The rise of parallel computing hardware has made it increasingly important to understand which nonlinear state space models can be efficiently parallelized. Recent advances like DEER (arXiv:2309.12252) or DeepPCR (arXiv:2309.16318) have shown that evaluating a state space model can be recast as solving a parallelizable optimization problem, and sometimes this approach can yield dramatic speed-ups in evaluation time. However, the factors that govern the difficulty of these optimization problems remain unclear, limiting the larger adoption of the technique. In this work, we establish a precise relationship between the dynamics of a nonlinear system and the conditioning of its corresponding optimization formulation. We show that the predictability of a system, defined as the degree to which small perturbations in state influence future behavior, impacts the number of optimization steps required for evaluation. In predictable systems, the state trajectory can be computed in $O((\log T)^2)$ time, where $T$ is the sequence length, a major improvement over the conventional sequential approach. In contrast, chaotic or unpredictable systems exhibit poor conditioning, with the consequence that parallel evaluation converges too slowly to be useful. Importantly, our theoretical analysis demonstrates that for predictable systems, the optimization problem is always well-conditioned, whereas for unpredictable systems, the conditioning degrades exponentially as a function of the sequence length. We validate our claims through extensive experiments, providing practical guidance on when nonlinear dynamical systems can be efficiently parallelized, and highlighting predictability as a key design principle for parallelizable models.

This paper proposes a frequency-domain system identification method for learning low-order systems. The identification problem is formulated as the minimization of the l2 norm between the identified and measured frequency responses, with the nuclear norm of the Loewner matrix serving as a regularization term. This formulation results in an optimization problem that can be efficiently solved using standard convex optimization techniques. We derive an upper bound on the sampled-frequency complexity of the identification process and subsequently extend this bound to characterize the identification error over all frequencies. A detailed analysis of the sample complexity is provided, along with a thorough interpretation of its terms and dependencies. Finally, the efficacy of the proposed method is demonstrated through an example, along with numerical simulations validating the growth rate of the sample complexity bound.

These lecture notes evolve around mathematical concepts arising in inverse problems. We start by introducing inverse problems through examples such as differentiation, deconvolution, computed tomography and phase retrieval. This then leads us to the framework of well-posedness and first considerations regarding reconstruction and inversion approaches. The second chapter then first deals with classical regularization theory of inverse problems in Hilbert spaces. After introducing the pseudo-inverse, we review the concept of convergent regularization. Within this chapter we then proceed to ask the question of how to realize practical reconstruction algorithms. Here, we mainly focus on Tikhonov and sparsity promoting regularization in finite dimensional spaces. In the third chapter, we dive into modern deep-learning methods, which allow solving inverse problems in a data-dependent approach. The intersection between inverse problems and machine learning is a rapidly growing field and our exposition here restricts itself to a very limited selection of topics. Among them are learned regularization, fully-learned Bayesian estimation, post-processing strategies and plug-n-play methods.

While powerful, large language models (LLMs) present significant fine-tuning challenges due to their size. Parameter-efficient fine-tuning (PEFT) methods like LoRA provide solutions, yet suffer from critical optimizer inefficiencies; notably basis redundancy in LoRA's $B$ matrix when using AdamW, which fundamentally limits performance. We address this by optimizing the $B$ matrix on the Stiefel manifold, imposing explicit orthogonality constraints that achieve near-perfect orthogonality and full effective rank. This geometric approach dramatically enhances parameter efficiency and representational capacity. Our Stiefel optimizer consistently outperforms AdamW across benchmarks with both LoRA and DoRA, demonstrating that geometric constraints are the key to unlocking LoRA's full potential for effective LLM fine-tuning.

DPO (Direct Preference Optimization) has become a widely used offline preference optimization algorithm due to its simplicity and training stability. However, DPO is prone to overfitting and collapse. To address these challenges, we propose Linear Preference Optimization (LPO), a novel alignment framework featuring three key innovations. First, we introduce gradient decoupling by replacing the log-sigmoid function with an absolute difference loss, thereby isolating the optimization dynamics. Second, we improve stability through an offset constraint combined with a positive regularization term to preserve the chosen response quality. Third, we implement controllable rejection suppression using gradient separation with straightforward estimation and a tunable coefficient that linearly regulates the descent of the rejection probability. Through extensive experiments, we demonstrate that LPO consistently improves performance on various tasks, including general text tasks, math tasks, and text-to-speech (TTS) tasks. These results establish LPO as a robust and tunable paradigm for preference alignment, and we release the source code, models, and training data publicly.

Decision-focused learning (DFL) trains a machine learning (ML) model to predict parameters of an optimization problem, to directly minimize decision regret, i.e., maximize decision quality. Gradient-based DFL requires computing the derivative of the solution to the optimization problem with respect to the predicted parameters. However, for many optimization problems, such as linear programs (LPs), the gradient of the regret with respect to the predicted parameters is zero almost everywhere. Existing gradient-based DFL approaches for LPs try to circumvent this issue in one of two ways: (a) smoothing the LP into a differentiable optimization problem by adding a quadratic regularizer and then minimizing the regret directly or (b) minimizing surrogate losses that have informative (sub)gradients. In this paper, we show that the former approach still results in zero gradients, because even after smoothing the regret remains constant across large regions of the parameter space. To address this, we propose minimizing surrogate losses -- even when a differentiable optimization layer is used and regret can be minimized directly. Our experiments demonstrate that minimizing surrogate losses allows differentiable optimization layers to achieve regret comparable to or better than surrogate-loss based DFL methods. Further, we demonstrate that this also holds for DYS-Net, a recently proposed differentiable optimization technique for LPs, that computes approximate solutions and gradients through operations that can be performed using feedforward neural network layers. Because DYS-Net executes the forward and the backward pass very efficiently, by minimizing surrogate losses using DYS-Net, we are able to attain regret on par with the state-of-the-art while reducing training time by a significant margin.

