https://probabilize.dev/ Project-based courses from Linear Algebra and Optimization through RL, Robot Learning, Physical AI, Generative Models, Probability & Statistics, and Advanced Optimization — short concepts paired with hands-on builds. Sat, 19 Sep 2026 18:13:47 GMT https://probabilize.dev/courses/linear-algebra/lessons/start-here/ https://probabilize.dev/courses/linear-algebra/lessons/start-here/ Your first step in the Guided Foundations Path: run a browser Python cell, read its output and a traceback, use variables, lists, loops, if/else and functions, build numpy arrays, and read a sum, an average, an area under a curve, and a slope. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/linear-algebra/lessons/la-w1/ https://probabilize.dev/courses/linear-algebra/lessons/la-w1/ Represent vectors in R^n, combine them linearly, and use the dot product to measure length, angle, and orthogonality. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/linear-algebra/lessons/la-w2/ https://probabilize.dev/courses/linear-algebra/lessons/la-w2/ See matrices as linear transformations, matrix-vector multiplication as applying one, and matrix multiplication as composing two. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/linear-algebra/lessons/la-w3/ https://probabilize.dev/courses/linear-algebra/lessons/la-w3/ Solve linear systems by Gaussian elimination, see elimination as matrix multiplication, and factor A = LU. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/linear-algebra/lessons/la-w4/ https://probabilize.dev/courses/linear-algebra/lessons/la-w4/ Learn the defining properties of the determinant, compute it by cofactor expansion, and use it to test invertibility and measure volume. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/linear-algebra/lessons/la-w5/ https://probabilize.dev/courses/linear-algebra/lessons/la-w5/ Test vector sets for linear independence, and learn how a basis pins down the dimension of a vector space. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/linear-algebra/lessons/la-w6/ https://probabilize.dev/courses/linear-algebra/lessons/la-w6/ Meet the column space, null space, row space, and left null space of a matrix, and the rank-nullity formula that ties their dimensions together. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/linear-algebra/lessons/la-w7/ https://probabilize.dev/courses/linear-algebra/lessons/la-w7/ Project a vector onto a line or a subspace, and build the projection matrix that does it automatically. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/linear-algebra/lessons/la-w8/ https://probabilize.dev/courses/linear-algebra/lessons/la-w8/ Fit lines to noisy data with least squares, build orthonormal bases with Gram-Schmidt, and factor A = QR. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/linear-algebra/lessons/la-w9/ https://probabilize.dev/courses/linear-algebra/lessons/la-w9/ Find the directions a matrix only stretches, not rotates — eigenvalues and eigenvectors via the characteristic equation. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/linear-algebra/lessons/la-w10/ https://probabilize.dev/courses/linear-algebra/lessons/la-w10/ Diagonalize a matrix to compute powers efficiently, and use eigenvalues to find the steady state of a Markov chain. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/linear-algebra/lessons/la-w11/ https://probabilize.dev/courses/linear-algebra/lessons/la-w11/ Solve systems of linear differential equations using eigenvalues and eigenvectors, and read off stability from their signs. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/linear-algebra/lessons/la-w12/ https://probabilize.dev/courses/linear-algebra/lessons/la-w12/ Meet the spectral theorem for symmetric matrices, and learn three equivalent tests for positive definiteness. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/linear-algebra/lessons/la-w13/ https://probabilize.dev/courses/linear-algebra/lessons/la-w13/ Generalize eigendecomposition to any matrix with the singular value decomposition, and meet the complex analogues of symmetric and orthogonal matrices. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/linear-algebra/lessons/la-w14/ https://probabilize.dev/courses/linear-algebra/lessons/la-w14/ Meet the Fourier matrix and the FFT that makes it fast, then close the course with PCA — variance-maximizing directions found via the SVD. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/optimization/lessons/opt-w1/ https://probabilize.dev/courses/optimization/lessons/opt-w1/ Formulate empirical risk minimization, define loss surfaces, and implement gradient descent on a simple quadratic using multivariable calculus. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/optimization/lessons/opt-w2/ https://probabilize.dev/courses/optimization/lessons/opt-w2/ Recognize convex sets and functions, and understand why a local minimum of a convex function is automatically global. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/optimization/lessons/opt-w3/ https://probabilize.dev/courses/optimization/lessons/opt-w3/ Understand why GD converges: L-smoothness, convexity, and step-size rules that guarantee monotonic progress. