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Optimization · Non-Convex

Kwangjun Ahn, Ali Jadbabaie, Suvrit Sra

Modern machine learning applications have witnessed the remarkable success of optimization algorithms that are designed to find flat minima. Motivated by this design choice, we undertake a formal study that (i) formulates the notion of flat minima, and (ii) studies the complexity of finding them. Specifically, we adopt the trace of the Hessian of the cost function as a measure of flatness, and use it to formally define the notion of approximate flat minima. Under this notion, we then analyze algorithms that find approximate flat minima efficiently. For general cost functions, we discuss a gradient-based algorithm that finds an approximate flat local minimum efficiently. The main component of the algorithm is to use gradients computed from randomly perturbed iterates to estimate a direction that leads to flatter minima. For the setting where the cost function is an empirical risk over training data, we present a faster algorithm that is inspired by a recently proposed practical algorithm called sharpness-aware minimization, supporting its success in practice.

Deep Learning · Large Language Models

Gustaf Ahdritz, Tian Qin, Nikhil Vyas, Boaz Barak, Benjamin Edelman

We study the feasibility of identifying *epistemic* uncertainty (reflecting a lack of knowledge), as opposed to *aleatoric* uncertainty (reflecting entropy in the underlying distribution), in the outputs of large language models (LLMs) over free-form text. In the absence of ground-truth probabilities, we explore a setting where, in order to (approximately) disentangle a given LLM's uncertainty, a significantly larger model stands in as a proxy for the ground truth. We show that small linear probes trained on the embeddings of frozen, pretrained models accurately predict when larger models will be more confident at the token level and that probes trained on one text domain generalize to others. Going further, we propose a fully unsupervised method that achieves non-trivial accuracy on the same task. Taken together, we interpret these results as evidence that LLMs naturally contain internal representations of different types of uncertainty that could potentially be leveraged to devise more informative indicators of model confidence in diverse practical settings. Code can be found at: https://github.com/KempnerInstitute/llm_uncertainty

Deep Learning · Attention Mechanisms

Victor Agostinelli III, Sanghyun Hong, Lizhong Chen

A promising approach to preserving model performance in linearized transformers is to employ position-based re-weighting functions. However, state-of-the-art re-weighting functions rely heavily on target sequence lengths, making it difficult or impossible to apply them to autoregressive and simultaneous tasks, where the target and sometimes even the input sequence length are unknown. To address this issue, we propose Learned Proportions (LeaP) and LeaPformers. Our contribution is built on two major components. First, we generalize the dependence on explicit positional representations and sequence lengths into dependence on sequence proportions for re-weighting. Second, we replace static positional representations with dynamic proportions derived via a compact module, enabling more flexible attention concentration patterns. We evaluate LeaPformer against eight representative efficient transformers on the Long-Range Arena benchmark, where we show that LeaPformer achieves the best quality-throughput trade-off, as well as apply LeaPformer to Wikitext-103b autoregressive language modeling and simultaneous speech-to-text translation for two language pairs, achieving competitive results in both tasks.

Theory · Learning Theory

Alekh Agarwal, Jian Qian, Alexander Rakhlin, Tong Zhang

We propose a generalization of the classical G-optimal design concept to non-linear function classes. The criterion, termed F -design, coincides with G-design in the linear case. We compute the value of the optimal design, termed the F-condition number, for several non-linear function classes. We further provide algorithms to construct designs with a bounded F -condition number. Finally, we employ the F-design in a variety of interactive machine learning tasks, where the design is naturally useful for data collection or exploration. We show that in four diverse settings of confidence band construction, contextual bandits, model-free reinforcement learning, and active learning, F-design can be combined with existing approaches in a black-box manner to yield state-of-the-art results in known problem settings as well as to generalize to novel ones.

