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Entropic Risk-Averse Generalized Momentum Methods

Bugra Can, Mert Gürbüzbalaban

TL;DR

The paper addresses optimization with stochastic gradient noise by unifying momentum methods into a generalized momentum framework (GMM) and introducing entropic risk and EV@R as tail-risk measures for suboptimality. It derives explicit non-asymptotic convergence and risk bounds for quadratic and general strongly convex smooth objectives, and shows how to design risk-averse momentum parameters (RA-GMM/RA-AGD) to trade off tail risk against convergence speed. The results yield closed-form risk expressions and tractable optimization procedures that improve tail behavior while controlling rate, demonstrated on quadratic and logistic regression tasks. The work provides a principled methodology to tune momentum components under uncertainty, with potential impact on large-scale learning where gradient noise is persistent and tail guarantees are valuable.

Abstract

In the context of first-order algorithms subject to random gradient noise, we study the trade-offs between the convergence rate (which quantifies how fast the initial conditions are forgotten) and the "risk" of suboptimality, i.e. deviations from the expected suboptimality. We focus on a general class of momentum methods (GMM) which recover popular methods such as gradient descent (GD), accelerated gradient descent (AGD), and heavy-ball (HB) method as special cases depending on the choice of GMM parameters. We use well-known risk measures "entropic risk" and "entropic value at risk" to quantify the risk of suboptimality. For strongly convex smooth minimization, we first obtain new convergence rate results for GMM with a unified theory that is also applicable to both AGD and HB, improving some of the existing results for HB. We then provide explicit bounds on the entropic risk and entropic value at risk of suboptimality at a given iterate which also provides direct bounds on the probability that the suboptimality exceeds a given threshold based on Chernoff's inequality. Our results unveil fundamental trade-offs between the convergence rate and the risk of suboptimality. We then plug the entropic risk and convergence rate estimates we obtained in a computationally tractable optimization framework and propose entropic risk-averse GMM (RA-GMM) and entropic risk-averse AGD (RA-AGD) methods which can select the GMM parameters to systematically trade-off the entropic value at risk with the convergence rate. We show that RA-AGD and RA-GMM lead to improved performance on quadratic optimization and logistic regression problems compared to the standard choice of parameters. To our knowledge, our work is the first to resort to coherent measures to design the parameters of momentum methods in a systematic manner.

Entropic Risk-Averse Generalized Momentum Methods

TL;DR

The paper addresses optimization with stochastic gradient noise by unifying momentum methods into a generalized momentum framework (GMM) and introducing entropic risk and EV@R as tail-risk measures for suboptimality. It derives explicit non-asymptotic convergence and risk bounds for quadratic and general strongly convex smooth objectives, and shows how to design risk-averse momentum parameters (RA-GMM/RA-AGD) to trade off tail risk against convergence speed. The results yield closed-form risk expressions and tractable optimization procedures that improve tail behavior while controlling rate, demonstrated on quadratic and logistic regression tasks. The work provides a principled methodology to tune momentum components under uncertainty, with potential impact on large-scale learning where gradient noise is persistent and tail guarantees are valuable.

Abstract

In the context of first-order algorithms subject to random gradient noise, we study the trade-offs between the convergence rate (which quantifies how fast the initial conditions are forgotten) and the "risk" of suboptimality, i.e. deviations from the expected suboptimality. We focus on a general class of momentum methods (GMM) which recover popular methods such as gradient descent (GD), accelerated gradient descent (AGD), and heavy-ball (HB) method as special cases depending on the choice of GMM parameters. We use well-known risk measures "entropic risk" and "entropic value at risk" to quantify the risk of suboptimality. For strongly convex smooth minimization, we first obtain new convergence rate results for GMM with a unified theory that is also applicable to both AGD and HB, improving some of the existing results for HB. We then provide explicit bounds on the entropic risk and entropic value at risk of suboptimality at a given iterate which also provides direct bounds on the probability that the suboptimality exceeds a given threshold based on Chernoff's inequality. Our results unveil fundamental trade-offs between the convergence rate and the risk of suboptimality. We then plug the entropic risk and convergence rate estimates we obtained in a computationally tractable optimization framework and propose entropic risk-averse GMM (RA-GMM) and entropic risk-averse AGD (RA-AGD) methods which can select the GMM parameters to systematically trade-off the entropic value at risk with the convergence rate. We show that RA-AGD and RA-GMM lead to improved performance on quadratic optimization and logistic regression problems compared to the standard choice of parameters. To our knowledge, our work is the first to resort to coherent measures to design the parameters of momentum methods in a systematic manner.
Paper Structure (34 sections, 25 theorems, 344 equations, 7 figures)