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/optimization/lessons/opt-w4/ https://probabilize.dev/courses/optimization/lessons/opt-w4/ Solve least squares iteratively with GD and see how the condition number of the design matrix controls convergence speed. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/optimization/lessons/opt-w5/ https://probabilize.dev/courses/optimization/lessons/opt-w5/ Differentiate logistic, softmax, and cross-entropy losses, and see why convexity extends from regression to classification. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/optimization/lessons/opt-w6/ https://probabilize.dev/courses/optimization/lessons/opt-w6/ Replace full-batch gradients with mini-batch estimates, trade noise for speed, and tune learning rates under stochasticity. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/optimization/lessons/opt-w7/ https://probabilize.dev/courses/optimization/lessons/opt-w7/ Add velocity to gradient steps with heavy-ball momentum and see why Nesterov acceleration improves convergence on convex problems. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/optimization/lessons/opt-w8/ https://probabilize.dev/courses/optimization/lessons/opt-w8/ Let each parameter learn at its own rate with AdaGrad, RMSProp, and Adam, the methods that train deep neural networks. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/optimization/lessons/opt-w9/ https://probabilize.dev/courses/optimization/lessons/opt-w9/ Diagnose training runs with learning rate schedules, warmup, gradient clipping, and the art of reading loss curves. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/optimization/lessons/opt-w10/ https://probabilize.dev/courses/optimization/lessons/opt-w10/ Handle ReLU, L1, and other nondifferentiable losses with subgradients, and see how early stopping acts as implicit regularization. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/optimization/lessons/opt-w11/ https://probabilize.dev/courses/optimization/lessons/opt-w11/ Enforce constraints with projected gradient descent and soft penalties, and glimpse Lagrange multipliers as constraint pricing. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/optimization/lessons/opt-w12/ https://probabilize.dev/courses/optimization/lessons/opt-w12/ Build the Lagrangian dual, state weak and strong duality, and compute the soft-margin SVM dual as a concrete ML example. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/optimization/lessons/opt-w13/ https://probabilize.dev/courses/optimization/lessons/opt-w13/ Cross from convex land into the reality of deep net training: saddles, overparameterization, initialization, and normalization. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/optimization/lessons/opt-w14/ https://probabilize.dev/courses/optimization/lessons/opt-w14/ Diagnose a full training pipeline, choose methods and learning rates from first principles, and map course tools onto RL and generative model training. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/reinforcement-learning/lessons/rl-agents-rewards-and-mdps/ https://probabilize.dev/courses/reinforcement-learning/lessons/rl-agents-rewards-and-mdps/ Meet the reinforcement-learning loop in plain language — an agent acts, the world answers with a reward and a next state — and learn the five ingredients that make the problem solvable. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/reinforcement-learning/lessons/rl-exploration-and-regret/ https://probabilize.dev/courses/reinforcement-learning/lessons/rl-exploration-and-regret/ The simplest reinforcement-learning problem: a row of slot machines, no states, and one unavoidable trade-off. Learn why always choosing the current best is not enough — and how to measure the reward that exploring costs you. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/reinforcement-learning/lessons/rl-returns-and-value/ https://probabilize.dev/courses/reinforcement-learning/lessons/rl-returns-and-value/ Turn a stream of rewards into a single number: discounting, the return, value functions, and the one recursive equation that makes infinite-horizon problems solvable. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/reinforcement-learning/lessons/rl-ucb-and-thompson/ https://probabilize.dev/courses/reinforcement-learning/lessons/rl-ucb-and-thompson/ Two principled answers to the exploration problem: trust the optimistic edge of what you know (UCB1), or sample from what you believe (Thompson sampling). Both explore automatically, with no rate to tune. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w1/ https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w1/ Formalize sequential decision making as Markov Decision Processes, derive value functions and the Bellman equations, and understand the reward hypothesis and its limits. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w2/ https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w2/ Formalize the exploration-exploitation trade-off as a multi-armed bandit problem, derive the Lai-Robbins lower bound, and implement UCB1 and Thompson Sampling from first principles. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w3/ https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w3/ Master policy evaluation, policy iteration, and value iteration for finite MDPs. Understand the contraction mapping property that guarantees DP convergence, and trace the precise mappings from DP to Q-learning, actor-critic, and RLHF. