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Constrained optimization offers a powerful framework to prescribe desired behaviors in neural network models. Typically, constrained problems are solved via their min-max Lagrangian formulations, which exhibit unstable oscillatory dynamics when optimized using gradient descent-ascent. The adoption of constrained optimization techniques in the machine learning community is currently limited by the lack of reliable, general-purpose update schemes for the Lagrange multipliers. This paper proposes the $\nu$PI algorithm and contributes an optimization perspective on Lagrange multiplier updates based on PI controllers, extending the work of Stooke, Achiam and Abbeel (2020). We provide theoretical and empirical insights explaining the inability of momentum methods to address the shortcomings of gradient descent-ascent, and contrast this with the empirical success of our proposed $\nu$PI controller. Moreover, we prove that $\nu$PI generalizes popular momentum methods for single-objective minimization. Our experiments demonstrate that $\nu$PI reliably stabilizes the multiplier dynamics and its hyperparameters enjoy robust and predictable behavior., Comment: Published at ICML 2024. Code available at https://github.com/motahareh-sohrabi/nuPI

We study Thompson Sampling-based algorithms for stochastic bandits with bounded rewards. As the existing problem-dependent regret bound for Thompson Sampling with Gaussian priors [Agrawal and Goyal, 2017] is vacuous when $T \le 288 e^{64}$, we derive a more practical bound that tightens the coefficient of the leading term %from $288 e^{64}$ to $1270$. Additionally, motivated by large-scale real-world applications that require scalability, adaptive computational resource allocation, and a balance in utility and computation, we propose two parameterized Thompson Sampling-based algorithms: Thompson Sampling with Model Aggregation (TS-MA-$\alpha$) and Thompson Sampling with Timestamp Duelling (TS-TD-$\alpha$), where $\alpha \in [0,1]$ controls the trade-off between utility and computation. Both algorithms achieve $O \left(K\ln^{\alpha+1}(T)/\Delta \right)$ regret bound, where $K$ is the number of arms, $T$ is the finite learning horizon, and $\Delta$ denotes the single round performance loss when pulling a sub-optimal arm.

Artificial nanostructures with ultrafine and deep-subwavelength feature sizes have emerged as a paradigm-shifting platform to advanced light field management, becoming a key building block for high-performance integrated optoelectronics and flat optics. However, direct optical inspection of such integrated chips with densely packed complex and small features remains a missing metrology gap that hinders quick feedback between design and fabrications. Here, we demonstrate that photothermal nonlinear scattering microscopy can be utilized for direct imaging and resolving of integrated optoelectronic chips beyond the diffraction limit. We reveal that the inherent coupling among deep-subwavelength nanostructures supporting leaky resonances allows for the pronounced heating effect to access reversible nonlinear modulations of the confocal reflection intensity, leading to optical resolving power down to 80 nm (~lambda/7). The versatility of this approach has been exemplified by direct imaging of silicon grating couplers and metalens with a minimum critical dimension of 100 nm, as well as central processing unit (CPU) chip with 45 nm technology, unfolding the long-sought possibility of in-situ, non-destructive, high-throughput optical inspection of integrated optoelectronic chips and nanophotonic chips.

Featured with a plethora of electric and magnetic Mie resonances, high index dielectric nanostructures offer a versatile platform to concentrate light-matter interactions at the nanoscale. By integrating unique features of far-field scattering control and near-field concentration from radiationless anapole states, here, we demonstrate a giant photothermal nonlinearity in single subwavelength-sized silicon nanodisks. The nanoscale energy concentration and consequent near-field enhancements mediated by the anapole mode yield a reversible nonlinear scattering with a large modulation depth and a broad dynamic range, unveiling a record-high nonlinear index change up to 0.5 at mild incident light intensities on the order of MW/cm2. The observed photothermal nonlinearity showcases three orders of magnitude enhancement compared with that of unstructured bulk silicon, as well as nearly one order of magnitude higher than that through the radiative electric dipolar mode. Such nonlinear scattering can empower distinctive point spread functions in confocal reflectance imaging, offering the potential for far-field localization of nanostructured Si with an accuracy approaching 40 nm. Our findings shed new light on active silicon photonics based on optical anapoles.