Theory · Probabilistic Methods

Sanyam Agarwal, Markus Bläser

Zhang et al. (ICML 2021, PLMR 139, pp. 12447–12457) introduced probabilistic generating circuits (PGCs) as a probabilistic model to unify probabilistic circuits (PCs) and determinantal point processes (DPPs). At a first glance, PGCs store a distribution in a very different way, they compute the probability generating polynomial instead of the probability mass function and it seems that this is the main reason why PGCs are more powerful than PCs or DPPs. However, PGCs also allow for negative weights, whereas classical PCs assume that all weights are nonnegative. One main insight of this work is that the negative weights are the cause for the power of PGCs and not the different representation. PGCs are PCs in disguise: we show how to transform any PGC on binary variables into a PC with negative weights with only polynomial blowup. PGCs were defined by Zhang et al. only for binary random variables. As our second main result, we show that there is a good reason for this: we prove that PGCs for categorical variables with larger image size do not support tractable marginalization unless NP=P. On the other hand, we show that we can model categorical variables with larger image size as PC with negative weights computing set-multilinear polynomials. These allow for tractable marginalization. In this sense, PCs with negative weights strictly subsume PGCs.

Reinforcement Learning

Pranav Agarwal, Sheldon Andrews, Samira Ebrahimi Kahou

Model-based reinforcement learning agents utilizing transformers have shown improved sample efficiency due to their ability to model extended context, resulting in more accurate world models. However, for complex reasoning and planning tasks, these methods primarily rely on continuous representations. This complicates modeling of discrete properties of the real world such as disjoint object classes between which interpolation is not plausible. In this work, we introduce discrete abstract representations for transformer-based learning (DART), a sample-efficient method utilizing discrete representations for modeling both the world and learning behavior. We incorporate a transformer-decoder for auto-regressive world modeling and a transformer-encoder for learning behavior by attending to task-relevant cues in the discrete representation of the world model. For handling partial observability, we aggregate information from past time steps as memory tokens. DART outperforms previous state-of-the-art methods that do not use look-ahead search on the Atari 100k sample efficiency benchmark with a median human-normalized score of 0.790 and beats humans in 9 out of 26 games. We release our code at https://pranaval.github.io/DART/.

Theory · Learning Theory

Peyman Afshani, Chris Schwiegelshohn

We investigate coresets for approximating the cost with respect to median queries. In this problem, we are given a set of points $P\subset \mathbb{R}^d$ and median queries are $\sum_{p\in P} ||p-c||$ for any point $c\in \mathbb{R}^d$. Our goal is to compute a small weighted summary $S\subset P$ such that the cost of any median query is approximated within a multiplicative $(1\pm\varepsilon)$ factor. We provide matching upper and lower bounds on the number of points contained in $S$ of the order $\tilde{\Theta}\left(\varepsilon^{-d/(d+1)}\right)$.

Reinforcement Learning · Deep RL

Sandesh Adhikary, Anqi Li, Byron Boots

Training reinforcement learning (RL) agents directly from high-dimensional image observations continues to be a challenging problem. Recent line of work on behavioral distances proposes to learn representations that encode behavioral similarities quantified by the bisimulation metric. By learning an isometric mapping to a lower dimensional space that preserves this metric, such methods attempt to learn representations that group together functionally similar states. However, such an isometric mapping may not exist, making the learning objective ill-defined. We propose an alternative objective that allows distortions in long-range distances, while preserving *local* metric structure -- inducing representations that highlight natural clusters in the state space. This leads to new representations, which we term Behavioral Eigenmaps (BeigeMaps), corresponding to the eigenfunctions of similarity kernels induced by behavioral distances. We empirically demonstrate that when added as a drop-in modification, BeigeMaps improve the policy performance of prior behavioral distance based RL algorithms.