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w4/ https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w4/ Master Monte Carlo and Temporal-Difference methods for learning value functions from experience. Understand the bias-variance tradeoff that defines every modern deep RL algorithm, from SARSA and Q-learning to PPO and DQN. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w5/ https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w5/ How function approximation breaks tabular RL convergence — the MSVE objective, the deadly triad mechanism, Baird's counterexample, overestimation bias, and the neural network instabilities that DQN's target networks fix. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w6/ https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w6/ How DQN stabilizes deep RL with experience replay, target networks, and reward clipping, plus Double DQN, Dueling, Prioritized Replay, Distributional RL, and Rainbow. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w7/ https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w7/ Derive the policy gradient theorem from first principles, build REINFORCE to actor-critic, develop GAE, and connect PPO to RLHF — the algorithmic backbone of modern language model alignment. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w8/ https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w8/ Explore the off-policy actor-critic family — DDPG, TD3, and SAC — from deterministic policy gradients to maximum-entropy RL, with practical guidance on choosing algorithms for real-robot deployment. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w9/ https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w9/ Exploration strategies (count-based, ICM, RND), partial observability via POMDPs and recurrent policies, and multi-agent RL with CTDE — connecting all three to modern agentic AI systems. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w10/ https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w10/ From Dyna to Dreamer to MuZero — the model-based RL design space: learned world models, receding-horizon MPC, latent-space planning, and how these connect to tree-of-thought reasoning in modern LLMs. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w11/ https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w11/ Why standard RL fails on fixed datasets — and how conservative Q-learning (CQL), implicit Q-learning (IQL), behavior regularization, and doubly robust off-policy evaluation enable learning from offline data without environment interaction. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w12/ https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w12/ RLHF pipeline: supervised fine-tuning, reward modeling via Bradley-Terry preferences, and KL-regularized PPO for aligning language models with human values. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w13/ https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w13/ DPO eliminates the reward model by inverting the KL-regularized optimal policy; GRPO replaces the value critic with a group-normalized empirical baseline for verifiable reasoning tasks. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w14/ https://probabilize.dev/courses/reinforcement-learning/lessons/rl-w14/ Capstone: map agentic AI architectures (ReAct, Reflexion, plan-and-execute) onto the MDP formalism, covering memory, safety constraints, process rewards, and open problems. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/robot-learning/lessons/rb-b1/ https://probabilize.dev/courses/robot-learning/lessons/rb-b1/ The sense → decide → act loop, the parts of a robot arm, and how two joint angles decide where the hand lands — a beginner robotics lesson with a runnable 2-DOF arm lab. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/robot-learning/lessons/rb-w1/ https://probabilize.dev/courses/robot-learning/lessons/rb-w1/ Configuration spaces, forward and inverse kinematics, and the Jacobian — the geometric foundations that map joint angles to end-effector pose in SE(3). Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/robot-learning/lessons/rb-b2/ https://probabilize.dev/courses/robot-learning/lessons/rb-b2/ Why a sensor reading is not the truth: prediction, measurement and the weighted estimate that combines them, explained in plain language with a runnable 1-D position-tracking lab. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/robot-learning/lessons/rb-w2/ https://probabilize.dev/courses/robot-learning/lessons/rb-w2/ From geometry to forces: rigid-body dynamics and state estimation, including the Kalman and extended Kalman filter for tracking robot state under noise. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/robot-learning/lessons/rb-w3/ https://probabilize.dev/courses/robot-learning/lessons/rb-w3/ Feedback control for robots — PID, computed-torque, and the trajectory-tracking foundations that turn desired motions into stable actuator commands. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/robot-learning/lessons/rb-w4/ https://probabilize.dev/courses/robot-learning/lessons/rb-w4/ How expert demonstration data is gathered — teleoperation interfaces, synchronized state-action logging, and the data-quality choices that determine what a policy can learn. