Energy-based models (EBMs) define a distribution through a scalar energy, $p_θ(x)\propto\exp(-E_θ(x))$, and are usually presented as one model family among many. This tutorial has a broader goal: beyond introducing EBMs, it uses them as a single lens that connects the different aspects of machine learning. Any EBM is specified by three choices, the energy, which determines what the model represents; the inference procedure, which samples from or optimizes the energy; and the learning method, which fits the energy to data. We first develop these components, including sampling, variational inference, and learning by maximum likelihood, noise contrastive estimation, pseudolikelihood, score matching, and adversarial training, and show that the learning methods share one contrastive structure. We then show that discriminative models, generative models such as GANs, normalizing flows, diffusion and autoregressive models, representation learning methods such as contrastive learning, word2vec, BERT, denoising auto-encoders, and diffusion representations, reinforcement learning with EBM policies and world models, inverse reinforcement learning, and planning, as well as inference-time computation with verifiers, are all EBMs that differ in these three choices, mostly in how they handle the partition function and sampling. This view explains existing methods, organizes them into a common framework, and suggests new algorithms by replacing one component while keeping the others, with composition of energies as a principled way to combine learned models at inference time.
Controllable diffusion generation often relies on various heuristics that are seemingly disconnected without a unified understanding. We bridge this gap with Diffusion Controller (DiffCon), a unified control-theoretic view that casts reverse diffusion sampling as state-only stochastic control within (generalized) linearly-solvable Markov Decision Processes (LS-MDPs). Under this framework, control acts by reweighting the pretrained reverse-time transition kernels, balancing terminal objectives against an $f$-divergence cost. From the resulting optimality conditions, we derive practical reinforcement learning methods for diffusion fine-tuning: (i) f-divergence-regularized policy-gradient updates, including a PPO-style rule, and (ii) a regularizer-determined reward-weighted regression objective with a minimizer-preservation guarantee under the Kullback-Leibler (KL) divergence. The LS-MDP framework further implies a principled model form: the optimal score decomposes into a fixed pretrained baseline plus a lightweight control correction, motivating a side-network parameterization conditioned on exposed intermediate denoising outputs, enabling effective gray-box adaptation with a frozen backbone. Experiments on Stable Diffusion v1.4 across supervised and reward-driven finetuning show consistent gains in preference-alignment win rates and improved quality-efficiency trade-offs versus gray-box baselines and even the parameter-efficient white-box adapter LoRA.