The prevalent use of Transformer-like models, exemplified by ChatGPT in modern language processing applications, underscores the critical need for enabling private inference essential for many cloud-based services reliant on such models. However, current privacy-preserving frameworks impose significant communication burden, especially for non-linear computation in Transformer model. In this paper, we introduce a novel plug-in method Comet to effectively reduce the communication cost without compromising the inference performance. We second introduce an efficient approximation method to eliminate the heavy communication in finding good initial approximation. We evaluate our Comet on Bert and RoBERTa models with GLUE benchmark datasets, showing up to 3.9$\times$ less communication and 3.5$\times$ speedups while keep competitive model performance compared to the prior art.

Models in which the covariance matrix has the structure of a sparse matrix plus a low rank perturbation are ubiquitous in data science applications. It is often desirable for algorithms to take advantage of such structures, avoiding costly matrix computations that often require cubic time and quadratic storage. This is often accomplished by performing operations that maintain such structures, e.g. matrix inversion via the Sherman-Morrison-Woodbury formula. In this paper we consider the matrix square root and inverse square root operations. Given a low rank perturbation to a matrix, we argue that a low-rank approximate correction to the (inverse) square root exists. We do so by establishing a geometric decay bound on the true correction's eigenvalues. We then proceed to frame the correction as the solution of an algebraic Riccati equation, and discuss how a low-rank solution to that equation can be computed. We analyze the approximation error incurred when approximately solving the algebraic Riccati equation, providing spectral and Frobenius norm forward and backward error bounds. Finally, we describe several applications of our algorithms, and demonstrate their utility in numerical experiments.

Activation functions play a significant role in the performance of deep learning algorithms. In particular, the Swish activation function tends to outperform ReLU on deeper models, including deep reinforcement learning models, across challenging tasks. Despite this progress, ReLU is the preferred function partly because it is more efficient than Swish. Furthermore, in contrast to the fields of computer vision and natural language processing, the deep reinforcement learning and robotics domains have seen less inclination to adopt new activation functions, such as Swish, and instead continue to use more traditional functions, like ReLU. To tackle those issues, we propose Swim, a general-purpose, efficient, and high-performing alternative to Swish, and then provide an analysis of its properties as well as an explanation for its high-performance relative to Swish, in terms of both reward-achievement and efficiency. We focus on testing Swim on MuJoCo's locomotion continuous control tasks since they exhibit more complex dynamics and would therefore benefit most from a high-performing and efficient activation function. We also use the TD3 algorithm in conjunction with Swim and explain this choice in the context of the robot locomotion domain. We then conclude that Swim is a state-of-the-art activation function for continuous control locomotion tasks and recommend using it with TD3 as a working framework.

Computing the matrix square root and its inverse in a differentiable manner is important in a variety of computer vision tasks. Previous methods either adopt the Singular Value Decomposition (SVD) to explicitly factorize the matrix or use the Newton-Schulz iteration (NS iteration) to derive the approximate solution. However, both methods are not computationally efficient enough in either the forward pass or the backward pass. In this paper, we propose two more efficient variants to compute the differentiable matrix square root and the inverse square root. For the forward propagation, one method is to use Matrix Taylor Polynomial (MTP), and the other method is to use Matrix Pad\'e Approximants (MPA). The backward gradient is computed by iteratively solving the continuous-time Lyapunov equation using the matrix sign function. A series of numerical tests show that both methods yield considerable speed-up compared with the SVD or the NS iteration. Moreover, we validate the effectiveness of our methods in several real-world applications, including de-correlated batch normalization, second-order vision transformer, global covariance pooling for large-scale and fine-grained recognition, attentive covariance pooling for video recognition, and neural style transfer. The experimental results demonstrate that our methods can also achieve competitive and even slightly better performances. The Pytorch implementation is available at https://github.com/KingJamesSong/FastDifferentiableMatSqrt

Since ML algorithms have proven their success in many different applications, there is also a big interest in privacy preserving (PP) ML methods for building models on sensitive data. Moreover, the increase in the number of data sources and the high computational power required by those algorithms force individuals to outsource the training and/or the inference of a ML model to the clouds providing such services. To address this, we propose a secure 3-party computation framework, CECILIA, offering PP building blocks to enable complex operations privately. In addition to the adapted and common operations like addition and multiplication, it offers multiplexer, most significant bit and modulus conversion. The first two are novel in terms of methodology and the last one is novel in terms of both functionality and methodology. CECILIA also has two complex novel methods, which are the exact exponential of a public base raised to the power of a secret value and the inverse square root of a secret Gram matrix. We use CECILIA to realize the private inference on pre-trained RKNs, which require more complex operations than most other DNNs, on the structural classification of proteins as the first study ever accomplishing the PP inference on RKNs. In addition to the successful private computation of basic building blocks, the results demonstrate that we perform the exact and fully private exponential computation, which is done by approximation in the literature so far. Moreover, they also show that we compute the exact inverse square root of a secret Gram matrix up to a certain privacy level, which has not been addressed in the literature at all. We also analyze the scalability of CECILIA to various settings on a synthetic dataset. The framework shows a great promise to make other ML algorithms as well as further computations privately computable by the building blocks of the framework.

Capacity analysis has been introduced in [2] as a way to analyze which dependencies a linear model is focussing its modelling capacity on, when trained on a given task. The concept was then extended in [3] to neural networks with nonlinear activations, where capacity propagation through layers was studied. When the layers are residual (or differential), and in one limiting case with extremely irregular activations (which was called the pseudo-random limit), it has been shown that capacity propagation through layers follows a discrete Markov equation. This discrete equation can then be approximated by a continuous Kolmogorov forward equation in the deep limit, provided some specific scaling relation holds between the network depth and the scale of its residual connections - more precisely, the residual weights must scale as the inverse square root of the number of layers. Following [1], it was then hypothesized that the success of residual networks lies in their ability to propagate capacity through a large number of layers in a non-degenerate manner. It is interesting to note that the inverse square root scaling mentioned above is the only scaling relation that leads to a non-degenerate propagation PDE in that case: larger weights would lead to shattering, while smaller ones would lead to no spatial propagation at all. In this paper, we take this idea one step further and formulate the conjecture that enforcing the right scaling relations - i.e. the ones that lead to a non-degenerate continuous limit for capacity propagation - is key to avoiding the shattering problem: we call this the neural network scaling conjecture. In the example above, this would mean that the inverse square root scaling must be enforced if one wants to use residual networks at their full power. In the second part of this paper, we use the PDE capacity propagation framework to study a number of commonly used network architectures, and determine the scaling relations that are required for a non-degenerate capacity propagation to happen in each case.

Advanced optimization algorithms such as Newton method and AdaGrad benefit from second order derivative or second order statistics to achieve better descent directions and faster convergence rates. At their heart, such algorithms need to compute the inverse or inverse square root of a matrix whose size is quadratic of the dimensionality of the search space. For high dimensional search spaces, the matrix inversion or inversion of square root becomes overwhelming which in turn demands for approximate methods. In this work, we propose a new matrix approximation method which divides a matrix into blocks and represents each block by one or two numbers. The method allows efficient computation of matrix inverse and inverse square root. We apply our method to AdaGrad in training deep neural networks. Experiments show encouraging results compared to the diagonal approximation.