The Profiled V ehicle Routing Problem (PVRP) extends the classical VRP by incorporating vehicle-client-specific preferences and constraints, reflecting real-world requirements such as zone restrictions and service-level preferences. While recent reinforcement-learning solvers have shown promising performance, they require retraining for each new profile distribution, suffer from poor representation ability, and struggle to generalize to out-of-distribution instances. In this paper, we address these limitations by introducing U nified Solver for Profiled R outing (USPR), a novel framework that natively handles arbitrary profile types. USPR introduces on three key innovations: (i) Profile Embeddings (PE) to encode any combination of profile types; (ii) Multi-Head Profiled Attention (MHP A), an attention mechanism that models rich interactions between vehicles and clients; (iii) Profile-aware Score Reshaping (PSR), which dynamically adjusts decoder logits using profile scores to improve generalization. Empirical results on diverse PVRP benchmarks demonstrate that USPR achieves state-of-the-art results among learning-based methods while offering significant gains in flexibility and computational efficiency. We make our source code publicly available to foster future research.

The data mixture used in the pre-training of a language model is a cornerstone of its final performance. However, a static mixing strategy is suboptimal, as the model's learning preferences for various data domains shift dynamically throughout training. Crucially, observing these evolving preferences in a computationally efficient manner remains a significant challenge. To address this, we propose TiKMiX, a method that dynamically adjusts the data mixture according to the model's evolving preferences. TiKMiX introduces Group Influence, an efficient metric for evaluating the impact of data domains on the model. This metric enables the formulation of the data mixing problem as a search for an optimal, influence-maximizing distribution. We solve this via two approaches: TiKMiX-D for direct optimization, and TiKMiX-M, which uses a regression model to predict a superior mixture. We trained models with different numbers of parameters, on up to 1 trillion tokens. TiKMiX-D exceeds the performance of state-of-the-art methods like REGMIX while using just 20% of the computational resources. TiKMiX-M leads to an average performance gain of 2% across 9 downstream benchmarks. Our experiments reveal that a model's data preferences evolve with training progress and scale, and we demonstrate that dynamically adjusting the data mixture based on Group Influence, a direct measure of these preferences, significantly improves performance by mitigating the underdigestion of data seen with static ratios.

In structure-based drug discovery, virtual screening using conventional molecular docking methods can be performed rapidly but suffers from limitations in prediction accuracy. Recently, Boltz-2 was proposed, achieving extremely high accuracy in binding affinity prediction, but requiring approximately 20 seconds per compound per GPU, making it difficult to apply to large-scale screening of hundreds of thousands to millions of compounds. This study proposes Boltzina, a novel framework that leverages Boltz-2's high accuracy while significantly improving computational efficiency. Boltzina achieves both accuracy and speed by omitting the rate-limiting structure prediction from Boltz-2's architecture and directly predicting affinity from AutoDock Vina docking poses. We evaluate on eight assays from the MF-PCBA dataset and show that while Boltzina performs below Boltz-2, it provides significantly higher screening performance compared to AutoDock Vina and GNINA. Additionally, Boltzina achieved up to 11.8$\times$ faster through reduced recycling iterations and batch processing. Furthermore, we investigated multi-pose selection strategies and two-stage screening combining Boltzina and Boltz-2, presenting optimization methods for accuracy and efficiency according to application requirements. This study represents the first attempt to apply Boltz-2's high-accuracy predictions to practical-scale screening, offering a pipeline that combines both accuracy and efficiency in computational biology. The Boltzina is available on github; https://github.com/ohuelab/boltzina.

Brain-body co-optimization remains a challenging problem, despite increasing interest from the community in recent years. To understand and overcome the challenges, we propose exhaustively mapping a morphology-fitness landscape to study it. To this end, we train controllers for each feasible morphology in a design space of 1,305,840 distinct morphologies, constrained by a computational budget. First, we show that this design space constitutes a good model for studying the brain-body co-optimization problem, and our attempt to exhaustively map it roughly captures the landscape. We then proceed to analyze how evolutionary brain-body co-optimization algorithms work in this design space. The complete knowledge of the morphology-fitness landscape facilitates a better understanding of the results of evolutionary brain-body co-optimization algorithms and how they unfold over evolutionary time in the morphology space. This investigation shows that the experimented algorithms cannot consistently find near-optimal solutions. The search, at times, gets stuck on morphologies that are sometimes one mutation away from better morphologies, and the algorithms cannot efficiently track the fitness gradient in the morphology-fitness landscape. We provide evidence that experimented algorithms regularly undervalue the fitness of individuals with newly mutated bodies and, as a result, eliminate promising morphologies throughout evolution. Our work provides the most concrete demonstration of the challenges of evolutionary brain-body co-optimization. Our findings ground the trends in the literature and provide valuable insights for future work.