Alexandr Katrutsa

Alexandr Katrutsa, Talgat Daulbaev, Ivan Oseledets

This paper proposes the method to optimize restriction and prolongation operators in the two-grid method. The proposed method is straightforwardly extended to the geometric multigrid method (GMM). GMM is used in solving discretized partial differential equation (PDE) and based on the restriction and prolongation operators. The operators are crucial for fast convergence of GMM, but they are unknown. To find them we propose a reformulation of the two-grid method in terms of a deep neural network with a specific architecture. This architecture is based on the idea that every operation in the two-grid method can be considered as a layer of a deep neural network. The parameters of layers correspond to the restriction and prolongation operators. Therefore, we state an optimization problem with respect to these operators and get optimal ones through backpropagation approach. To illustrate the performance of the proposed approach, we carry out experiments on the discretized Laplace equation, Helmholtz equation and singularly perturbed convection-diffusion equation and demonstrate that proposed approach gives operators, which lead to faster convergence.

Roman Korkin, Ivan Oseledets, Aleksandr Katrutsa

This study considers the object localization problem and proposes a novel multiparticle Kalman filter to solve it in complex and symmetric environments. Two well-known classes of filtering algorithms to solve the localization problem are Kalman filter-based methods and particle filter-based methods. We consider these classes, demonstrate their complementary properties, and propose a novel filtering algorithm that takes the best from two classes. We evaluate the multiparticle Kalman filter in symmetric and noisy environments. Such environments are especially challenging for both classes of classical methods. We compare the proposed approach with the particle filter since only this method is feasible if the initial state is unknown. In the considered challenging environments, our method outperforms the particle filter in terms of both localization error and runtime.

Valentin Leplat, Daniil Merkulov, Aleksandr Katrutsa et al.

Classical machine learning models such as deep neural networks are usually trained by using Stochastic Gradient Descent-based (SGD) algorithms. The classical SGD can be interpreted as a discretization of the stochastic gradient flow. In this paper we propose a novel, robust and accelerated stochastic optimizer that relies on two key elements: (1) an accelerated Nesterov-like Stochastic Differential Equation (SDE) and (2) its semi-implicit Gauss-Seidel type discretization. The convergence and stability of the obtained method, referred to as NAG-GS, are first studied extensively in the case of the minimization of a quadratic function. This analysis allows us to come up with an optimal learning rate in terms of the convergence rate while ensuring the stability of NAG-GS. This is achieved by the careful analysis of the spectral radius of the iteration matrix and the covariance matrix at stationarity with respect to all hyperparameters of our method. Further, we show that NAG- GS is competitive with state-of-the-art methods such as momentum SGD with weight decay and AdamW for the training of machine learning models such as the logistic regression model, the residual networks models on standard computer vision datasets, Transformers in the frame of the GLUE benchmark and the recent Vision Transformers.

Alexandr Katrutsa, Mike Botchev, George Ovchinnikov et al.

The conjugate gradient method (CG) is typically used with a preconditioner which improves efficiency and robustness of the method. Many preconditioners include parameters and a proper choice of a preconditioner and its parameters is often not a trivial task. Although many convergence estimates exist which can be used for optimizing preconditioners, these estimates typically hold for all initial guess vectors, in other words, they reflect the worst convergence rate. To account for the mean convergence rate instead, in this paper, we follow a stochastic approach. It is based on trial runs with random initial guess vectors and leads to a functional which can be used to monitor convergence and to optimize preconditioner parameters in CG. Presented numerical experiments show that optimization of this new functional with respect to preconditioner parameters usually yields a better parameter value than optimization of the functional based on the spectral condition number.

Albert Sayapin, Gleb Balitskiy, Daniel Bershatsky et al.

