Nikita Doikov

Nikita Doikov, Konstantin Mishchenko, Yurii Nesterov

We analyze the performance of a variant of Newton method with quadratic regularization for solving composite convex minimization problems. At each step of our method, we choose regularization parameter proportional to a certain power of the gradient norm at the current point. We introduce a family of problem classes characterized by Hölder continuity of either the second or third derivative. Then we present the method with a simple adaptive search procedure allowing an automatic adjustment to the problem class with the best global complexity bounds, without knowing specific parameters of the problem. In particular, for the class of functions with Lipschitz continuous third derivative, we get the global $O(1/k^3)$ rate, which was previously attributed to third-order tensor methods. When the objective function is uniformly convex, we justify an automatic acceleration of our scheme, resulting in a faster global rate and local superlinear convergence. The switching between the different rates (sublinear, linear, and superlinear) is automatic. Again, for that, no a priori knowledge of parameters is needed.

Nikita Doikov, El Mahdi Chayti, Martin Jaggi

We analyze Newton's method with lazy Hessian updates for solving general possibly non-convex optimization problems. We propose to reuse a previously seen Hessian for several iterations while computing new gradients at each step of the method. This significantly reduces the overall arithmetical complexity of second-order optimization schemes. By using the cubic regularization technique, we establish fast global convergence of our method to a second-order stationary point, while the Hessian does not need to be updated each iteration. For convex problems, we justify global and local superlinear rates for lazy Newton steps with quadratic regularization, which is easier to compute. The optimal frequency for updating the Hessian is once every $d$ iterations, where $d$ is the dimension of the problem. This provably improves the total arithmetical complexity of second-order algorithms by a factor $\sqrt{d}$.

El Mahdi Chayti, Nikita Doikov, Martin Jaggi

We study stochastic Cubic Newton methods for solving general possibly non-convex minimization problems. We propose a new framework, which we call the helper framework, that provides a unified view of the stochastic and variance-reduced second-order algorithms equipped with global complexity guarantees. It can also be applied to learning with auxiliary information. Our helper framework offers the algorithm designer high flexibility for constructing and analyzing the stochastic Cubic Newton methods, allowing arbitrary size batches, and the use of noisy and possibly biased estimates of the gradients and Hessians, incorporating both the variance reduction and the lazy Hessian updates. We recover the best-known complexities for the stochastic and variance-reduced Cubic Newton, under weak assumptions on the noise. A direct consequence of our theory is the new lazy stochastic second-order method, which significantly improves the arithmetic complexity for large dimension problems. We also establish complexity bounds for the classes of gradient-dominated objectives, that include convex and strongly convex problems. For Auxiliary Learning, we show that using a helper (auxiliary function) can outperform training alone if a given similarity measure is small.

Dongyang Fan, Bettina Messmer, Nikita Doikov et al.

On-device LLMs have gained increasing attention for their ability to enhance privacy and provide a personalized user experience. To facilitate private learning with scarce data, Federated Learning has become a standard approach. However, it faces challenges such as computational resource heterogeneity and data heterogeneity among end users. We propose CoMiGS ($\textbf{Co}$llaborative learning with a $\textbf{Mi}$xture of $\textbf{G}$eneralists and $\textbf{S}$pecialists), the first approach to address both challenges. A key innovation of our method is the bi-level optimization formulation of the Mixture-of-Experts learning objective, where the router is optimized using a separate validation set to ensure alignment with the target distribution. We solve our objective with alternating minimization, for which we provide a theoretical analysis. Our method shares generalist experts across users while localizing a varying number of specialist experts, thereby adapting to users' computational resources and preserving privacy. Through extensive experiments, we show CoMiGS effectively balances general and personalized knowledge for each token generation. We demonstrate that CoMiGS remains robust against overfitting-due to the generalists' regularizing effect-while adapting to local data through specialist expertise. We open source our codebase for collaborative LLMs.

Maria-Luiza Vladarean, Nikita Doikov, Martin Jaggi et al.

This paper studies first-order algorithms for solving fully composite optimization problems over convex and compact sets. We leverage the structure of the objective by handling its differentiable and non-differentiable components separately, linearizing only the smooth parts. This provides us with new generalizations of the classical Frank-Wolfe method and the Conditional Gradient Sliding algorithm, that cater to a subclass of non-differentiable problems. Our algorithms rely on a stronger version of the linear minimization oracle, which can be efficiently implemented in several practical applications. We provide the basic version of our method with an affine-invariant analysis and prove global convergence rates for both convex and non-convex objectives. Furthermore, in the convex case, we propose an accelerated method with correspondingly improved complexity. Finally, we provide illustrative experiments to support our theoretical results.