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/robot-learning/lessons/rb-w5/ https://probabilize.dev/courses/robot-learning/lessons/rb-w5/ Learning controllers from demonstration — behavior cloning, its compounding-error failure mode, and the foundations of imitation learning from state-action trajectories. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/robot-learning/lessons/rb-w6/ https://probabilize.dev/courses/robot-learning/lessons/rb-w6/ Connecting RL theory to physical robots — why imitation alone is insufficient, and how policy-gradient methods let robots improve beyond demonstrated behavior. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/robot-learning/lessons/rb-w7/ https://probabilize.dev/courses/robot-learning/lessons/rb-w7/ Bridging the simulation-to-reality gap with domain randomization and the IsaacLab pipeline for training transferable robot policies at scale. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/robot-learning/lessons/rb-w8/ https://probabilize.dev/courses/robot-learning/lessons/rb-w8/ Why reactive per-timestep policies fail on long-horizon manipulation, and how Action Chunking Transformers (ACT) predict coherent sequences of future actions. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/robot-learning/lessons/rb-w9/ https://probabilize.dev/courses/robot-learning/lessons/rb-w9/ Generative robot policies — diffusion and flow-matching models that sample multimodal action trajectories for dexterous manipulation. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/robot-learning/lessons/rb-w10/ https://probabilize.dev/courses/robot-learning/lessons/rb-w10/ Vision-Language-Action models that fuse perception, language, and action into a single policy — architecture, training, and how language grounds robot behavior. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/robot-learning/lessons/rb-w11/ https://probabilize.dev/courses/robot-learning/lessons/rb-w11/ Adapting pretrained VLA models to new robots and tasks — fine-tuning strategies that preserve broad manipulation priors while specializing behavior. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/robot-learning/lessons/rb-w12/ https://probabilize.dev/courses/robot-learning/lessons/rb-w12/ Formal safety for learned controllers — control barrier functions, constraint enforcement, and reliability guarantees for robots acting near people. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/robot-learning/lessons/rb-w13/ https://probabilize.dev/courses/robot-learning/lessons/rb-w13/ Scaling beyond a single robot and task — multi-task policy learning and coordination strategies across multiple robots. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/robot-learning/lessons/rb-w14/ https://probabilize.dev/courses/robot-learning/lessons/rb-w14/ A capstone synthesizing modeling, control, imitation, RL, and sim2real into an end-to-end pipeline for training and deploying a real robot policy. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/physical-ai/lessons/pa-w1/ https://probabilize.dev/courses/physical-ai/lessons/pa-w1/ Trace the evolution from convolutional networks to Vision Transformers, formalize self-attention over image patches, and survey native 3D representations (NeRF, Gaussian Splatting) that ground vision models in physical space. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/physical-ai/lessons/pa-w2/ https://probabilize.dev/courses/physical-ai/lessons/pa-w2/ How VLM vision encoders are pretrained without labels — contrastive learning, masked autoencoding (MAE), and self-distillation (DINO), and why each objective yields representations with different geometric properties. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/physical-ai/lessons/pa-w3/ https://probabilize.dev/courses/physical-ai/lessons/pa-w3/ Derive the CLIP architecture and its symmetric contrastive objective, see how zero-shot classification emerges from vector similarity, and understand the limits that motivated later vision-language models. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/physical-ai/lessons/pa-w4/ https://probabilize.dev/courses/physical-ai/lessons/pa-w4/ Move from discriminative to generative perception: encoder-decoder captioning and cross-attention grounding that let models describe what they see and localize where objects are. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/physical-ai/lessons/pa-w5/ https://probabilize.dev/courses/physical-ai/lessons/pa-w5/ Unify contrastive alignment with generative captioning through BLIP, then meet the Q-Former — the low-parameter bottleneck bridging frozen vision encoders and frozen LLMs without catastrophic forgetting. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/physical-ai/lessons/pa-w6/ https://probabilize.dev/courses/physical-ai/lessons/pa-w6/ Multimodal instruction tuning — how LLaVA's minimalist projector design and supervised instruction-response pairs turn a captioner into a conversational visual assistant. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/physical-ai/lessons/pa-w7/ https://probabilize.dev/courses/physical-ai/lessons/pa-w7/ Beyond projector-based VLMs: Flamingo's gated cross-attention, the Perceiver's latent bottleneck, PaLI's unified encoder-decoder, and extensions to tactile and non-visual modalities. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/physical-ai/lessons/pa-w8/ https://probabilize.dev/courses/physical-ai/lessons/pa-w8/ Parameter-efficient fine-tuning for billion-parameter VLMs — the mathematics of LoRA and QLoRA, memory budgeting, and continuous learning on edge robotics hardware. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/physical-ai/lessons/pa-w9/ https://probabilize.dev/courses/physical-ai/lessons/pa-w9/ Why benchmark accuracy misleads — captioning, VQA, and grounding metrics, distribution shift and spurious correlations, and the adversarial testing needed before physical deployment. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/physical-ai/lessons/pa-w10/ https://probabilize.dev/courses/physical-ai/lessons/pa-w10/ Precise geometric control over diffusion models via ControlNet and zero-initialized convolutions, and how VLMs act as System-2 controllers translating language into structured conditioning signals. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/physical-ai/lessons/pa-w11/ https://probabilize.dev/courses/physical-ai/lessons/pa-w11/ From passive perception to action — VLMs as the policy engine of a POMDP, grounding language intentions into tool calls and affordances to complete multi-step tasks. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/physical-ai/lessons/pa-w12/ https://probabilize.dev/courses/physical-ai/lessons/pa-w12/ Embodying VLMs in robots: perceiving unstructured environments, interpreting natural-language instructions, and producing continuous motor commands under real-time and safety constraints. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/physical-ai/lessons/pa-w13/ https://probabilize.dev/courses/physical-ai/lessons/pa-w13/ How web-scale training imprints systematic representation bias, how text-to-image models amplify stereotypes, and the alignment methods (RLHF, DPO) used to constrain VLM behavior before deployment. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/physical-ai/lessons/pa-w14/ https://probabilize.dev/courses/physical-ai/lessons/pa-w14/ Integrate thirteen weeks of perception, generation, and embodiment into a coherent vision-language system design — architecture, alignment, and safety for a deployed VLM. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/generative-models/lessons/gm-w1/ https://probabilize.dev/courses/generative-models/lessons/gm-w1/ The probabilistic vocabulary shared by every generative model — maximum likelihood, latent-variable models, the evidence lower bound, amortized inference, and score functions — and where each model class sits on the expressiveness–tractability frontier. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/generative-models/lessons/gm-w2/ https://probabilize.dev/courses/generative-models/lessons/gm-w2/ The VAE operationalizes the ELBO — the reparameterization trick, posterior collapse and its mitigations, β-VAE disentanglement, VQ-VAE, and hierarchical variants — the template for every latent-variable generative model. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/generative-models/lessons/gm-w3/ https://probabilize.dev/courses/generative-models/lessons/gm-w3/ The adversarial game and its divergence geometry — the min-max objective and its Jensen-Shannon interpretation, mode collapse and training instability, the Wasserstein reformulation with gradient penalty, spectral normalization, conditional and StyleGAN architectures, and FID/precision/recall evaluation. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/generative-models/lessons/gm-w4/ https://probabilize.dev/courses/generative-models/lessons/gm-w4/ Energy-based models and the partition-function problem — contrastive divergence, Langevin dynamics sampling and the mixing challenge, score matching and denoising score matching, and NCSN multi-scale score estimation as the direct mathematical precursor to diffusion models. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/generative-models/lessons/gm-w5/ https://probabilize.dev/courses/generative-models/lessons/gm-w5/ Exact-likelihood generative modeling through invertible maps — the change-of-variables formula, affine coupling layers (RealNVP, Glow), autoregressive flows and the training/sampling speed duality, continuous normalizing flows via neural ODEs, and the link to Schrödinger bridges and flow matching. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/generative-models/lessons/gm-w6/ https://probabilize.dev/courses/generative-models/lessons/gm-w6/ Denoising diffusion from first principles — the fixed Gaussian forward process, the variational bound on the reverse process, the simplified noise-prediction objective, the time-conditioned U-Net, ancestral and accelerated DDIM sampling, and the SDE/ODE unification with score matching. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/generative-models/lessons/gm-w7/ https://probabilize.dev/courses/generative-models/lessons/gm-w7/ Generative modeling as regression on a vector field — conditional flow matching, rectified flow and straight-line trajectories, optimal-transport coupling via mini-batch Sinkhorn, stochastic interpolants unifying flow matching and DDPM, and consistency models for single-step generation. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/generative-models/lessons/gm-w8/ https://probabilize.dev/courses/generative-models/lessons/gm-w8/ Steering diffusion generation — classifier and classifier-free guidance, the implicit energy-model view and manifold-departure artifacts, cross-attention for text conditioning and its memory advantage, ControlNet structural conditioning with zero convolutions, and adaptive layer normalization. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/generative-models/lessons/gm-w9/ https://probabilize.dev/courses/generative-models/lessons/gm-w9/ Scaling diffusion to high resolution and beyond images — latent diffusion (Stable Diffusion), text conditioning via cross-attention, the diffusion transformer (DiT), and extensions to audio (CLAP), video (temporal attention, 3D convolutions), and multimodal generation. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/generative-models/lessons/gm-w10/ https://probabilize.dev/courses/generative-models/lessons/gm-w10/ How to tell whether a generative model is good — bits-per-dimension, FID and its finite-sample bias, Inception Score, precision/recall, text-to-image faithfulness metrics (TIFA, VQAScore), task-success evaluation for robotics, and Goodhart's-law failure modes. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/generative-models/lessons/gm-w11/ https://probabilize.dev/courses/generative-models/lessons/gm-w11/ What generative models learn internally — generative vs. contrastive self-supervision, masked autoencoders, diffusion-feature correspondence (DIFT), feature-extraction protocols, and why spatially-detailed generative representations excel at robotics perception and planning. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/generative-models/lessons/gm-w12/ https://probabilize.dev/courses/generative-models/lessons/gm-w12/ Generative models as learned simulators — the world-model architecture, the recurrent state-space model (RSSM), Dreamer-style latent imagination for policy optimization, latent-space model predictive control, and the sample-efficiency tradeoffs of model-based vs. model-free RL. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/generative-models/lessons/gm-w13/ https://probabilize.dev/courses/generative-models/lessons/gm-w13/ The dual-use risks of generative models — deepfakes and detection, training-data memorization and extraction, differential privacy (DP-SGD), red-teaming and adversarial robustness, and the RLHF/DPO alignment techniques that steer model behavior. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/generative-models/lessons/gm-w14/ https://probabilize.dev/courses/generative-models/lessons/gm-w14/ Building and evaluating generative systems with scientific rigor — the generative pipeline as a system, controlled ablations and baselines, scaling laws, multi-dimensional and failure-mode evaluation, deployment engineering, and capstone project tracks. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/probability-statistics/lessons/ps-w1/ https://probabilize.dev/courses/probability-statistics/lessons/ps-w1/ Build probability from sample spaces, events, and Kolmogorov's axioms, and set up discrete vs continuous models. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/probability-statistics/lessons/ps-w2/ https://probabilize.dev/courses/probability-statistics/lessons/ps-w2/ Map outcomes to numbers with random variables, and work with PMFs, PDFs, CDFs, and four named distributions. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/probability-statistics/lessons/ps-w3/ https://probabilize.dev/courses/probability-statistics/lessons/ps-w3/ Compute expectation, variance, and covariance, and use linearity of expectation without requiring independence. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/probability-statistics/lessons/ps-w4/ https://probabilize.dev/courses/probability-statistics/lessons/ps-w4/ Condition on information, factor joints with the chain rule, and separate independence from conditional independence. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/probability-statistics/lessons/ps-w5/ https://probabilize.dev/courses/probability-statistics/lessons/ps-w5/ Update beliefs with Bayes' rule and make MAP decisions in simple generative classification settings. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/probability-statistics/lessons/ps-w6/ https://probabilize.dev/courses/probability-statistics/lessons/ps-w6/ Parameterize joint Gaussians with mean and covariance, and compute marginals, conditionals, and whitening transforms. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/probability-statistics/lessons/ps-w7/ https://probabilize.dev/courses/probability-statistics/lessons/ps-w7/ See sample averages concentrate with the LLN and become approximately normal with the central limit theorem. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/probability-statistics/lessons/ps-w8/ https://probabilize.dev/courses/probability-statistics/lessons/ps-w8/ Form the likelihood, maximize it for Bernoulli and Gaussian models, and connect MLE to empirical risk. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/probability-statistics/lessons/ps-w9/ https://probabilize.dev/courses/probability-statistics/lessons/ps-w9/ Treat parameters as uncertain, update posteriors, and compare MAP estimates to MLE with conjugate examples. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/probability-statistics/lessons/ps-w10/ https://probabilize.dev/courses/probability-statistics/lessons/ps-w10/ Decompose estimation error into bias and variance, form simple confidence intervals, and use the bootstrap as a computational uncertainty tool. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/probability-statistics/lessons/ps-w11/ https://probabilize.dev/courses/probability-statistics/lessons/ps-w11/ Define entropy, cross-entropy, and KL divergence, and connect them to maximum likelihood and classification losses. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/probability-statistics/lessons/ps-w12/ https://probabilize.dev/courses/probability-statistics/lessons/ps-w12/ See Bernoulli, Gaussian, and categorical models as exponential families with natural parameters and sufficient statistics. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/probability-statistics/lessons/ps-w13/ https://probabilize.dev/courses/probability-statistics/lessons/ps-w13/ Factor joint distributions with directed and undirected graphs, and read conditional independence from graph structure at a cartoon level. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/probability-statistics/lessons/ps-w14/ https://probabilize.dev/courses/probability-statistics/lessons/ps-w14/ Map probability tools onto reinforcement learning and generative modeling, and audit common probabilistic modeling mistakes. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/advanced-optimization/lessons/ao-w1/ https://probabilize.dev/courses/advanced-optimization/lessons/ao-w1/ Elevate intermediate first-order intuition into rigorous stationarity conditions and constraint qualifications that unlock KKT and duality. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/advanced-optimization/lessons/ao-w2/ https://probabilize.dev/courses/advanced-optimization/lessons/ao-w2/ Build the Lagrangian dual function, prove weak duality, and state Slater's condition for strong duality. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/advanced-optimization/lessons/ao-w3/ https://probabilize.dev/courses/advanced-optimization/lessons/ao-w3/ State the full KKT system, use complementary slackness, and read dual variables as shadow prices. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/advanced-optimization/lessons/ao-w4/ https://probabilize.dev/courses/advanced-optimization/lessons/ao-w4/ Write linear and quadratic programs in standard form and see second-order cone constraints as a modeling tool. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/advanced-optimization/lessons/ao-w5/ https://probabilize.dev/courses/advanced-optimization/lessons/ao-w5/ Formulate semidefinite programs, visualise spectrahedra at a high level, and see SDP relaxations for hard combinatorial problems. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/advanced-optimization/lessons/ao-w6/ https://probabilize.dev/courses/advanced-optimization/lessons/ao-w6/ Follow the central path with log barriers, sketch self-concordance at a practical level, and interpret commercial IPM solvers. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/advanced-optimization/lessons/ao-w7/ https://probabilize.dev/courses/advanced-optimization/lessons/ao-w7/ Use proximal operators for composite objectives, run ISTA/FISTA, and split structure with ADMM. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/advanced-optimization/lessons/ao-w8/ https://probabilize.dev/courses/advanced-optimization/lessons/ao-w8/ Derive the Newton step, add damping and line search, and use BFGS/L-BFGS when Hessians are unavailable. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/advanced-optimization/lessons/ao-w9/ https://probabilize.dev/courses/advanced-optimization/lessons/ao-w9/ Exploit finite-sum structure with SVRG and SAGA to reduce gradient noise without full-batch cost every step. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/advanced-optimization/lessons/ao-w10/ https://probabilize.dev/courses/advanced-optimization/lessons/ao-w10/ Scale minibatches across workers with data-parallel gradients, and separate solid theory from large-batch learning-rate folklore. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/advanced-optimization/lessons/ao-w11/ https://probabilize.dev/courses/advanced-optimization/lessons/ao-w11/ Contrast forward and reverse mode AD, build vector-Jacobian products on a tiny graph, and cost out reverse mode for deep nets. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/advanced-optimization/lessons/ao-w12/ https://probabilize.dev/courses/advanced-optimization/lessons/ao-w12/ Define sharpness of minima, connect large-batch geometry, and implement a SAM-style ascent-descent step. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/advanced-optimization/lessons/ao-w13/ https://probabilize.dev/courses/advanced-optimization/lessons/ao-w13/ Navigate nonconvex landscapes, saddles, and overparameterisation, and see how GD/SGD prefer particular solutions among many minimisers. Thu, 17 Sep 2026 00:00:00 GMT https://probabilize.dev/courses/advanced-optimization/lessons/ao-w14/ https://probabilize.dev/courses/advanced-optimization/lessons/ao-w14/ Diagnose a hard problem and choose either a convex cone reformulation path or a large-scale first/second-order training recipe. Thu, 17 Sep 2026 00:00:00 GMT