In this study, we propose a novel SeqMF model to solve the problem of predicting the next app launch during mobile device usage. Although this problem can be represented as a classical collaborative filtering problem, it requires proper modification since the data are sequential, the user feedback is distributed among devices and the transmission of users' data to aggregate common patterns must be protected against leakage. According to such requirements, we modify the structure of the classical matrix factorization model and update the training procedure to sequential learning. Since the data about user experience are distributed among devices, the federated learning setup is used to train the proposed sequential matrix factorization model. One more ingredient of the proposed approach is a new privacy mechanism that guarantees the protection of the sent data from the users to the remote server. To demonstrate the efficiency of the proposed model we use publicly available mobile user behavior data. We compare our model with sequential rules and models based on the frequency of app launches. The comparison is conducted in static and dynamic environments. The static environment evaluates how our model processes sequential data compared to competitors. Therefore, the standard train-validation-test evaluation procedure is used. The dynamic environment emulates the real-world scenario, where users generate new data by running apps on devices, and evaluates our model in this case. Our experiments show that the proposed model provides comparable quality with other methods in the static environment. However, more importantly, our method achieves a better privacy-utility trade-off than competitors in the dynamic environment, which provides more accurate simulations of real-world usage.

Alexandr Katrutsa, Ivan Oseledets

We propose a new method for preconditioning Kaczmarz method by sketching. Kaczmarz method is a stochastic method for solving overdetermined linear systems based on a sampling of matrix rows. The standard approach to speed up convergence of iterative methods is using preconditioner. As we show the best possible preconditioner for this method can be constructed from QR decomposition of the system matrix, but the complexity of this procedure is too high. Therefore, to reduce this complexity, we use random sketching and compare it with the Kaczmarz method without preconditioning. The developed method is applicable for different modifications of classical Kaczmarz method that were proposed recently. We provide numerical experiments to show the performance of the developed methods on solving both random and real overdetermined linear systems.

Roman Korkin, Ivan Oseledets, Aleksandr Katrutsa

This study proposes a novel memory-efficient recurrent neural network (RNN) architecture specified to solve the object localization problem. This problem is to recover the object states along with its movement in a noisy environment. We take the idea of the classical particle filter and combine it with GRU RNN architecture. The key feature of the resulting memory-efficient particle filter RNN model (mePFRNN) is that it requires the same number of parameters to process environments of different sizes. Thus, the proposed mePFRNN architecture consumes less memory to store parameters compared to the previously proposed PFRNN model. To demonstrate the performance of our model, we test it on symmetric and noisy environments that are incredibly challenging for filtering algorithms. In our experiments, the mePFRNN model provides more precise localization than the considered competitors and requires fewer trained parameters.

Andrey Pudovikov, Alexandra Khirianova, Ekaterina Solodneva et al.

Advertisement auctions play a crucial role in revenue generation for e-commerce companies. To make the bidding procedure scalable to thousands of auctions, the automatic bidding (autobidding) algorithms are actively developed in the industry. Therefore, the fair and reproducible evaluation of autobidding algorithms is an important problem. We present a standardized and transparent evaluation protocol for comparing classical and reinforcement learning (RL) autobidding algorithms. We consider the most efficient autobidding algorithms from different classes, e.g., ones based on the controllers, RL, optimal formulas, etc., and benchmark them in the bidding environment. We utilize the most recent open-source environment developed in the industry, which accurately emulates the bidding process. Our work demonstrates the most promising use cases for the considered autobidding algorithms, highlights their surprising drawbacks, and evaluates them according to multiple metrics. We select the evaluation metrics that illustrate the performance of the autobidding algorithms, the corresponding costs, and track the budget pacing. Such a choice of metrics makes our results applicable to the broad range of platforms where autobidding is effective. The presented comparison results help practitioners to evaluate the candidate autobidding algorithms from different perspectives and select ones that are efficient according to their companies' targets.

Julia Gusak, Alexandr Katrutsa, Talgat Daulbaev et al.

