Working Notes: a commonplace notebook for recording & exploring ideas.
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Hierarchical Reasoning Model

• Paper

Firstp ass on aper

• Recurrent architecture

• Sequentioal reasoning tasks in a single forward pass without supervision

  • 2 interdependent recurrent modules
  • high level module for slow, abstract planning
  • low level module for rapid, detailed computations

• 27mn parameters + 1k samples -- exceptional performance (!)

• Transformer complexity is AC^0 or TC^0

• rely on breaking the task into simple intermediate steps

• this paper uses latent reasonsing

• latent reasoning is constrained by the effective computational depth

• brains use

  • hierarchical computation
  • across different timescales
  • recurrent loops refine internal representations

• one step gradient approximation for HRM

• eliminates requirement for backpropagation through time (BPTT)

• HRM

  • hierarchical processing: higher areas over longer timescales, more abstract
  • lower level: more immediate
  • temporal separation
  • recurrent connectivity

• HRM model

  • input network f_i(.; theta_i)
  • low level recurrent module f_l(.; theta_l)
  • high level recurrent module f_h(.; theta_h)
  • output network f_o(.; theta_o)

• model dynamics

  • N high level cycles
    • T low level timestamps
  • indexed by 1, ..., N x T
  • f_l & f_h keep a hidden state z^i_L for f_l
  • initialized with z^0_L for z^0_H

• project input x into a working representation using f

• at each timestep i, l updates state based on previous state, h module's state and input rep

• h updates every T steps using L modules final state

• z^i-L = f_L(z^i-1_L, z^i-1_H, theta_L)

• z^i_H = { f_H(z^i-1_H, z^i-1_L, theta_H) if i == 0; z^i-1_H }

• finally, y = output(z)

• need to find a balance to converge correctly

  • want convergence to proceed slowly

• H resets the L in cycles

• graphs are amazing around how the inner and outer module behavie

• forward residual == main stream everything adds to

• choice of PCA and residual to identify behavior of the model

• simplify gradient across multiple steps just based on the last state of the H& Lmodules

• implement by disabing grad for the intermediate steps on time

• deep supervision

  • based on neural oscillations regulating when learning happens in the brain
  • run multiple forward passes for a data sample - each forward pass is a segment
  • M segments executed before termination
  • collect state of h, l at conclusion of each segment m
  • supervision state:
    • compute next state given previous state + forward pass
    • compute loss for current segment
    • update parameters
  • hidden state is detached before being used as input state, creating 1 step approximation

• adaptive computational time

  • Q-head uses final state of H to predict Q values of halt, continue actions
  • choose M_min randomly between 2...M_max occasionally or 1
  • halt action is selected when we cross M_max (fixed hyperparam)
  • or halt value exceeds estimated continue value, and we're at M_min
  • Q learning algorithm
    • episodic markov decision process
    • state at segment m is z^m
    • use this to calculate loss if stopping or continuing

• overall loss combines q-head loss and seq to seq loss

• this helps train stopping decisions

• inference time scaling is simply increasing M_max

• use RMS with post rnorm and AdamW, keeping params bounded by 1/lambda (=?)

• stabelmax for small sample experiments

• fl/h ar eimplemented as encoder only transformer blocks that are identical

• using Rope, gated linear unit,s rms norm, and no bias in linear layers

• lecun normal initalization

• use adam atan2, with a constant learning rate with a linear warmup

• inspect intermediate state by printing outputs at each step

• from the brain

  • in dynamic s5ystems for reasoning & decision makning
  • higher dimensional state space trajectories, for more computations
  • dimensionality hierarchy
  • Participation Ratio -- effective dimensionality of a higher dimension representation
    • (sum (eigenvalues of covariance matrix of neural trajectories))^2 / sum of squares
    • higher PR == variance evenly distributed across more dimensions, otherwise lower dimensioal structuore

• also shows up in the test

• rest

  • hrm is also turing complete given enough time & memory
  • takes feedback from dense gradients instead of sparse reward
  • recurrence can be more sequential / linear attention

Follow up references

• computational complexity of transformers
• deep equilibrium models, implicit function theory
• adamw
• q learning
• stablemax

• notes/neural_nets
• notes/gradient_descent
• notes/transformers/inference
• notes/stochastic_gradient_descent
• notes/work_efficient
• notes/softmax
• notes/deep_learning_nyu/deep_learning_nyu_week_5
• notes/parametrized_models
• notes/deep_learning_nyu/deep_learning_nyu_week_1
• notes/deepseekv3