Deep Learning · Large Language Models

Harshavardhan Adepu, Zhanpeng Zeng, Li Zhang, Vikas Singh

Transformers are the backbone of powerful foundation models for many Vision and Natural Language Processing tasks. But their compute and memory/storage footprint is large, and so, serving such models is expensive often requiring high-end hardware. To mitigate this difficulty, Post-Training Quantization seeks to modify a pre-trained model and quantize it to eight bits or lower, significantly boosting compute/memory/latency efficiency. Such models have been successfully quantized to four bits with some performance loss. In this work, we outline a simple scheme to quantize Transformer-based models to just two bits (plus some overhead) with only a small drop in accuracy. Key to our formulation is a concept borrowed from Harmonic analysis called Fusion Frames. Our main finding is that the quantization must take place not in the original weight space, but instead in the Fusion Frame representations. If quantization is interpreted as the addition of noise, our casting of the problem allows invoking an extensive body of known consistent recovery and noise robustness guarantees. Further, if desired, de-noising filters are known in closed form. We show empirically, via a variety of experiments, that (almost) two-bit quantization for Transformer models promises sizable efficiency gains. The code is available at https://github.com/vsingh-group/FrameQuant

Social Aspects · Trustworthy Machine Learning

Reduan Achtibat, Sayed Mohammad Vakilzadeh Hatefi, Maximilian Dreyer, Aakriti Jain, Thomas Wiegand, Sebastian Lapuschkin, Wojciech Samek

Large Language Models are prone to biased predictions and hallucinations, underlining the paramount importance of understanding their model-internal reasoning process. However, achieving faithful attributions for the entirety of a black-box transformer model and maintaining computational efficiency is an unsolved challenge. By extending the Layer-wise Relevance Propagation attribution method to handle attention layers, we address these challenges effectively. While partial solutions exist, our method is the first to faithfully and holistically attribute not only input but also latent representations of transformer models with the computational efficiency similar to a single backward pass. Through extensive evaluations against existing methods on LLaMa 2, Mixtral 8x7b, Flan-T5 and vision transformer architectures, we demonstrate that our proposed approach surpasses alternative methods in terms of faithfulness and enables the understanding of latent representations, opening up the door for concept-based explanations. We provide an LRP library at https://github.com/rachtibat/LRP-eXplains-Transformers.

Deep Learning · Everything Else

Idan Achituve, Idit Diamant, Arnon Netzer, Gal Chechik, Ethan Fetaya

As machine learning becomes more prominent there is a growing demand to perform several inference tasks in parallel. Multi-task learning (MTL) addresses this challenge by learning a single model that solves several tasks simultaneously and efficiently. Often optimizing MTL models entails first computing the gradient of the loss for each task, and then aggregating all the gradients to obtain a combined update direction. However, common methods following this approach do not consider an important aspect, the sensitivity in the dimensions of the gradients. Some dimensions may be more lenient for changes while others may be more restrictive. Here, we introduce a novel gradient aggregation procedure using Bayesian inference. We place a probability distribution over the task-specific parameters, which in turn induce a *distribution* over the gradients of the tasks. This valuable information allows us to quantify the uncertainty associated with each of the gradients' dimensions which is factored in when aggregating them. We empirically demonstrate the benefits of our approach in a variety of datasets, achieving state-of-the-art performance.

Theory · Game Theory

Kenshi Abe, Kaito Ariu, Mitsuki Sakamoto, Atsushi Iwasaki

This paper proposes a payoff perturbation technique for the Mirror Descent (MD) algorithm in games where the gradient of the payoff functions is monotone in the strategy profile space, potentially containing additive noise. The optimistic family of learning algorithms, exemplified by optimistic MD, successfully achieves *last-iterate* convergence in scenarios devoid of noise, leading the dynamics to a Nash equilibrium. A recent re-emerging trend underscores the promise of the perturbation approach, where payoff functions are perturbed based on the distance from an anchoring, or *slingshot*, strategy. In response, we propose *Adaptively Perturbed MD* (APMD), which adjusts the magnitude of the perturbation by repeatedly updating the slingshot strategy at a predefined interval. This innovation empowers us to find a Nash equilibrium of the underlying game with guaranteed rates. Empirical demonstrations affirm that our algorithm exhibits significantly accelerated convergence.