A conventional approach to train neural ordinary differential equations (ODEs) is to fix an ODE solver and then learn the neural network's weights to optimize a target loss function. However, such an approach is tailored for a specific discretization method and its properties, which may not be optimal for the selected application and yield the overfitting to the given solver. In our paper, we investigate how the variability in solvers' space can improve neural ODEs performance. We consider a family of Runge-Kutta methods that are parameterized by no more than two scalar variables. Based on the solvers' properties, we propose an approach to decrease neural ODEs overfitting to the pre-defined solver, along with a criterion to evaluate such behaviour. Moreover, we show that the right choice of solver parameterization can significantly affect neural ODEs models in terms of robustness to adversarial attacks. Recently it was shown that neural ODEs demonstrate superiority over conventional CNNs in terms of robustness. Our work demonstrates that the model robustness can be further improved by optimizing solver choice for a given task. The source code to reproduce our experiments is available at https://github.com/juliagusak/neural-ode-metasolver.

Ivan Zhigalskii, Andrey Pudovikov, Aleksandr Katrutsa et al.

Modern e-commerce platforms employ various auction mechanisms to allocate paid slots for a given item. To scale this approach to the millions of auctions, the platforms suggest promotion tools based on the autobidding algorithms. These algorithms typically depend on the Click-Through-Rate (CTR) and Conversion-Rate (CVR) estimates provided by a pre-trained machine learning model. However, the predictions of such models are uncertain and can significantly affect the performance of the autobidding algorithm. To address this issue, we propose the DenoiseBid method, which corrects the generated CTRs and CVRs to make the resulting bids more efficient in auctions. The underlying idea of our method is to employ a Bayesian approach and replace noisy CTR or CVR estimates with those from recovered distributions. To demonstrate the performance of the proposed approach, we perform extensive experiments on the synthetic, iPinYou, and BAT datasets. To evaluate the robustness of our approach to the noise scale, we use synthetic noise and noise estimated from the predictions of the pre-trained machine learning model.

Yuriy Dorn, Aleksandr Katrutsa, Ilgam Latypov et al.

In this study, we propose a new method for constructing UCB-type algorithms for stochastic multi-armed bandits based on general convex optimization methods with an inexact oracle. We derive the regret bounds corresponding to the convergence rates of the optimization methods. We propose a new algorithm Clipped-SGD-UCB and show, both theoretically and empirically, that in the case of symmetric noise in the reward, we can achieve an $O(\log T\sqrt{KT\log T})$ regret bound instead of $O\left (T^{\frac{1}{1+α}} K^{\fracα{1+α}} \right)$ for the case when the reward distribution satisfies $\mathbb{E}_{X \in D}[|X|^{1+α}] \leq σ^{1+α}$ ($α\in (0, 1])$, i.e. perform better than it is assumed by the general lower bound for bandits with heavy-tails. Moreover, the same bound holds even when the reward distribution does not have the expectation, that is, when $α<0$.

Tatyana Matveeva, Aleksandr Katrutsa, Evgeny Frolov

Adaptive gradient methods like Adagrad and its variants are widespread in large-scale optimization. However, their use of diagonal preconditioning matrices limits the ability to capture parameter correlations. Full-matrix adaptive methods, approximating the exact Hessian, can model these correlations and may enable faster convergence. At the same time, their computational and memory costs are often prohibitive for large-scale models. To address this limitation, we propose AdaGram, an optimizer that enables efficient full-matrix adaptive gradient updates. To reduce memory and computational overhead, we utilize fast symmetric factorization for computing the preconditioned update direction at each iteration. Additionally, we maintain the low-rank structure of a preconditioner along the optimization trajectory using matrix integrator methods. Numerical experiments on standard machine learning tasks show that AdaGram converges faster or matches the performance of diagonal adaptive optimizers when using rank five and smaller rank approximations. This demonstrates AdaGram's potential as a scalable solution for adaptive optimization in large models.

Yuriy Dorn, Aleksandr Katrutsa, Ilgam Latypov et al.

