-blocks of modest size. Our novel approach utilizes the parallelism of GPUs to efficiently perform computations on the large number of matrices in each layer. Consequently, MBF's per-iteration computational cost is only slightly higher than it is for first-order methods. The performance of our proposed method is compared to that of several baseline methods, on both autoencoder and CNN problems, to validate its effectiveness both in terms of time efficiency and generalization power. Finally, it is proved that an idealized version of MBF converges linearly.

Yi Ren, Achraf Bahamou, Donald Goldfarb

Second-order methods have the capability of accelerating optimization by using much richer curvature information than first-order methods. However, most are impractical for deep learning, where the number of training parameters is huge. In Goldfarb et al. (2020), practical quasi-Newton methods were proposed that approximate the Hessian of a multilayer perceptron (MLP) model by a layer-wise block diagonal matrix where each layer's block is further approximated by a Kronecker product corresponding to the structure of the Hessian restricted to that layer. Here, we extend these methods to enable them to be applied to convolutional neural networks (CNNs), by analyzing the Kronecker-factored structure of the Hessian matrix of convolutional layers. Several improvements to the methods in Goldfarb et al. (2020) are also proposed that can be applied to both MLPs and CNNs. These new methods have memory requirements comparable to first-order methods and much less per-iteration time complexity than those in Goldfarb et al. (2020). Moreover, convergence results are proved for a variant under relatively mild conditions. Finally, we compared the performance of our new methods against several state-of-the-art (SOTA) methods on MLP autoencoder and CNN problems, and found that they outperformed the first-order SOTA methods and performed comparably to the second-order SOTA methods.

Donald Goldfarb, Yi Ren, Achraf Bahamou

We consider the development of practical stochastic quasi-Newton, and in particular Kronecker-factored block-diagonal BFGS and L-BFGS methods, for training deep neural networks (DNNs). In DNN training, the number of variables and components of the gradient $n$ is often of the order of tens of millions and the Hessian has $n^2$ elements. Consequently, computing and storing a full $n \times n$ BFGS approximation or storing a modest number of (step, change in gradient) vector pairs for use in an L-BFGS implementation is out of the question. In our proposed methods, we approximate the Hessian by a block-diagonal matrix and use the structure of the gradient and Hessian to further approximate these blocks, each of which corresponds to a layer, as the Kronecker product of two much smaller matrices. This is analogous to the approach in KFAC, which computes a Kronecker-factored block-diagonal approximation to the Fisher matrix in a stochastic natural gradient method. Because the indefinite and highly variable nature of the Hessian in a DNN, we also propose a new damping approach to keep the upper as well as the lower bounds of the BFGS and L-BFGS approximations bounded. In tests on autoencoder feed-forward neural network models with either nine or thirteen layers applied to three datasets, our methods outperformed or performed comparably to KFAC and state-of-the-art first-order stochastic methods.

Krzysztof Choromanski, David Cheikhi, Jared Davis et al.

We present a new class of stochastic, geometrically-driven optimization algorithms on the orthogonal group $O(d)$ and naturally reductive homogeneous manifolds obtained from the action of the rotation group $SO(d)$. We theoretically and experimentally demonstrate that our methods can be applied in various fields of machine learning including deep, convolutional and recurrent neural networks, reinforcement learning, normalizing flows and metric learning. We show an intriguing connection between efficient stochastic optimization on the orthogonal group and graph theory (e.g. matching problem, partition functions over graphs, graph-coloring). We leverage the theory of Lie groups and provide theoretical results for the designed class of algorithms. We demonstrate broad applicability of our methods by showing strong performance on the seemingly unrelated tasks of learning world models to obtain stable policies for the most difficult $\mathrm{Humanoid}$ agent from $\mathrm{OpenAI}$ $\mathrm{Gym}$ and improving convolutional neural networks.

Achraf Bahamou, Donald Goldfarb

We propose a stochastic optimization method for minimizing loss functions, expressed as an expected value, that adaptively controls the batch size used in the computation of gradient approximations and the step size used to move along such directions, eliminating the need for the user to tune the learning rate. The proposed method exploits local curvature information and ensures that search directions are descent directions with high probability using an acute-angle test and can be used as a method that has a global linear rate of convergence on self-concordant functions with high probability. Numerical experiments show that this method is able to choose the best learning rates and compares favorably to fine-tuned SGD for training logistic regression and DNNs. We also propose an adaptive version of ADAM that eliminates the need to tune the base learning rate and compares favorably to fine-tuned ADAM on training DNNs. In our DNN experiments, we rarely encountered negative curvature at the current point along the step direction in DNNs.