Theory · Learning Theory

Werner Zellinger, Stefan Kindermann, Sergei V. Pereverzyev

Estimating the ratio of two probability densities from finitely many observations of the densities is a central problem in machine learning and statistics with applications in two-sample testing, divergence estimation, generative modeling, covariate shift adaptation, conditional density estimation, and novelty detection. In this work, we analyze a large class of density ratio estimation methods that minimize a regularized Bregman divergence between the true density ratio and a model in a reproducing kernel Hilbert space (RKHS). We derive new finite-sample error bounds, and we propose a Lepskii type parameter choice principle that minimizes the bounds without knowledge of the regularity of the density ratio. In the special case of square loss, our method adaptively achieves a minimax optimal error rate. A numerical illustration is provided.

Theory · Deep Learning

Blake Bordelon, Alexander Atanasov, Cengiz Pehlevan

On a variety of tasks, the performance of neural networks predictably improves with training time, dataset size and model size across many orders of magnitude. This phenomenon is known as a neural scaling law. Of fundamental importance is the compute-optimal scaling law, which reports the performance as a function of units of compute when choosing model sizes optimally. We analyze a random feature model trained with gradient descent as a solvable model of network training and generalization. This reproduces many observations about neural scaling laws. First, our model makes a prediction about why the scaling of performance with training time and with model size have different power law exponents. Consequently, the theory predicts an asymmetric compute-optimal scaling rule where the number of training steps are increased faster than model parameters, consistent with recent empirical observations. Second, it has been observed that early in training, networks converge to their infinite-width dynamics at a rate $1/\text{width}$ but at late time exhibit a rate $\text{width}^{-c}$, where $c$ depends on the structure of the architecture and task. We show that our model exhibits this behavior. Lastly, our theory shows how the gap between training and test loss can gradually build up over time due to repeated reuse of data.

Masahiro Kato, Oga Akihiro, Wataru Komatsubara, Ryo Inokuchi

This study designs an adaptive experiment for efficiently estimating *average treatment effects* (ATEs). In each round of our adaptive experiment, an experimenter sequentially samples an experimental unit, assigns a treatment, and observes the corresponding outcome immediately. At the end of the experiment, the experimenter estimates an ATE using the gathered samples. The objective is to estimate the ATE with a smaller asymptotic variance. Existing studies have designed experiments that adaptively optimize the propensity score (treatment-assignment probability). As a generalization of such an approach, we propose optimizing the covariate density as well as the propensity score. First, we derive the efficient covariate density and propensity score that minimize the semiparametric efficiency bound and find that optimizing both covariate density and propensity score minimizes the semiparametric efficiency bound more effectively than optimizing only the propensity score. Next, we design an adaptive experiment using the efficient covariate density and propensity score sequentially estimated during the experiment. Lastly, we propose an ATE estimator whose asymptotic variance aligns with the minimized semiparametric efficiency bound.

Dora Zhao, Jerone Andrews, Orestis Papakyriakopoulos, Alice Xiang

Machine learning (ML) datasets, often perceived as neutral, inherently encapsulate abstract and disputed social constructs. Dataset curators frequently employ value-laden terms such as diversity, bias, and quality to characterize datasets. Despite their prevalence, these terms lack clear definitions and validation. Our research explores the implications of this issue by analyzing "diversity" across 135 image and text datasets. Drawing from social sciences, we apply principles from measurement theory to identify considerations and offer recommendations for conceptualizing, operationalizing, and evaluating diversity in datasets. Our findings have broader implications for ML research, advocating for a more nuanced and precise approach to handling value-laden properties in dataset construction.

Social Aspects · Fairness, Equity, Justice and Safety

Shizhou Xu, Thomas Strohmer

As machine learning powered decision-making becomes increasingly important in our daily lives, it is imperative to strive for fairness in the underlying data processing. We propose a pre-processing algorithm for fair data representation via which supervised learning results in estimations of the Pareto frontier between prediction error and statistical disparity. In particular, the present work applies the optimal affine transport to approach the post-processing Wasserstein barycenter characterization of the optimal fair $L^2$-objective supervised learning via a pre-processing data deformation. Furthermore, we show that the Wasserstein geodesics from the conditional (on sensitive information) distributions of the learning outcome to their barycenter characterize the Pareto frontier between $L^2$-loss and the average pairwise Wasserstein distance among sensitive groups on the learning outcome. Numerical simulations underscore the advantages: (1) the pre-processing step is compositive with arbitrary conditional expectation estimation supervised learning methods and unseen data; (2) the fair representation protects the sensitive information by limiting the inference capability of the remaining data with respect to the sensitive data; (3) the optimal affine maps are computationally efficient even for high-dimensional data.