Bandit optimization usually refers to the class of online optimization problems with limited feedback, namely, a decision maker uses only the objective value at the current point to make a new decision and does not have access to the gradient of the objective function. While this name accurately captures the limitation in feedback, it is somehow misleading since it does not have any connection with the multi-armed bandits (MAB) problem class. We propose two new classes of problems: the functional multi-armed bandit problem (FMAB) and the best function identification problem. They are modifications of a multi-armed bandit problem and the best arm identification problem, respectively, where each arm represents an unknown black-box function. These problem classes are a surprisingly good fit for modeling real-world problems such as competitive LLM training. To solve the problems from these classes, we propose a new reduction scheme to construct UCB-type algorithms, namely, the F-LCB algorithm, based on algorithms for nonlinear optimization with known convergence rates. We provide the regret upper bounds for this reduction scheme based on the base algorithms' convergence rates. We add numerical experiments that demonstrate the performance of the proposed scheme.

Aleksandr Katrutsa, Sergey Utyuzhnikov, Ivan Oseledets

The Dynamic Mode Decomposition has proved to be a very efficient technique to study dynamic data. This is entirely a data-driven approach that extracts all necessary information from data snapshots which are commonly supposed to be sampled from measurement. The application of this approach becomes problematic if the available data is incomplete because some dimensions of smaller scale either missing or unmeasured. Such setting occurs very often in modeling complex dynamical systems such as power grids, in particular with reduced-order modeling. To take into account the effect of unresolved variables the optimal prediction approach based on the Mori-Zwanzig formalism can be applied to obtain the most expected prediction under existing uncertainties. This effectively leads to the development of a time-predictive model accounting for the impact of missing data. In the present paper we provide a detailed derivation of the considered method from the Liouville equation and finalize it with the optimization problem that defines the optimal transition operator corresponding to the observed data. In contrast to the existing approach, we consider a first-order approximation of the Mori-Zwanzig decomposition, state the corresponding optimization problem and solve it with the gradient-based optimization method. The gradient of the obtained objective function is computed precisely through the automatic differentiation technique. The numerical experiments illustrate that the considered approach gives practically the same dynamics as the exact Mori-Zwanzig decomposition, but is less computationally intensive.

Daniel Bershatsky, Aleksandr Mikhalev, Alexandr Katrutsa et al.

In modern neural networks like Transformers, linear layers require significant memory to store activations during backward pass. This study proposes a memory reduction approach to perform backpropagation through linear layers. Since the gradients of linear layers are computed by matrix multiplications, we consider methods for randomized matrix multiplications and demonstrate that they require less memory with a moderate decrease of the test accuracy. Also, we investigate the variance of the gradient estimate induced by the randomized matrix multiplication. We compare this variance with the variance coming from gradient estimation based on the batch of samples. We demonstrate the benefits of the proposed method on the fine-tuning of the pre-trained RoBERTa model on GLUE tasks.

Alexandr Katrutsa, Daniil Merkulov, Nurislam Tursynbek et al.

This study presents a novel Equiangular Direction Method (EDM) to solve a multi-objective optimization problem. We consider optimization problems, where multiple differentiable losses have to be minimized. The presented method computes descent direction in every iteration to guarantee equal relative decrease of objective functions. This descent direction is based on the normalized gradients of the individual losses. Therefore, it is appropriate to solve multi-objective optimization problems with multi-scale losses. We test the proposed method on the imbalanced classification problem and multi-task learning problem, where standard datasets are used. EDM is compared with other methods to solve these problems.

Julia Gusak, Larisa Markeeva, Talgat Daulbaev et al.

Normalization is an important and vastly investigated technique in deep learning. However, its role for Ordinary Differential Equation based networks (neural ODEs) is still poorly understood. This paper investigates how different normalization techniques affect the performance of neural ODEs. Particularly, we show that it is possible to achieve 93% accuracy in the CIFAR-10 classification task, and to the best of our knowledge, this is the highest reported accuracy among neural ODEs tested on this problem.

Talgat Daulbaev, Alexandr Katrutsa, Larisa Markeeva et al.

We propose a simple interpolation-based method for the efficient approximation of gradients in neural ODE models. We compare it with the reverse dynamic method (known in the literature as "adjoint method") to train neural ODEs on classification, density estimation, and inference approximation tasks. We also propose a theoretical justification of our approach using logarithmic norm formalism. As a result, our method allows faster model training than the reverse dynamic method that was confirmed and validated by extensive numerical experiments for several standard benchmarks.