This paper contains 34 sections, 25 theorems, 344 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Other Related Work
  3. Preliminaries and Background
  4. Dynamical system representation of noisy GMM
  5. Entropic risk and entropic value at risk (EV@R)
  6. Main results for strongly convex quadratic objectives
  7. Main results for strongly convex smooth objectives
  8. Numerical Experiments
  9. Extensions to sub-Gaussian noise and to more general choice of parameters
  10. Conclusion
  11. Proofs of Section \ref{['sec: quad-obj']}
  12. Proof of Lemma \ref{['lem: non_asym_conv_quad_obj']}
  13. Proof of Proposition \ref{['prop: quad_qisk_meas_convergence']}
  14. Proof of Proposition \ref{['prop: quad-risk-meas-gauss-noise']}
  15. Proof of Theorem \ref{['thm: quad-evar-bound']}
  16. ...and 19 more sections

Key Result

Lemma 3.1

Consider the noisy GMM iterates ${\color{black} z}_k$ satisfying the recursion sys: TMM_quad for minimizing a quadratic function $f$ of the form def: quad-func where the gradient noise obeys Assumption Assump: Noise. Then, we have for any $k\geq 1$, where $\rho(A_Q)= \max_{i\in \{ 1,..,d\}}\{\rho_{i}\}$ is the spectral radius of $A_Q$ with and $\lambda_i(Q)$ are the eigenvalues of the Hessian m

Figures (7)

  • Figure 1: The feasible region $\mathcal{F}_{\theta}$ versus the stable set $\mathcal{S}_q$ for $f(x_{(1)},x_{(2)})=x_{(1)}^2+0.1x_{(2)}^2$ where $x_{(1)}, x_{(2)}\in\mathbb{R}$ and $\sigma^2=1$.
  • Figure 2: Illustration of the results of Proposition \ref{['prop: quad-risk-meas-gauss-noise']} and Theorem \ref{['thm: quad-evar-bound']} for noisy GMM and noisy GD on $f(x_{(1)},x_{(2)})=x_{(1)}^2+0.1 x_{(2)}^2$ with $\sigma^2 = 1$. Left: The convergence rate vs. optimal (smallest) infinite-horizon risk attainable at this convergence rate for $\theta \in \{0.01,1,2\}$, Right: The comparison of $EV@R_{1-\zeta}{\color{black} [f(x_\infty)-f(x_*)]}$ (straight lines) and its approximation $\bar{E}^q_{1-\zeta}(\alpha,\beta,\gamma)$ (dashed lines) at confidence levels $\zeta\in \{0.95, 0.5\}$.
  • Figure 3: The EV@R bound $\bar{E}_{1-\zeta}$ (Left) and the rate $\rho_{\vartheta,\psi}^2/\rho_*^2$ (Right) on stable set $\mathcal{S}_c$ at $\varphi=0.99$ and $\zeta=0.99$ confidence level where $x\in\mathbb{R}^{10}$, $L=1$, $\mu=0.1$ for the objective function and the Gaussian noise is $\mathcal{N}(0,I_{10})$.
  • Figure 4: (Left panel) The expected suboptimality versus iterations for GD, AGD, RA-AGD and RA- GMM. (Right panel) The cumulative distribution of the suboptimality of the last iterates for GD, AGD, RA-AGD and RA- GMM after $k=300$ iterations.
  • Figure 5: The empirical finite-horizon risk measure, $\tilde{r}_{k,1}(5)$ versus iterations on the quadratic optimization problem \ref{['opt-pbm-quad']}.
  • ...and 2 more figures

Theorems & Definitions (44)

  • Lemma 3.1
  • Remark 1
  • Proposition 1
  • proof
  • Proposition 2
  • Theorem 1
  • Example 1
  • Corollary 1
  • Theorem 2
  • Remark 2
  • ...and 34 more