Shuaicheng Niu, Chunyan Miao, Guohao Chen, Pengcheng Wu, Peilin Zhao

Test-time adaptation has proven effective in adapting a given trained model to unseen test samples with potential distribution shifts. However, in real-world scenarios, models are usually deployed on resource-limited devices, e.g., FPGAs, and are often quantized and hard-coded with non-modifiable parameters for acceleration. In light of this, existing methods are often infeasible since they heavily depend on computation-intensive backpropagation for model updating that may be not supported. To address this, we propose a test-time Forward-Optimization Adaptation (FOA) method. In FOA, we seek to solely learn a newly added prompt (as model's input) via a derivative-free covariance matrix adaptation evolution strategy. To make this strategy work stably under our online unsupervised setting, we devise a novel fitness function by measuring test-training statistic discrepancy and model prediction entropy. Moreover, we design an activation shifting scheme that directly tunes the model activations for shifted test samples, making them align with the source training domain, thereby further enhancing adaptation performance. Without using any backpropagation and altering model weights, FOA runs on quantized 8-bit ViT outperforms gradient-based TENT on full-precision 32-bit ViT, while achieving an up to *24*-fold memory reduction on ImageNet-C. The source code is available at: https://github.com/mr-eggplant/FOA.

Probabilistic Methods · Bayesian Models and Methods

Eliezer de Souza da Silva, Tomasz Kuśmierczyk, Marcelo Hartmann, Arto Klami

The behavior of many Bayesian models used in machine learning critically depends on the choice of prior distributions, controlled by some hyperparameters typically selected through Bayesian optimization or cross-validation. This requires repeated, costly, posterior inference. We provide an alternative for selecting good priors without carrying out posterior inference, building on the prior predictive distribution that marginalizes the model parameters. We estimate virtual statistics for data generated by the prior predictive distribution and then optimize over the hyperparameters to learn those for which the virtual statistics match the target values provided by the user or estimated from (a subset of) the observed data. We apply the principle for probabilistic matrix factorization, for which good solutions for prior selection have been missing. We show that for Poisson factorization models we can analytically determine the hyperparameters, including the number of factors, that best replicate the target statistics, and we empirically study the sensitivity of the approach for the model mismatch. We also present a model-independent procedure that determines the hyperparameters for general models by stochastic optimization and demonstrate this extension in the context of hierarchical matrix factorization models.

Yujia Huang, Adishree Ghatare, Yuanzhe Liu, ziniu hu, Qinsheng Zhang, Chandramouli Shama Sastry, Siddharth Gururani, Sageev Oore, Yisong Yue

We study the problem of symbolic music generation (e.g., generating piano rolls), with a technical focus on non-differentiable rule guidance. Musical rules are often expressed in symbolic form on note characteristics, such as note density or chord progression, many of which are non-differentiable which pose a challenge when using them for guided diffusion. We propose Stochastic Control Guidance (SCG), a novel guidance method that only requires forward evaluation of rule functions that can work with pre-trained diffusion models in a plug-and-play way, thus achieving training-free guidance for non-differentiable rules for the first time. Additionally, we introduce a latent diffusion architecture for symbolic music generation with high time resolution, which can be composed with SCG in a plug-and-play fashion. Compared to standard strong baselines in symbolic music generation, this framework demonstrates marked advancements in music quality and rule-based controllability, outperforming current state-of-the-art generators in a variety of settings. For detailed demonstrations, code and model checkpoints, please visit our [project website](https://scg-rule-guided-music.github.io/).