Nikita Doikov, Geovani Nunes Grapiglia

In this work, we develop first-order (Hessian-free) and zero-order (derivative-free) implementations of the Cubically regularized Newton method for solving general non-convex optimization problems. For that, we employ finite difference approximations of the derivatives. We use a special adaptive search procedure in our algorithms, which simultaneously fits both the regularization constant and the parameters of the finite difference approximations. It makes our schemes free from the need to know the actual Lipschitz constants. Additionally, we equip our algorithms with the lazy Hessian update that reuse a previously computed Hessian approximation matrix for several iterations. Specifically, we prove the global complexity bound of $\mathcal{O}( n^{1/2} ε^{-3/2})$ function and gradient evaluations for our new Hessian-free method, and a bound of $\mathcal{O}( n^{3/2} ε^{-3/2} )$ function evaluations for the derivative-free method, where $n$ is the dimension of the problem and $ε$ is the desired accuracy for the gradient norm. These complexity bounds significantly improve the previously known ones in terms of the joint dependence on $n$ and $ε$, for the first-order and zeroth-order non-convex optimization.

Nikita Doikov

We study the composite convex optimization problems with a Quasi-Self-Concordant smooth component. This problem class naturally interpolates between classic Self-Concordant functions and functions with Lipschitz continuous Hessian. Previously, the best complexity bounds for this problem class were associated with trust-region schemes and implementations of a ball-minimization oracle. In this paper, we show that for minimizing Quasi-Self-Concordant functions we can use instead the basic Newton Method with Gradient Regularization. For unconstrained minimization, it only involves a simple matrix inversion operation (solving a linear system) at each step. We prove a fast global linear rate for this algorithm, matching the complexity bound of the trust-region scheme, while our method remains especially simple to implement. Then, we introduce the Dual Newton Method, and based on it, develop the corresponding Accelerated Newton Scheme for this problem class, which further improves the complexity factor of the basic method. As a direct consequence of our results, we establish fast global linear rates of simple variants of the Newton Method applied to several practical problems, including Logistic Regression, Soft Maximum, and Matrix Scaling, without requiring additional assumptions on strong or uniform convexity for the target objective.

Nikita Doikov, Anton Rodomanov

We study first-order methods with preconditioning for solving structured nonlinear convex optimization problems. We propose a new family of preconditioners generated by symmetric polynomials. They provide first-order optimization methods with a provable improvement of the condition number, cutting the gaps between highest eigenvalues, without explicit knowledge of the actual spectrum. We give a stochastic interpretation of this preconditioning in terms of coordinate volume sampling and compare it with other classical approaches, including the Chebyshev polynomials. We show how to incorporate a polynomial preconditioning into the Gradient and Fast Gradient Methods and establish the corresponding global complexity bounds. Finally, we propose a simple adaptive search procedure that automatically chooses the best possible polynomial preconditioning for the Gradient Method, minimizing the objective along a low-dimensional Krylov subspace. Numerical experiments confirm the efficiency of our preconditioning strategies for solving various machine learning problems.

Ramon Viñas Torné, Sílvia Fàbregas Salazar, Soyon Park et al.

Inferring the structure of directed acyclic graphs (DAGs) from data is a central challenge in causal discovery, particularly in modern high-dimensional settings where large-scale interventional data are increasingly available. While interventional data can improve identifiability, existing methods remain limited by soft acyclicity constraints, leading to optimization over invalid cyclic graphs, numerical instability, and reduced scalability. We introduce PACER (Perturbation-driven Acyclic Causal Edge Recovery), a scalable framework for causal discovery that guarantees acyclicity by construction. PACER parameterizes a distribution over DAGs through a joint model of variable permutations and edge probabilities, enabling direct optimization over valid causal structures without surrogate penalties. The framework supports a unified likelihood-based treatment of observational and interventional data, flexible conditional density models, and the incorporation of structural prior knowledge. For linear-Gaussian mechanisms, we derive closed-form expressions for the expected interventional log-likelihood and its gradients, yielding substantial computational gains. Empirically, PACER matches or exceeds state-of-the-art methods on protein signaling and large-scale genetic perturbation benchmarks, while scaling efficiently to networks with thousands of variables and achieving up to two orders of magnitude speedups over penalty-based differentiable approaches. These results demonstrate that exact and scalable causal discovery from high-dimensional perturbation data is achievable through principled search space design.

