arXiv subjects

Valerii Likhosherstov

Publications and source records attributed to Valerii Likhosherstov.

25 records · Page 2Linked to original sources

arXiv2020

Sub-Linear Memory: How to Make Performers SLiM

The Transformer architecture has revolutionized deep learning on sequential data, becoming ubiquitous in state-of-the-art solutions for a wide variety of applications. Yet vanilla Transformers are notoriously resource-expensive, requiring $O(L^2)$ in serial time and memory as functions of input length $L$. Recent works proposed various linear self-attention mechanisms, scaling only as $O(L)$ for serial computation. We perform a thorough analysis of recent Transformer mechanisms with linear self-attention, Performers, in terms of overall computational complexity. We observe a remarkable computational flexibility: forward and backward propagation can be performed with no approximations using sublinear memory as a function of $L$ (in addition to negligible storage for the input sequence), at a cost of greater time complexity in the parallel setting. In the extreme case, a Performer consumes only $O(1)$ memory during training, and still requires $O(L)$ time. This discovered time-memory tradeoff can be used for training or, due to complete backward-compatibility, for fine-tuning on a low-memory device, e.g. a smartphone or an earlier-generation GPU, thus contributing towards decentralized and democratized deep learning.

cs.LG↗

arXiv2020

Masked Language Modeling for Proteins via Linearly Scalable Long-Context Transformers

Transformer models have achieved state-of-the-art results across a diverse range of domains. However, concern over the cost of training the attention mechanism to learn complex dependencies between distant inputs continues to grow. In response, solutions that exploit the structure and sparsity of the learned attention matrix have blossomed. However, real-world applications that involve long sequences, such as biological sequence analysis, may fall short of meeting these assumptions, precluding exploration of these models. To address this challenge, we present a new Transformer architecture, Performer, based on Fast Attention Via Orthogonal Random features (FAVOR). Our mechanism scales linearly rather than quadratically in the number of tokens in the sequence, is characterized by sub-quadratic space complexity and does not incorporate any sparsity pattern priors. Furthermore, it provides strong theoretical guarantees: unbiased estimation of the attention matrix and uniform convergence. It is also backwards-compatible with pre-trained regular Transformers. We demonstrate its effectiveness on the challenging task of protein sequence modeling and provide detailed theoretical analysis.

cs.LG↗

arXiv2020

An Ode to an ODE

We present a new paradigm for Neural ODE algorithms, called ODEtoODE, where time-dependent parameters of the main flow evolve according to a matrix flow on the orthogonal group O(d). This nested system of two flows, where the parameter-flow is constrained to lie on the compact manifold, provides stability and effectiveness of training and provably solves the gradient vanishing-explosion problem which is intrinsically related to training deep neural network architectures such as Neural ODEs. Consequently, it leads to better downstream models, as we show on the example of training reinforcement learning policies with evolution strategies, and in the supervised learning setting, by comparing with previous SOTA baselines. We provide strong convergence results for our proposed mechanism that are independent of the depth of the network, supporting our empirical studies. Our results show an intriguing connection between the theory of deep neural networks and the field of matrix flows on compact manifolds.

cs.LG↗

arXiv2020

Stochastic Flows and Geometric Optimization on the Orthogonal Group

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.

cs.LG↗

arXiv2019

Tractable Minor-free Generalization of Planar Zero-field Ising Models

We present a new family of zero-field Ising models over $N$ binary variables/spins obtained by consecutive "gluing" of planar and $O(1)$-sized components and subsets of at most three vertices into a tree. The polynomial-time algorithm of the dynamic programming type for solving exact inference (computing partition function) and exact sampling (generating i.i.d. samples) consists in a sequential application of an efficient (for planar) or brute-force (for $O(1)$-sized) inference and sampling to the components as a black box. To illustrate the utility of the new family of tractable graphical models, we first build a polynomial algorithm for inference and sampling of zero-field Ising models over $K_{3,3}$-minor-free topologies and over $K_{5}$-minor-free topologies -- both are extensions of the planar zero-field Ising models -- which are neither genus - nor treewidth-bounded. Second, we demonstrate empirically an improvement in the approximation quality of the NP-hard problem of inference over the square-grid Ising model in a node-dependent non-zero "magnetic" field.

cs.DS↗

arXiv2019

A New Family of Tractable Ising Models

We present a new family of zero-field Ising models over N binary variables/spins obtained by consecutive "gluing" of planar and $O(1)$-sized components along with subsets of at most three vertices into a tree. The polynomial time algorithm of the dynamic programming type for solving exact inference (partition function computation) and sampling consists of a sequential application of an efficient (for planar) or brute-force (for $O(1)$-sized) inference and sampling to the components as a black box. To illustrate the utility of the new family of tractable graphical models, we first build an $O(N^{3/2})$ algorithm for inference and sampling of the K5-minor-free zero-field Ising models - an extension of the planar zero-field Ising models - which is neither genus- nor treewidth-bounded. Second, we demonstrate empirically an improvement in the approximation quality of the NP-hard problem of the square-grid Ising model (with non-zero field) inference.

cs.DS↗

arXiv2019

Inference and Sampling of $K_{33}$-free Ising Models

We call an Ising model tractable when it is possible to compute its partition function value (statistical inference) in polynomial time. The tractability also implies an ability to sample configurations of this model in polynomial time. The notion of tractability extends the basic case of planar zero-field Ising models. Our starting point is to describe algorithms for the basic case computing partition function and sampling efficiently. To derive the algorithms, we use an equivalent linear transition to perfect matching counting and sampling on an expanded dual graph. Then, we extend our tractable inference and sampling algorithms to models, whose triconnected components are either planar or graphs of $O(1)$ size. In particular, it results in a polynomial-time inference and sampling algorithms for $K_{33}$ (minor) free topologies of zero-field Ising models - a generalization of planar graphs with a potentially unbounded genus.

stat.CO↗