Session: Recent Progress on Bilevel Optimization for Machine Learning
Chair: Tong Zhang
Cluster: Nonlinear Optimization
Talk 1: Tuning-Free Bilevel Optimization: New Algorithms and Convergence Analysis
Speaker: Kaiyi Ji
Abstract: Bilevel optimization has recently attracted considerable attention due to its abundant applications in machine learning problems. However, existing methods rely on prior knowledge of problem parameters to determine stepsizes, resulting in significant effort in tuning stepsizes when these parameters are unknown. In this paper, we propose two novel tuning-free algorithms, D-TFBO and S-TFBO. D-TFBO employs a double-loop structure with stepsizes adaptively adjusted by the "inverse of cumulative gradient norms" strategy. S-TFBO features a simpler fully single-loop structure that updates three variables simultaneously with a theory-motivated joint design of adaptive stepsizes for all variables. We provide a comprehensive convergence analysis for both algorithms and show that D-TFBO and S-TFBO respectively require $\mathcal{O}(\frac{1}{\epsilon})$ and $\mathcal{O}(\frac{1}{\epsilon}\log^4(\frac{1}{\epsilon}))$ iterations to find an $\epsilon$-accurate stationary point, (nearly) matching their well-tuned counterparts using the information of problem parameters. Experiments on various problems show that our methods achieve performance comparable to existing well-tuned approaches, while being more robust to the selection of initial stepsizes. To the best of our knowledge, our methods are the first to completely eliminate the need for stepsize tuning, while achieving theoretical guarantees.
Talk 2: Optimal Algorithms for Stochastic Bilevel Optimization under Relaxed Smoothness Conditions
Speaker: Xuxing Chen
Abstract: Stochastic Bilevel optimization usually involves minimizing an upper-level (UL) function that is dependent on the arg-min of a strongly-convex lower-level (LL) function. Several algorithms utilize Neumann series to approximate certain matrix inverses involved in estimating the implicit gradient of the UL function (hypergradient). The state-of-the-art StOchastic Bilevel Algorithm (SOBA) instead uses stochastic gradient descent steps to solve the linear system associated with the explicit matrix inversion. This modification enables SOBA to match the lower bound of sample complexity for the single-level counterpart in non-convex settings. Unfortunately, the current analysis of SOBA relies on the assumption of higher-order smoothness for the UL and LL functions to achieve optimality. In this talk, I will introduce a novel fully single-loop and Hessian-inversion-free algorithmic framework for stochastic bilevel optimization and present a tighter analysis under standard smoothness assumptions (first-order Lipschitzness of the UL function and second-order Lipschitzness of the LL function). Furthermore, I will show that by a slight modification of our approach, our algorithm can handle a more general multi-objective robust bilevel optimization problem. For this case, we obtain the state-of-the-art oracle complexity results demonstrating the generality of both the proposed algorithmic and analytic frameworks. Numerical experiments demonstrate the performance gain of the proposed algorithms over existing ones.
Talk 3: Two-Timescale Linear Stochastic Approximation: Constant Stepsizes Go a Long Way
Speaker: Jeongyeol Kwon
Abstract: Previous studies on two-timescale stochastic approximation (SA) mainly focused on bounding mean-squared errors under diminishing stepsize schemes. In this work, we investigate {\it constant} stpesize schemes through the lens of Markov processes, proving that the iterates of both timescales converge to a unique joint stationary distribution in the Wasserstein metric. We derive explicit geometric and non-asymptotic convergence rates, as well as the variance and bias introduced by constant stepsizes in the presence of Markovian noise. Specifically, with two constant stepsizes $\alpha < \beta$, we show that the biases scale linearly with both stepsizes as $\Theta(\alpha)+\Theta(\beta)$ up to higher-order terms, while the variance of the slower iterate (resp., faster iterate) scales only with its own stepsize as $O(\alpha)$ (resp., $O(\beta)$). Unlike previous work, our results require no additional assumptions such as $\beta^2 \ll \alpha$ nor extra dependence on dimensions. These fine-grained characterizations allow tail-averaging and extrapolation techniques to reduce variance and bias, improving mean-squared error bound to $O(\beta^4 + \frac{1}{t})$ for both iterates.