El Mahdi Chayti, Nikita Doikov, Martin Jaggi

We study stochastic second-order methods for solving general non-convex optimization problems. We propose using a special version of momentum to stabilize the stochastic gradient and Hessian estimates in Newton's method. We show that momentum provably improves the variance of stochastic estimates and allows the method to converge for any noise level. Using the cubic regularization technique, we prove a global convergence rate for our method on general non-convex problems to a second-order stationary point, even when using only a single stochastic data sample per iteration. This starkly contrasts with all existing stochastic second-order methods for non-convex problems, which typically require large batches. Therefore, we are the first to demonstrate global convergence for batches of arbitrary size in the non-convex case for the Stochastic Cubic Newton. Additionally, we show improved speed on convex stochastic problems for our regularized Newton methods with momentum.

Andrei Semenov, Martin Jaggi, Nikita Doikov

In this work, we develop new optimization algorithms that use approximate second-order information combined with the gradient regularization technique to achieve fast global convergence rates for both convex and non-convex objectives. The key innovation of our analysis is a novel notion called Gradient-Normalized Smoothness, which characterizes the maximum radius of a ball around the current point that yields a good relative approximation of the gradient field. Our theory establishes a natural intrinsic connection between Hessian approximation and the linearization of the gradient. Importantly, Gradient-Normalized Smoothness does not depend on the specific problem class of the objective functions, while effectively translating local information about the gradient field and Hessian approximation into the global behavior of the method. This new concept equips approximate second-order algorithms with universal global convergence guarantees, recovering state-of-the-art rates for functions with Hölder-continuous Hessians and third derivatives, quasi-self-concordant functions, as well as smooth classes in first-order optimization. These rates are achieved automatically and extend to broader classes, such as generalized self-concordant functions. We demonstrate direct applications of our results for global linear rates in logistic regression and softmax problems with approximate Hessians, as well as in non-convex optimization using Fisher and Gauss-Newton approximations.

Jim Zhao, Aurelien Lucchi, Nikita Doikov

This paper addresses the optimization problem of minimizing non-convex continuous functions, which is relevant in the context of high-dimensional machine learning applications characterized by over-parametrization. We analyze a randomized coordinate second-order method named SSCN which can be interpreted as applying cubic regularization in random subspaces. This approach effectively reduces the computational complexity associated with utilizing second-order information, rendering it applicable in higher-dimensional scenarios. Theoretically, we establish convergence guarantees for non-convex functions, with interpolating rates for arbitrary subspace sizes and allowing inexact curvature estimation. When increasing subspace size, our complexity matches $\mathcal{O}(ε^{-3/2})$ of the cubic regularization (CR) rate. Additionally, we propose an adaptive sampling scheme ensuring exact convergence rate of $\mathcal{O}(ε^{-3/2}, ε^{-3})$ to a second-order stationary point, even without sampling all coordinates. Experimental results demonstrate substantial speed-ups achieved by SSCN compared to conventional first-order methods.

Anastasia Koloskova, Nikita Doikov, Sebastian U. Stich et al.

In machine learning and neural network optimization, algorithms like incremental gradient, and shuffle SGD are popular due to minimizing the number of cache misses and good practical convergence behavior. However, their optimization properties in theory, especially for non-convex smooth functions, remain incompletely explored. This paper delves into the convergence properties of SGD algorithms with arbitrary data ordering, within a broad framework for non-convex smooth functions. Our findings show enhanced convergence guarantees for incremental gradient and single shuffle SGD. Particularly if $n$ is the training set size, we improve $n$ times the optimization term of convergence guarantee to reach accuracy $\varepsilon$ from $O(n / \varepsilon)$ to $O(1 / \varepsilon)$.

Filip Hanzely, Nikita Doikov, Peter Richtárik et al.

In this paper, we propose a new randomized second-order optimization algorithm---Stochastic Subspace Cubic Newton (SSCN)---for minimizing a high dimensional convex function $f$. Our method can be seen both as a {\em stochastic} extension of the cubically-regularized Newton method of Nesterov and Polyak (2006), and a {\em second-order} enhancement of stochastic subspace descent of Kozak et al. (2019). We prove that as we vary the minibatch size, the global convergence rate of SSCN interpolates between the rate of stochastic coordinate descent (CD) and the rate of cubic regularized Newton, thus giving new insights into the connection between first and second-order methods. Remarkably, the local convergence rate of SSCN matches the rate of stochastic subspace descent applied to the problem of minimizing the quadratic function $\frac12 (x-x^*)^\top \nabla^2f(x^*)(x-x^*)$, where $x^*$ is the minimizer of $f$, and hence depends on the properties of $f$ at the optimum only. Our numerical experiments show that SSCN outperforms non-accelerated first-order CD algorithms while being competitive to their accelerated variants.